malachite_float/float/arithmetic/

atan.rs

// Copyright © 2026 Mikhail Hogrefe
//
// Uses code adopted from the GNU MPFR Library.
//
//      Copyright © 2001-2025 Free Software Foundation, Inc.
//
//      Contributed by the Pascaline and Caramba projects, INRIA.
//
// This file is part of Malachite.
//
// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.

// Port of MPFR's arctangent (`mpfr_atan`, `atan.c`). An input above 1 in magnitude is inverted and
// its arctangent taken from pi/2. The argument is then reduced by atan(x) = 2 atan((sqrt(1 + x^2)
// - 1)/x) until it is below about 1/sqrt(prec), and what remains is split into binary chunks, each
// of whose arctangent is summed by binary splitting of the series for atan(x)/x (with a table of
// the twenty most common small chunks for low precisions), the pieces being combined through
// atan(a) + atan(b) = atan((a + b)/(1 - ab)). The arctangent is bounded, so it never overflows; it
// underflows only for an input at the very bottom of the exponent range, where it is the input
// itself rounded toward zero, which MPFR's small-input shortcut decides directly.

use crate::InnerFloat::{Finite, Infinity, NaN, Zero};
use crate::float::arithmetic::round_near_x::{
    LEADING_TERM_MIN_EXPONENT, round_rational_leading_term, small_input_shortcut,
};
use crate::float::arithmetic::sin::{
    SCALE, SCALED_INPUT_EXPONENT, UNDERFLOW_EXPONENT, scaled_underflow, underflowed,
};
use crate::float::arithmetic::tan::round_bracket_signed_by;
use crate::{Float, emulate_float_to_float_fn, emulate_rational_to_float_fn};
use alloc::vec;
use core::cmp::Ordering::{self, Equal, Greater};

use core::mem::take;
use malachite_base::num::arithmetic::traits::{
    Abs, Atan, AtanAssign, CeilingLogBase2, IsPowerOf2, Parity, PowerOf2, Reciprocal, Square,
    SquareAssign,
};
use malachite_base::num::basic::floats::PrimitiveFloat;
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::basic::traits::{
    NaN as NaNTrait, NegativeZero as NegativeZeroTrait, One, Zero as ZeroTrait,
};
use malachite_base::num::conversion::traits::{ExactFrom, RoundingFrom};
use malachite_base::num::logic::traits::SignificantBits;
use malachite_base::rounding_modes::RoundingMode::{self, *};
use malachite_base::split_into_chunks_mut;
use malachite_nz::integer::Integer;
use malachite_nz::natural::Natural;
use malachite_nz::natural::arithmetic::float::round::float_can_round;
use malachite_nz::platform::Limb;
use malachite_q::Rational;

// For each pair (r, p), a 192-bit truncation of atan(x)/x for x = p/2^r, as three 64-bit words with
// the lowest first; these are the chunks that a precision of at most 192 bits meets most often, and
// that are the most expensive to sum. MPFR only compiles this table for 64-bit limbs; the values do
// not depend on the limb width, so it is used unconditionally here.
//
// In Sage: for p in range(1, 2^ceil(r/2)), with x = p/2^r, the words are floor(2^192*n(atan(x)/x,
// 300)).digits(2^64).
const ATAN_TABLE: [[u64; 3]; 20] = [
    [0x6e141587261cdf00, 0x6fe445ecbc3a8d03, 0xed63382b0dda7b45], // (1,1)
    [0xaa7fa90388b3836b, 0x6dc79ef5f7a217e5, 0xfadbafc96406eb15], // (2,1)
    [0x319c12cf59d4b2dc, 0xcb2792dc0e2e0d51, 0xffaaddb967ef4e36], // (4,1)
    [0x8b3957d95d9ad922, 0xc897989f3e888ef7, 0xfeadd4d5617b6e32], // (4,2)
    [0xc4e6abc8af62e439, 0x4eb9bf602625f0b4, 0xfd0fcdd343cac19b], // (4,3)
    [0x7c18baeb9bc95789, 0xb12afb6b6d4f7e16, 0xffffaaaaddddb94b], // (8,1)
    [0x6856a0171a2f001a, 0x62351fbbe60af47, 0xfffeaaadddd4b968],  // (8,2)
    [0x69164c094f49da06, 0xd517294f7373d07a, 0xfffd001032cb1179], // (8,3)
    [0x20ef65c10deef460, 0xe78c564015f76048, 0xfffaaadddb94d5bb], // (8,4)
    [0x3ce233aa002f0344, 0x9dd8ea342a65d4cc, 0xfff7ab27a1f32f95], // (8,5)
    [0xa37f403c7279c5cb, 0x13ab53a1c8db8497, 0xfff40103192ce74d], // (8,6)
    [0xe5a85657103c1aa8, 0xb8409e6c914191d3, 0xffefac8a9c40a26b], // (8,7)
    [0x806d0294c0db8816, 0x779d776dda8c6213, 0xffeaaddd4bb12542], // (8,8)
    [0x5545d1914ef21478, 0x3aea58d6660f5a12, 0xffe5051f0aebf73a], // (8,9)
    [0x6e47a91d015f4133, 0xc085ab6b490b7f02, 0xffdeb2787d4adac1], // (8,10)
    [0x4efc1f931f7ec9b3, 0xb7f43cd16195ef4b, 0xffd7b61702b09aad], // (8,11)
    [0xd27d1dbf55fed60d, 0xd812c11d7d473e5e, 0xffd0102cb3c1bfbe], // (8,12)
    [0xca629e927383fe97, 0x8c61aedf58e42206, 0xffc7c0f05db9d1b6], // (8,13)
    [0x4eff0b53d4e905b7, 0x28ac1e800ca31e9d, 0xffbec89d7dddd7e9], // (8,14)
    [0xb0a7931deec6fe60, 0xb46feea78588554b, 0xffb527743c8cdd8f], // (8,15)
];

// A table entry, in [1/2, 1), truncated to `prec` bits.
//
// This is `set_table` from atan.c, MPFR 4.2.2.
fn set_table(prec: u64, x: &[u64; 3]) -> Float {
    let n = (Natural::from(x[2]) << 128u32) | (Natural::from(x[1]) << 64u32) | Natural::from(x[0]);
    Float::from_rational_prec_round(Rational::from(n) >> 192u32, prec, Down).0
}

// Multiplies `xs[k - 1]` by `xs[k]` in place.
fn mul_prev(xs: &mut [Integer], k: usize) {
    let (lo, hi) = xs.split_at_mut(k);
    lo[k - 1] *= &hi[0];
}

// If x = p/2^r, computes an approximation to atan(x)/x using 2^m terms of the series, with an error
// of at most 1 ulp at precision `precy`. Assumes 0 < x < 1, so 1 <= p < 2^r. More precisely, p
// consists of the floor(r/2) bits of the binary expansion of a number 0 < s < 1: the bit of weight
// 2^-1 is for r = 1, so p <= 1; the bit of weight 2^-2 is for r = 2, so p <= 1; the two bits of
// weight 2^-3 and 2^-4 are for r = 4, so p <= 3; and in general p < 2^(r/2).
//
// With X = x^2 = p/2^r the series is 1 - X/3 + X^2/5 - ... + (-1)^k X^k/(2k+1) + ..., summed by
// binary splitting: P(a,b) = p if a+1 = b, else P(a,c) P(c,b); Q(a,b) = (2a+1) 2^r if a+1 = b
// (except Q(0,1) = 1), else Q(a,c) Q(c,b); S(a,b) = p (2a+1) if a+1 = b, else Q(c,b) S(a,c) +
// Q(a,c) P(a,c) S(c,b). Then atan(x)/x ~ S(0,i)/Q(0,i) for an i making (p/2^r)^i/i small enough.
// The factor 2^(r(b-a)) in Q(a,b) is implicit, and is applied when Q is used.
//
// This is `mpfr_atan_aux` from atan.c, MPFR 4.2.2.
fn atan_aux(mut p: Integer, mut r: u64, m: usize, precy: u64) -> Float {
    debug_assert!(p > 0u32);
    debug_assert!(m > 0);
    // tabulate values for small precision and small r, which are the most expensive to compute
    if precy <= 192 {
        let index = match r {
            // p has 1 bit: necessarily p = 1
            1 => Some(0),
            2 => Some(1),
            // p has at most 2 bits: 1 <= p <= 3
            4 => Some(1 + usize::exact_from(&p)),
            // p has at most 4 bits: 1 <= p <= 15
            8 => Some(4 + usize::exact_from(&p)),
            _ => None,
        };
        if let Some(index) = index {
            return set_table(precy, &ATAN_TABLE[index]);
        }
    }
    // From p to p^2, and r to 2r
    p.square_assign();
    r <<= 1;
    // Normalize p
    let n = p.trailing_zeros().unwrap();
    if n > 0 {
        p >>= n;
        r -= n;
    }
    // Since |p/2^r| < 1, and p is a nonzero integer, necessarily r > 0.
    debug_assert!(r > 0);
    // MPFR lays the three tables out in one array of 3(m+1) entries; the p = 1 loop can run one
    // step past 2^m terms, so S and Q get a spare slot each here
    let len = m + 2;
    let mut scratch = vec![Integer::ZERO; (len << 1) + m + 1];
    // ptoj[j] = p^(2^j)
    split_into_chunks_mut!(scratch, len, [s, q], ptoj);
    let mut log2_nb_terms = vec![0u64; len + 1];
    // accu[k] = mult[0] + ... + mult[k], where mult[j] is the number of bits of the corresponding
    // term S[j]/Q[j]
    let mut accu = vec![0i64; len + 1];
    let ri = i64::exact_from(r);
    let mut i = 0u64;
    let mut k = 0usize;
    let p_is_1 = p == 1u32;
    if p_is_1 {
        // special case p = 1: the i-th term being X^i/(2i+1) with X = 1/2^r, we can stop when r i >
        // precy, i.e. i > precy/r
        let mut n = u64::power_of_2(u64::exact_from(m));
        if precy / r <= n {
            n = precy / r + 1;
        }
        while i < n {
            q[k + 1] = Integer::from((i << 1) + 3);
            s[k] = (&q[k + 1] << r) - Integer::from((i << 1) + 1);
            q[k] = &q[k + 1] * Integer::from((i << 1) + 1);
            log2_nb_terms[k] = 1; // S[k]/Q[k] corresponds to 2 terms
            let mut j = (i + 2) >> 1;
            let mut l = 1u64;
            while j.even() {
                debug_assert!(k > 0);
                s[k] *= &q[k - 1];
                let mut t = &s[k - 1] * &q[k];
                t <<= r << l;
                t += &s[k];
                s[k - 1] = t;
                mul_prev(q, k);
                log2_nb_terms[k - 1] = l + 1;
                l += 1;
                j >>= 1;
                k -= 1;
            }
            i += 2;
            k += 1;
        }
    } else {
        // p != 1: precompute the ptoj table
        ptoj[0] = p.clone();
        for im in 1..=m {
            ptoj[im] = (&ptoj[im - 1]).square();
        }
        // main loop: the i-th term being X^i/(2i+1) with X = p/2^r, we can stop when p^i/2^(ri) <
        // 2^-precy, i.e. r i > precy + log2(p^i)
        let n = u64::power_of_2(u64::exact_from(m));
        let mut done = false;
        while i < n && !done {
            // initialize both S[k], Q[k] and S[k+1], Q[k+1]
            q[k + 1] = Integer::from((i << 1) + 3); // Q(i+1,i+2)
            s[k + 1] = &p * Integer::from((i << 1) + 1); // S(i+1,i+2)
            s[k] = (&q[k + 1] << r) - &s[k + 1]; // S(i,i+2)
            q[k] = &q[k + 1] * Integer::from((i << 1) + 1); // Q(i,i+2)
            log2_nb_terms[k] = 1; // S[k]/Q[k] corresponds to 2 terms
            let mut j = (i + 2) >> 1;
            let mut l = 1u64;
            while j.even() {
                // invariant: S[k-1]/Q[k-1] and S[k]/Q[k] correspond to 2^l terms each; combine them
                // into S[k-1]/Q[k-1]
                debug_assert!(k > 0);
                s[k] *= &q[k - 1];
                s[k] *= &ptoj[usize::exact_from(l)];
                let mut t = &s[k - 1] * &q[k];
                t <<= r << l;
                t += &s[k];
                s[k - 1] = t;
                mul_prev(q, k);
                log2_nb_terms[k - 1] = l + 1;
                // now S[k-1]/Q[k-1] corresponds to 2^(l+1) terms
                let mult = (ri << (l + 1))
                    - i64::exact_from(ptoj[usize::exact_from(l + 1)].significant_bits())
                    - 1;
                accu[k - 1] = if k == 1 { mult } else { accu[k - 2] + mult };
                if accu[k - 1] > i64::exact_from(precy) {
                    done = true;
                }
                l += 1;
                j >>= 1;
                k -= 1;
            }
            i += 2;
            k += 1;
        }
    }
    // we need to combine S[0]/Q[0] ... S[k-1]/Q[k-1]
    let mut h = 0u64; // number of terms accumulated in S[k]/Q[k]
    while k > 1 {
        k -= 1;
        // combine S[k-1]/Q[k-1] and S[k]/Q[k]
        s[k] *= &q[k - 1];
        if !p_is_1 {
            s[k] *= &ptoj[usize::exact_from(log2_nb_terms[k - 1])];
        }
        let mut t = &s[k - 1] * &q[k];
        h += u64::power_of_2(log2_nb_terms[k]);
        t <<= r * h;
        t += &s[k];
        s[k - 1] = t;
        mul_prev(q, k);
    }
    let mut s0 = take(&mut s[0]);
    let mut q0 = take(&mut q[0]);
    let precy_i = i64::exact_from(precy);
    let mut diff = i64::exact_from(s0.significant_bits()) - (precy_i << 1);
    let mut expo = diff;
    // a negative shift is a left shift, covering MPFR's mul_2exp branch
    s0 >>= diff;
    diff = i64::exact_from(q0.significant_bits()) - precy_i;
    expo -= diff;
    q0 >>= diff;
    s0 /= q0; // truncating division (both positive)
    // y = (S[0] rounded down to precy) * 2^(expo - r(i-1)); MPFR sets the mantissa via set_z then
    // overrides the exponent, which is exactly this scaling. The scaled value is atan(x)/x, of
    // ordinary size, so attaching the scaling before the conversion keeps every exponent in range.
    Float::from_rational_prec_round(
        Rational::from(s0) << (expo - ri * (i64::exact_from(i) - 1)),
        precy,
        Floor,
    )
    .0
}

// Rounds the sum of an alternating series x - c_1 x^3/3 + c_2 x^5/5 - ... for a nonzero `Rational`
// x with |x| <= 1/2, whose terms decrease in magnitude, so that successive partial sums bracket the
// sum. The coefficients start at c_0 = 1 and c_k is c_(k-1) times `ratio(k)`, or c_(k-1) itself if
// `ratio(k)` is `None`. With all coefficients 1 this is the arctangent; with ratio (2k - 1)/(2k) it
// is the inverse hyperbolic sine.
pub(crate) fn alternating_odd_series<F: Fn(u64) -> Option<Rational>>(
    x: &Rational,
    prec: u64,
    rm: RoundingMode,
    ratio: F,
) -> (Float, Ordering) {
    let negative = *x < 0u32;
    let ax = x.abs();
    let x2 = (&ax).square();
    // hi and lo are the partial sums with an odd and an even number of terms
    let mut power = ax.clone();
    let mut coefficient = Rational::ONE;
    let mut hi = ax;
    let mut lo = Rational::ZERO;
    let mut k = 1u64;
    loop {
        power *= &x2;
        if let Some(r) = ratio(k) {
            coefficient *= r;
        }
        let t = &coefficient * &power / Rational::from((k << 1) + 1);
        if k.odd() {
            lo = &hi - t;
        } else {
            hi = &lo + t;
        }
        if let Some(result) = round_bracket_signed_by(negative, lo.clone(), hi.clone(), prec, rm) {
            return result;
        }
        k += 1;
    }
}

// atan x for a tiny nonzero `Rational` x, bracketed by consecutive partial sums of its alternating
// series: x - x^3/3 < atan x < x - x^3/3 + x^5/5 < x, and so on, a bracket that narrows without
// bound; the arctangent is transcendental, so it eventually rounds unambiguously. This is the
// fallback for the tiny inputs that MPFR's small-input shortcut declines, which are those at the
// very bottom of the exponent range whose result underflows; the general algorithm would otherwise
// work at a precision of about 2^30 bits for them.
fn atan_series(x: &Rational, prec: u64, rm: RoundingMode) -> (Float, Ordering) {
    alternating_odd_series(x, prec, rm, |_| None)
}

// One step of `atan_rational_huge`: pi to w bits gives pi/2 to within 2^(1 - w), and [lo, hi]
// brackets atan(1/|x|), so pi/2 - atan(1/|x|) lies between the two ends below.
fn atan_huge_step(
    negative: bool,
    lo: &Rational,
    hi: &Rational,
    w: u64,
    prec: u64,
    rm: RoundingMode,
) -> Option<(Float, Ordering)> {
    // pi_lo <= pi <= pi_lo + 2^(2 - w)
    let pi_lo = Rational::exact_from(&Float::pi_prec_round(w, Floor).0);
    let pi_hi = &pi_lo + Rational::power_of_2(2 - i64::exact_from(w));
    round_bracket_signed_by(
        negative,
        (pi_lo >> 1u32) - hi,
        (pi_hi >> 1u32) - lo,
        prec,
        rm,
    )
}

// atan x for a `Rational` x beyond the top of the exponent range, where neither x nor 1/x is a
// `Float`: atan x = pi/2 - atan(1/x), and atan(1/x) is bracketed by the partial sums of its
// alternating series, so pi/2 minus that bracket settles the result.
fn atan_rational_huge(x: &Rational, prec: u64, rm: RoundingMode) -> (Float, Ordering) {
    let negative = *x < 0u32;
    // 0 < t < 2^(-2^30)
    let t = x.abs().reciprocal();
    let mut lo = Rational::ZERO;
    let mut hi = t.clone();
    let mut w = prec + 64;
    // The first bracket, atan(t) in (0, t), is already far narrower than an ulp of pi/2 for a t
    // this small, and needs none of the powers of t, each of which is as long as the input itself.
    if let Some(result) = atan_huge_step(negative, &lo, &hi, w, prec, rm) {
        return result;
    }
    let t2 = (&t).square();
    let mut term = t;
    let mut k = 1u64;
    loop {
        w <<= 1;
        term *= &t2;
        let d = &term / Rational::from((k << 1) + 1);
        if k.odd() {
            lo = &hi - d;
        } else {
            hi = &lo + d;
        }
        if let Some(result) = atan_huge_step(negative, &lo, &hi, w, prec, rm) {
            return result;
        }
        k += 1;
    }
}

// Computes atan(x) for a nonzero `Rational` x, rounded to precision `prec` with rounding mode `rm`.
// (x = 0 is handled by the caller.) The result is never exactly representable, so `rm` must not be
// `Exact`.
//
// The general case rounds the input once and takes its `Float` arctangent at a working precision.
// That is sound because the arctangent is 1-Lipschitz, so the half-ulp of the input carries to the
// result unmagnified, and because the result is never much smaller than the input: |atan t| > |t|/2
// for |t| <= 1, while for |t| > 1 the result lies in (pi/4, pi/2) and the input error is damped by
// 1/(1 + t^2). Both errors together stay below 2^(EXP(a) - m + 1). The two ends of the exponent
// range, where the input itself is not a `Float`, are bracketed instead.
pub(crate) fn atan_rational_helper(x: &Rational, prec: u64, rm: RoundingMode) -> (Float, Ordering) {
    assert_ne!(rm, Exact, "Inexact atan");
    let exp_x = x.floor_log_base_2_abs() + 1; // the MPFR-style exponent of x
    if exp_x < UNDERFLOW_EXPONENT {
        // |atan x| < |x| < 2^(MIN_EXPONENT - 2), below half the smallest positive `Float`, so the
        // result is zero or that `Float` by the rounding mode alone, and no 2^30-bit arithmetic is
        // needed
        return underflowed(*x > 0u32, prec, rm);
    }
    // atan(x) = x(1 - x^2/3 + ...), so |x| exceeds |atan x| by less than 2^(3 EXP(x) - 1). Once
    // that is below the distance from x to the nearest (prec + 1)-bit dyadic -- at least 2^(EXP(x)
    // - prec - 1)/d for a denominator of d, the two coinciding only when x is itself such a dyadic,
    // the exact and tie cases -- x's own rounding is the answer, nudged toward zero. The series
    // below would say the same, but its partial sums are formed exactly, and for a tiny x they are
    // dense `Rational`s of about 2 |EXP(x)| bits: 7 seconds for x = 2^-536870908, against
    // microseconds here. (`acot_rational` leans on this too, taking the arctangent of a
    // reciprocal.) Inputs at the bottom of the exponent range are left to the series below: there
    // the nudge and the tie test would be working with `Float`s that underflow.
    if exp_x > LEADING_TERM_MIN_EXPONENT
        && -(exp_x << 1) > i64::exact_from(prec + x.denominator_ref().significant_bits()) + 4
    {
        return round_rational_leading_term(x.abs(), *x > 0u32, false, prec, rm);
    }
    // For |x| <= 1/2, |x| - |x|^3/3 <= |atan x| <= |x| - |x|^3/3 + |x|^5/5, a bracket of relative
    // width below x^4, which decides the rounding once x^4 is below 2^-(prec + 3); a handful of
    // terms is cheaper than a `Float` arctangent at the working precision.
    if exp_x < 0 && -(exp_x << 2) > i64::exact_from(prec) + 3 {
        return atan_series(x, prec, rm);
    }
    if exp_x > Float::MAX_EXPONENT_I64 {
        return atan_rational_huge(x, prec, rm);
    }
    let mut m = prec + prec.ceiling_log_base_2() + 8;
    let mut increment = Limb::WIDTH;
    loop {
        let (f, o_f) = Float::from_rational_prec_ref(x, m);
        if o_f == Equal {
            // x is exactly representable at m bits, so its arctangent is simply the `Float` one
            return atan_prec_round_normal_ref(&f, prec, rm);
        }
        let a = (&f).atan();
        if float_can_round(a.significand_ref().unwrap(), m - 2, prec, rm) {
            return Float::from_float_prec_round(a, prec, rm);
        }
        m += increment;
        increment = m >> 1;
    }
}

// This is mpfr_atan from atan.c, MPFR 4.2.2, for a finite nonzero input, with a series bracket for
// the tiny inputs whose result underflows.
fn atan_prec_round_normal_ref(x: &Float, prec: u64, rm: RoundingMode) -> (Float, Ordering) {
    assert_ne!(rm, Exact, "Inexact atan");
    let exp_x = i64::from(x.get_exponent().unwrap());
    // atan(x) = x - x^3/3 + x^5/5 ..., so the error is < 2^(3 EXP(x) - 1), and `EXP(x) - (3 EXP(x)
    // - 1)` = -2 EXP(x) + 1
    //
    // MPFR_FAST_COMPUTE_IF_SMALL_INPUT (atan, x, -2 * MPFR_GET_EXP (x), 1, 0, rnd_mode, {});
    // atan(x) = x - x^3/3 + ..., so the correction is below 2^(3 EXP(x) - 1) and carries the value
    // toward zero
    if let Some(result) = small_input_shortcut(x, -(exp_x << 1), 1, false, prec, rm) {
        return result;
    }
    let negative = *x < 0u32;
    let xp = x.clone().abs();
    // Other simple case: atan(±1) = ±pi/4
    let comparison = xp.partial_cmp(&1u32).unwrap();
    if comparison == Equal {
        let (pi, o) = Float::pi_prec_round(prec, if negative { -rm } else { rm });
        // exact
        let quarter = pi >> 2u32;
        return if negative {
            (-quarter, o.reverse())
        } else {
            (quarter, o)
        };
    }
    let mut realprec = prec + prec.ceiling_log_base_2() + 4;
    let mut increment = Limb::WIDTH;
    // If |x| < 1, we need more precision to be able to round
    let sup = if exp_x < 0 {
        u64::exact_from(2 - exp_x)
    } else {
        1
    };
    loop {
        // n0 = ceil(log(prec_requested + 2 + 1 + ln(2.4)/ln(2))/log(2))
        let n0 = (realprec + sup + 3).ceiling_log_base_2();
        // since realprec >= 4, n0 >= ceil(log2(8)) >= 3, so 3 n0 > 2
        let mut wp = realprec + sup + 1 + (3 * n0 - 2).ceiling_log_base_2();
        // The number of lost bits due to argument reduction is 9 - 2 EXP(sk), estimated by 9 + 2
        // ceil(log2(p)), since we manage that sk < 1/p. Below 100 bits the argument is not reduced.
        let (log2p, est_lost) = if prec > 100 {
            let log2p = (wp.ceiling_log_base_2() >> 1) - 3;
            (log2p, 9 + (log2p << 1))
        } else {
            (0, 0)
        };
        wp += est_lost;
        // use atan(xp) = pi/2 - atan(1/xp) for xp > 1; now 0 < sk <= 1
        let mut sk = if comparison == Greater {
            xp.reciprocal_prec_ref(wp).0
        } else {
            Float::from_float_prec_ref(&xp, wp).0
        };
        // Argument reduction: atan(x) = 2 atan((sqrt(1 + x^2) - 1)/x), applied until |sk| <
        // k/sqrt(p), where p is the target precision. It keeps 0 < sk <= 1, and is applied at least
        // once when sk = 1, after which sk < 1 (see atan.c for the interval-arithmetic argument).
        let mut lost = 0u64;
        let mut red = 0u64;
        let log2p_i = i64::exact_from(log2p);
        loop {
            let exp_sk = i64::from(sk.get_exponent().unwrap());
            if exp_sk <= -log2p_i {
                break;
            }
            lost = u64::exact_from(9 - (exp_sk << 1));
            let mut tmp = sk.square_prec_ref(wp).0;
            tmp.add_prec_assign_ref(&Float::ONE, wp);
            tmp.sqrt_prec_assign(wp);
            tmp.sub_prec_assign_ref(&Float::ONE, wp);
            sk = if red == 0 && comparison == Greater {
                // use xp = 1/sk
                tmp.mul_prec_val_ref(&xp, wp).0
            } else {
                tmp.div_prec_val_ref(&sk, wp).0
            };
            red += 1;
        }
        debug_assert!(sk < 1u32);
        // Assignation
        let mut arctgt = Float::ZERO;
        let mut twopoweri = 1u64;
        for i in 0..n0 {
            if sk == 0u32 {
                break;
            }
            // trunc(sk 2^twopoweri) as an integer; since the s_k are decreasing (see
            // algorithms.tex) and s_0 = min(|x|, 1/|x|) < 1, sk < 1
            let ukz = Integer::rounding_from(&(&sk << twopoweri), Down).0;
            if ukz != 0u32 {
                // tmp = ukz / 2^twopoweri
                let tmp = Float::from_integer_prec_ref(&ukz, wp).0 >> twopoweri;
                // atan(A_k)
                let tmp2 = atan_aux(ukz, twopoweri, usize::exact_from(n0 - i), wp)
                    .mul_prec_val_ref(&tmp, wp)
                    .0;
                // Addition
                arctgt.add_prec_assign(tmp2, wp);
                // Next iteration
                let tmp2 = sk.sub_prec_ref_ref(&tmp, wp).0;
                sk.mul_prec_assign_ref(&tmp, wp);
                sk.add_prec_assign_ref(&Float::ONE, wp);
                sk = tmp2.div_prec(sk, wp).0;
            }
            twopoweri <<= 1;
        }
        // Add the last step: atan(sk) ~= sk
        arctgt.add_prec_assign(sk, wp);
        // undo the argument reduction
        arctgt <<= red;
        if comparison == Greater {
            // atan(x) = pi/2 - atan(1/x) for x > 0
            arctgt = (Float::pi_prec(wp).0 >> 1u32).sub_prec(arctgt, wp).0;
        }
        debug_assert!(arctgt > 0u32);
        let err = i64::exact_from(realprec + est_lost) - i64::exact_from(lost);
        if err > 0
            && float_can_round(
                arctgt.significand_ref().unwrap(),
                u64::exact_from(err),
                prec,
                rm,
            )
        {
            return Float::from_float_prec_round(if negative { -arctgt } else { arctgt }, prec, rm);
        }
        realprec += increment;
        increment = realprec >> 1;
    }
}

// u/2^k with the sign of `positive`, rounded to `prec` with `rm`. The shift is exact, so the
// ternary value is the conversion's, reversed along with the sign.
pub(crate) fn scaled_unsigned(
    u: u64,
    k: u32,
    positive: bool,
    prec: u64,
    rm: RoundingMode,
) -> (Float, Ordering) {
    let (f, o) = Float::from_unsigned_prec_round(u, prec, if positive { rm } else { -rm });
    let f = f >> k;
    if positive { (f, o) } else { (-f, o.reverse()) }
}

// Computes atan(x) u/(2 pi) for a finite nonzero `Float` x and a nonzero u, rounded to precision
// `prec` with rounding mode `rm`. `rm` may be `Exact` only for |x| = 1, where the result is u/8.
//
// This is mpfr_atanu from atanu.c, MPFR 4.2.2. The quotient is formed with the numerator scaled up
// by 2^SCALE, since atan(x) u/(2 pi) can fall below the smallest positive `Float` for a tiny x and
// a small u, which MPFR, computing inside a temporarily extended exponent range, never sees; a
// result below it is then decided by the rounding mode alone, as in `sin_with_period`.
fn atan_with_period_prec_round_normal_ref(
    x: &Float,
    u: u64,
    prec: u64,
    rm: RoundingMode,
) -> (Float, Ordering) {
    let positive = *x > 0u32;
    let exp_x = i64::from(x.get_exponent().unwrap());
    // |x| = 1: atanu(1, u) = u/8, atanu(-1, u) = -u/8, both exact
    if exp_x == 1 && x.significand_ref().unwrap().is_power_of_2() {
        return scaled_unsigned(u, 3, positive, prec, rm);
    }
    // Only |x| = 1 can be rounded exactly
    assert_ne!(rm, Exact, "Inexact atan_with_period");
    // For x >= 1, pi/2 - 1/x < atan(x) < pi/2, so u/4 - u/(2 pi x) < atanu(x, u) < u/4, and the
    // relative difference from u/4 is below 2/(pi x) < 1/x <= 2^(1 - EXP(x)). Once that is at most
    // 2^(-prec - 2), the difference is at most a quarter ulp of u/4, hence at most half an ulp of
    // u/4 at the target precision, so the result rounds like the value one ulp below u/4 at a
    // working precision above the target's. Requiring EXP(x) >= 65 also gives x > 2u/pi, which
    // keeps that value above (u - 1)/4.
    if exp_x >= 65 && exp_x > i64::exact_from(prec) + 2 {
        let w = if prec <= 63 { 65 } else { prec + 2 };
        // exact, since w >= 64
        let mut t = Float::from_unsigned_prec_round(u, w, Exact).0;
        t.decrement();
        // the last bit of t is 1 and w exceeds the target precision, so t is not representable
        // there, which pins the ternary value below
        t >>= 2u32;
        return Float::from_float_prec_round(if positive { t } else { -t }, prec, rm);
    }
    arc_with_period_scale(
        // scaling by a power of 2 is exact, and atan(x) u 2^SCALE stays far below the top of the
        // range, since |atan x| < pi/2 and u < 2^64
        |w| x.atan_prec_round_ref(w, Up).0 << SCALE,
        u,
        positive,
        prec,
        rm,
    )
}

// The Ziv loop shared by the inverse trigonometric functions with a period: given a way to compute
// f(x) 2^SCALE rounded away from zero at a working precision, forms f(x) u/(2 pi). It is used by
// the `Float` and `Rational` arctangents and by the arcsine, whose error analyses agree.
//
// The numerator is scaled up by 2^SCALE throughout, since f(x) u/(2 pi) can fall below the smallest
// positive `Float` for a tiny x and a small u, which MPFR, computing inside a temporarily extended
// exponent range, never sees; a result below it is then decided by the rounding mode alone, as in
// `sin_with_period`. Scaling before the multiplication rather than after also lets a `Rational` x
// too small to be a `Float` at all reach the loop, which is why the shift belongs to the caller.
pub(crate) fn arc_with_period_scale<F: FnMut(u64) -> Float>(
    mut scaled_f: F,
    u: u64,
    positive: bool,
    prec: u64,
    rm: RoundingMode,
) -> (Float, Ordering) {
    let mut w = prec + prec.ceiling_log_base_2() + 10;
    let mut increment = Limb::WIDTH;
    let u_float = Float::from(u);
    loop {
        // In the error analysis below, each theta denotes a value with |theta| <= 2^(1 - w).
        //
        // t = f(x) 2^SCALE (1 + theta), nonzero since we rounded away from zero and x is not
        let mut t = scaled_f(w);
        // t = f(x) u 2^SCALE (1 + theta)^2
        t.mul_prec_round_assign_ref(&u_float, w, Up);
        // 2 pi rounded toward zero, so that the quotient rounds away
        let two_pi = Float::pi_prec_round(w, Down).0 << 1u32;
        // t = f(x) u 2^SCALE/(2 pi) (1 + theta)^4, whose relative error is below 2^(4 - w) since
        // |(1 + theta)^4 - 1| <= 8 |theta| for w >= 3
        t.div_prec_round_assign(two_pi, w, Up);
        if let Some(result) = scaled_underflow(&t, positive, prec, rm) {
            return result;
        }
        let t = t >> SCALE;
        if float_can_round(t.significand_ref().unwrap(), w - 4, prec, rm) {
            return Float::from_float_prec_round(t, prec, rm);
        }
        w += increment;
        increment = w >> 1;
    }
}

// Computes atan(x) u/(2 pi) for a nonzero `Rational` x and a nonzero u, rounded to precision `prec`
// with rounding mode `rm`. (x = 0 and u = 0 are handled by the caller.) `rm` may be `Exact` only
// for |x| = 1, where the result is u/8.
//
// The three branches match the `Float` case, with one addition: an x below the bottom of the
// exponent range is not a `Float`, but its arctangent is its own leading term, so the quotient is
// formed from x itself. That substitution neglects a relative x^2/3, which for such an x is below
// 2^(2 SCALED_INPUT_EXPONENT) and so far beneath any working precision the loop can reach. It is
// also needed rather than merely cheaper: `atan_rational_helper` reports such an x as an underflow,
// and a large u can lift the quotient back into the range, where that answer would be wrong.
pub(crate) fn atan_with_period_rational_helper(
    x: &Rational,
    u: u64,
    prec: u64,
    rm: RoundingMode,
) -> (Float, Ordering) {
    let positive = *x > 0u32;
    let exp_x = x.floor_log_base_2_abs() + 1; // the MPFR-style exponent of x
    // |x| = 1: atanu(1, u) = u/8, atanu(-1, u) = -u/8, both exact
    if exp_x == 1 && x.denominator_ref() == &1u32 {
        return scaled_unsigned(u, 3, positive, prec, rm);
    }
    // Only |x| = 1 can be rounded exactly
    assert_ne!(rm, Exact, "Inexact atan_with_period_rational");
    // as in the `Float` case, the result is one ulp below u/4 once x is large enough
    if exp_x >= 65 && exp_x > i64::exact_from(prec) + 2 {
        let w = if prec <= 63 { 65 } else { prec + 2 };
        // exact, since w >= 64
        let mut t = Float::from_unsigned_prec_round(u, w, Exact).0;
        t.decrement();
        t >>= 2u32;
        return Float::from_float_prec_round(if positive { t } else { -t }, prec, rm);
    }
    if exp_x <= SCALED_INPUT_EXPONENT {
        let scaled = x << SCALE;
        return arc_with_period_scale(
            |w| Float::from_rational_prec_round_ref(&scaled, w, Up).0,
            u,
            positive,
            prec,
            rm,
        );
    }
    arc_with_period_scale(
        |w| atan_rational_helper(x, w, Up).0 << SCALE,
        u,
        positive,
        prec,
        rm,
    )
}

impl Float {
    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result to the specified
    /// precision and with the specified rounding mode. The [`Float`] is taken by value. An
    /// [`Ordering`] is also returned, indicating whether the rounded arctangent is less than, equal
    /// to, or greater than the exact arctangent. Although `NaN`s are not comparable to any
    /// [`Float`], whenever this function returns a `NaN` it also returns `Equal`.
    ///
    /// See [`RoundingMode`] for a description of the possible rounding modes.
    ///
    /// $$
    /// f(x,p,m) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN and $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan x|\rfloor-p+1}$.
    /// - If $x$ is not NaN and $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan
    ///   x|\rfloor-p}$.
    ///
    /// If the output has a precision, it is `prec`.
    ///
    /// Special cases:
    /// - $f(\text{NaN},p,m)=\text{NaN}$
    /// - $f(\pm\infty,p,m)=\pm\pi/2$, rounded
    /// - $f(\pm0.0,p,m)=\pm0.0$
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Floor` or `Down`, $0.0$ is returned instead.
    /// - If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Ceiling` or `Up`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $0<f(x,p,m)\leq2^{-2^{30}-1}$, and $m$ is `Nearest`, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Nearest`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is `Ceiling` or `Down`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is `Floor` or `Up`, $-2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,p,m)<0$, and $m$ is `Nearest`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<-2^{-2^{30}-1}$, and $m$ is `Nearest`, $-2^{-2^{30}}$ is
    ///   returned instead.
    ///
    /// Underflow requires an input of magnitude $2^{-2^{30}}$, the smallest positive [`Float`],
    /// rounded toward zero: since $|\arctan x| < |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you know you'll be using `Nearest`, consider using [`Float::atan_prec`] instead. If you
    /// know that your target precision is the precision of the input, consider using
    /// [`Float::atan_round`] instead. If both of these things are true, consider using
    /// [`Float::atan`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `self.significant_bits()`: an input above 1 in magnitude is first inverted, and the argument
    /// is then halved a logarithmic number of times and split into chunks whose arctangents are
    /// summed by binary splitting, all at a working precision of about $n$; the summation is the
    /// first term, and the inversion of the $m$-bit input the second. The magnitude of the input
    /// does not drive the cost.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` and `self` is nonzero and not NaN, since the arctangent of a
    /// finite nonzero [`Float`] is never exactly representable and neither is $\pm\pi/2$, or if
    /// `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100)
    ///     .0
    ///     .atan_prec_round(5, Floor);
    /// assert_eq!(c.to_string(), "0.781");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100)
    ///     .0
    ///     .atan_prec_round(5, Ceiling);
    /// assert_eq!(c.to_string(), "0.812");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100)
    ///     .0
    ///     .atan_prec_round(5, Nearest);
    /// assert_eq!(c.to_string(), "0.781");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100)
    ///     .0
    ///     .atan_prec_round(20, Floor);
    /// assert_eq!(c.to_string(), "0.78539753");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100)
    ///     .0
    ///     .atan_prec_round(20, Ceiling);
    /// assert_eq!(c.to_string(), "0.78539848");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100)
    ///     .0
    ///     .atan_prec_round(20, Nearest);
    /// assert_eq!(c.to_string(), "0.78539848");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_prec_round(self, prec: u64, rm: RoundingMode) -> (Self, Ordering) {
        self.atan_prec_round_ref(prec, rm)
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result to the specified
    /// precision and with the specified rounding mode. The [`Float`] is taken by reference. An
    /// [`Ordering`] is also returned, indicating whether the rounded arctangent is less than, equal
    /// to, or greater than the exact arctangent. Although `NaN`s are not comparable to any
    /// [`Float`], whenever this function returns a `NaN` it also returns `Equal`.
    ///
    /// See [`RoundingMode`] for a description of the possible rounding modes.
    ///
    /// $$
    /// f(x,p,m) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN and $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan x|\rfloor-p+1}$.
    /// - If $x$ is not NaN and $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan
    ///   x|\rfloor-p}$.
    ///
    /// If the output has a precision, it is `prec`.
    ///
    /// Special cases:
    /// - $f(\text{NaN},p,m)=\text{NaN}$
    /// - $f(\pm\infty,p,m)=\pm\pi/2$, rounded
    /// - $f(\pm0.0,p,m)=\pm0.0$
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Floor` or `Down`, $0.0$ is returned instead.
    /// - If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Ceiling` or `Up`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $0<f(x,p,m)\leq2^{-2^{30}-1}$, and $m$ is `Nearest`, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Nearest`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is `Ceiling` or `Down`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is `Floor` or `Up`, $-2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,p,m)<0$, and $m$ is `Nearest`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<-2^{-2^{30}-1}$, and $m$ is `Nearest`, $-2^{-2^{30}}$ is
    ///   returned instead.
    ///
    /// Underflow requires an input of magnitude $2^{-2^{30}}$, the smallest positive [`Float`],
    /// rounded toward zero: since $|\arctan x| < |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you know you'll be using `Nearest`, consider using [`Float::atan_prec_ref`] instead. If
    /// you know that your target precision is the precision of the input, consider using
    /// [`Float::atan_round_ref`] instead. If both of these things are true, consider using
    /// `(&Float).atan()` instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `self.significant_bits()`: an input above 1 in magnitude is first inverted, and the argument
    /// is then halved a logarithmic number of times and split into chunks whose arctangents are
    /// summed by binary splitting, all at a working precision of about $n$; the summation is the
    /// first term, and the inversion of the $m$-bit input the second. The magnitude of the input
    /// does not drive the cost.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` and `self` is nonzero and not NaN, since the arctangent of a
    /// finite nonzero [`Float`] is never exactly representable and neither is $\pm\pi/2$, or if
    /// `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_round_ref(5, Floor);
    /// assert_eq!(c.to_string(), "0.781");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_round_ref(5, Ceiling);
    /// assert_eq!(c.to_string(), "0.812");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_round_ref(5, Nearest);
    /// assert_eq!(c.to_string(), "0.781");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_round_ref(20, Floor);
    /// assert_eq!(c.to_string(), "0.78539753");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_round_ref(20, Ceiling);
    /// assert_eq!(c.to_string(), "0.78539848");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_round_ref(20, Nearest);
    /// assert_eq!(c.to_string(), "0.78539848");
    /// assert_eq!(o, Greater);
    /// ```
    pub fn atan_prec_round_ref(&self, prec: u64, rm: RoundingMode) -> (Self, Ordering) {
        assert_ne!(prec, 0);
        match &self.0 {
            NaN => (Self::NAN, Equal),
            // atan(+infinity) = pi/2, atan(-infinity) = -pi/2
            Infinity { sign } => {
                assert_ne!(rm, Exact, "Inexact atan");
                let (pi, o) = Self::pi_prec_round(prec, if *sign { rm } else { -rm });
                // exact
                let half = pi >> 1u32;
                if *sign {
                    (half, o)
                } else {
                    (-half, o.reverse())
                }
            }
            // atan(+0) = +0, atan(-0) = -0
            Zero { .. } => (self.clone(), Equal),
            Finite { .. } => atan_prec_round_normal_ref(self, prec, rm),
        }
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result to the nearest
    /// value of the specified precision. The [`Float`] is taken by value. An [`Ordering`] is also
    /// returned, indicating whether the rounded arctangent is less than, equal to, or greater than
    /// the exact arctangent. Although `NaN`s are not comparable to any [`Float`], whenever this
    /// function returns a `NaN` it also returns `Equal`.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the specified precision, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// $$
    /// f(x,p) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$.
    ///
    /// If the output has a precision, it is `prec`.
    ///
    /// Special cases:
    /// - $f(\text{NaN},p)=\text{NaN}$
    /// - $f(\pm\infty,p)=\pm\pi/2$, rounded
    /// - $f(\pm0.0,p)=1.0$
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,p)<0$, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.
    ///
    /// Underflow requires an input of magnitude $2^{-2^{30}}$, the smallest positive [`Float`],
    /// rounded toward zero: since $|\arctan x| < |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_prec_round`] instead. If you know that your target precision is the precision
    /// of the input, consider using [`Float::atan`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `self.significant_bits()`: an input above 1 in magnitude is first inverted, and the argument
    /// is then halved a logarithmic number of times and split into chunks whose arctangents are
    /// summed by binary splitting, all at a working precision of about $n$; the summation is the
    /// first term, and the inversion of the $m$-bit input the second. The magnitude of the input
    /// does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100).0.atan_prec(5);
    /// assert_eq!(c.to_string(), "0.781");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100).0.atan_prec(20);
    /// assert_eq!(c.to_string(), "0.78539848");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_prec(self, prec: u64) -> (Self, Ordering) {
        self.atan_prec_round(prec, Nearest)
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result to the nearest
    /// value of the specified precision. The [`Float`] is taken by reference. An [`Ordering`] is
    /// also returned, indicating whether the rounded arctangent is less than, equal to, or greater
    /// than the exact arctangent. Although `NaN`s are not comparable to any [`Float`], whenever
    /// this function returns a `NaN` it also returns `Equal`.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the specified precision, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// $$
    /// f(x,p) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$.
    ///
    /// If the output has a precision, it is `prec`.
    ///
    /// Special cases:
    /// - $f(\text{NaN},p)=\text{NaN}$
    /// - $f(\pm\infty,p)=\pm\pi/2$, rounded
    /// - $f(\pm0.0,p)=1.0$
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,p)<0$, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.
    ///
    /// Underflow requires an input of magnitude $2^{-2^{30}}$, the smallest positive [`Float`],
    /// rounded toward zero: since $|\arctan x| < |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_prec_round_ref`] instead. If you know that your target precision is the
    /// precision of the input, consider using `(&Float).atan()` instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `self.significant_bits()`: an input above 1 in magnitude is first inverted, and the argument
    /// is then halved a logarithmic number of times and split into chunks whose arctangents are
    /// summed by binary splitting, all at a working precision of about $n$; the summation is the
    /// first term, and the inversion of the $m$-bit input the second. The magnitude of the input
    /// does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_ref(5);
    /// assert_eq!(c.to_string(), "0.781");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_prec_ref(20);
    /// assert_eq!(c.to_string(), "0.78539848");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_prec_ref(&self, prec: u64) -> (Self, Ordering) {
        self.atan_prec_round_ref(prec, Nearest)
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result with the specified
    /// rounding mode. The [`Float`] is taken by value. An [`Ordering`] is also returned, indicating
    /// whether the rounded arctangent is less than, equal to, or greater than the exact arctangent.
    /// Although `NaN`s are not comparable to any [`Float`], whenever this function returns a `NaN`
    /// it also returns `Equal`.
    ///
    /// The precision of the output is the precision of the input. See [`RoundingMode`] for a
    /// description of the possible rounding modes.
    ///
    /// $$
    /// f(x,m) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN and $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan x|\rfloor-p+1}$, where $p$ is the precision of the input.
    /// - If $x$ is not NaN and $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan
    ///   x|\rfloor-p}$, where $p$ is the precision of the input.
    ///
    /// If the output has a precision, it is the precision of the input.
    ///
    /// Special cases:
    /// - $f(\text{NaN},m)=\text{NaN}$
    /// - $f(\pm\infty,m)=\pm\pi/2$, rounded
    /// - $f(\pm0.0,m)=1.0$
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,m)<2^{-2^{30}}$, and $m$ is `Floor` or `Down`, $0.0$ is returned instead.
    /// - If $0<f(x,m)<2^{-2^{30}}$, and $m$ is `Ceiling` or `Up`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $0<f(x,m)\leq2^{-2^{30}-1}$, and $m$ is `Nearest`, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,m)<2^{-2^{30}}$, and $m$ is `Nearest`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is `Ceiling` or `Down`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is `Floor` or `Up`, $-2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,m)<0$, and $m$ is `Nearest`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,m)<-2^{-2^{30}-1}$, and $m$ is `Nearest`, $-2^{-2^{30}}$ is returned
    ///   instead.
    ///
    /// Underflow requires an input of magnitude $2^{-2^{30}}$, the smallest positive [`Float`],
    /// rounded toward zero: since $|\arctan x| < |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you want to specify an output precision, consider using [`Float::atan_prec_round`]
    /// instead. If you know you'll be using the `Nearest` rounding mode, consider using
    /// [`Float::atan`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n) = O(n (\log n)^3 \log\log n)$
    ///
    /// $M(n) = O(n \log n)$
    ///
    /// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`: an input
    /// above 1 in magnitude is first inverted, and the argument is then halved a logarithmic number
    /// of times and split into chunks whose arctangents are summed by binary splitting, all at a
    /// working precision of about $n$. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` and `self` is nonzero and not NaN, since the arctangent of a
    /// finite nonzero [`Float`] is never exactly representable and neither is $\pm\pi/2$.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100).0.atan_round(Floor);
    /// assert_eq!(c.to_string(), "0.78539816339744830961566084581983");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100).0.atan_round(Ceiling);
    /// assert_eq!(c.to_string(), "0.78539816339744830961566084582062");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) = Float::from_unsigned_prec(1u32, 100).0.atan_round(Nearest);
    /// assert_eq!(c.to_string(), "0.78539816339744830961566084581983");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_round(self, rm: RoundingMode) -> (Self, Ordering) {
        let prec = self.significant_bits();
        self.atan_prec_round(prec, rm)
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result with the specified
    /// rounding mode. The [`Float`] is taken by reference. An [`Ordering`] is also returned,
    /// indicating whether the rounded arctangent is less than, equal to, or greater than the exact
    /// arctangent. Although `NaN`s are not comparable to any [`Float`], whenever this function
    /// returns a `NaN` it also returns `Equal`.
    ///
    /// The precision of the output is the precision of the input. See [`RoundingMode`] for a
    /// description of the possible rounding modes.
    ///
    /// $$
    /// f(x,m) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN and $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan x|\rfloor-p+1}$, where $p$ is the precision of the input.
    /// - If $x$ is not NaN and $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan
    ///   x|\rfloor-p}$, where $p$ is the precision of the input.
    ///
    /// If the output has a precision, it is the precision of the input.
    ///
    /// Special cases:
    /// - $f(\text{NaN},m)=\text{NaN}$
    /// - $f(\pm\infty,m)=\pm\pi/2$, rounded
    /// - $f(\pm0.0,m)=1.0$
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,m)<2^{-2^{30}}$, and $m$ is `Floor` or `Down`, $0.0$ is returned instead.
    /// - If $0<f(x,m)<2^{-2^{30}}$, and $m$ is `Ceiling` or `Up`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $0<f(x,m)\leq2^{-2^{30}-1}$, and $m$ is `Nearest`, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,m)<2^{-2^{30}}$, and $m$ is `Nearest`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is `Ceiling` or `Down`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,m)<0$, and $m$ is `Floor` or `Up`, $-2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,m)<0$, and $m$ is `Nearest`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,m)<-2^{-2^{30}-1}$, and $m$ is `Nearest`, $-2^{-2^{30}}$ is returned
    ///   instead.
    ///
    /// Underflow requires an input of magnitude $2^{-2^{30}}$, the smallest positive [`Float`],
    /// rounded toward zero: since $|\arctan x| < |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you want to specify an output precision, consider using [`Float::atan_prec_round_ref`]
    /// instead. If you know you'll be using the `Nearest` rounding mode, consider using
    /// `(&Float).atan()` instead.
    ///
    /// # Worst-case complexity
    /// $T(n) = O(n (\log n)^3 \log\log n)$
    ///
    /// $M(n) = O(n \log n)$
    ///
    /// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`: an input
    /// above 1 in magnitude is first inverted, and the argument is then halved a logarithmic number
    /// of times and split into chunks whose arctangents are summed by binary splitting, all at a
    /// working precision of about $n$. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` and `self` is nonzero and not NaN, since the arctangent of a
    /// finite nonzero [`Float`] is never exactly representable and neither is $\pm\pi/2$.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_round_ref(Floor);
    /// assert_eq!(c.to_string(), "0.78539816339744830961566084581983");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_round_ref(Ceiling);
    /// assert_eq!(c.to_string(), "0.78539816339744830961566084582062");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) = (&Float::from_unsigned_prec(1u32, 100).0).atan_round_ref(Nearest);
    /// assert_eq!(c.to_string(), "0.78539816339744830961566084581983");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_round_ref(&self, rm: RoundingMode) -> (Self, Ordering) {
        self.atan_prec_round_ref(self.significant_bits(), rm)
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result to the specified
    /// precision and with the specified rounding mode. The [`Float`] is replaced by the result, and
    /// an [`Ordering`] is returned, indicating whether the rounded arctangent is less than, equal
    /// to, or greater than the exact arctangent. Although `NaN`s are not comparable to any
    /// [`Float`], whenever this function sets a `NaN` it also returns `Equal`.
    ///
    /// See [`RoundingMode`] for a description of the possible rounding modes.
    ///
    /// $$
    /// x \gets \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN and $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan x|\rfloor-p+1}$.
    /// - If $x$ is not NaN and $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan
    ///   x|\rfloor-p}$.
    ///
    /// If the output has a precision, it is `prec`.
    ///
    /// See the [`Float::atan_prec_round`] documentation for information on special cases, overflow,
    /// and underflow.
    ///
    /// If you know you'll be using `Nearest`, consider using [`Float::atan_prec_assign`] instead.
    /// If you know that your target precision is the precision of the input, consider using
    /// [`Float::atan_round_assign`] instead. If both of these things are true, consider using
    /// [`Float::atan_assign`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `self.significant_bits()`: an input above 1 in magnitude is first inverted, and the argument
    /// is then halved a logarithmic number of times and split into chunks whose arctangents are
    /// summed by binary splitting, all at a working precision of about $n$; the summation is the
    /// first term, and the inversion of the $m$-bit input the second. The magnitude of the input
    /// does not drive the cost.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` and `self` is nonzero and not NaN, since the arctangent of a
    /// finite nonzero [`Float`] is never exactly representable and neither is $\pm\pi/2$, or if
    /// `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_round_assign(5, Floor), Less);
    /// assert_eq!(x.to_string(), "0.781");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_round_assign(5, Ceiling), Greater);
    /// assert_eq!(x.to_string(), "0.812");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_round_assign(5, Nearest), Less);
    /// assert_eq!(x.to_string(), "0.781");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_round_assign(20, Floor), Less);
    /// assert_eq!(x.to_string(), "0.78539753");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_round_assign(20, Ceiling), Greater);
    /// assert_eq!(x.to_string(), "0.78539848");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_round_assign(20, Nearest), Greater);
    /// assert_eq!(x.to_string(), "0.78539848");
    /// ```
    #[inline]
    pub fn atan_prec_round_assign(&mut self, prec: u64, rm: RoundingMode) -> Ordering {
        let o;
        (*self, o) = self.atan_prec_round_ref(prec, rm);
        o
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result to the nearest
    /// value of the specified precision. The [`Float`] is replaced by the result, and an
    /// [`Ordering`] is returned, indicating whether the rounded arctangent is less than, equal to,
    /// or greater than the exact arctangent. Although `NaN`s are not comparable to any [`Float`],
    /// whenever this function sets a `NaN` it also returns `Equal`.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the specified precision, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// $$
    /// x \gets \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$.
    ///
    /// If the output has a precision, it is `prec`.
    ///
    /// See the [`Float::atan_prec`] documentation for information on special cases, overflow, and
    /// underflow.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_prec_round_assign`] instead. If you know that your target precision is the
    /// precision of the input, consider using [`Float::atan_assign`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `self.significant_bits()`: an input above 1 in magnitude is first inverted, and the argument
    /// is then halved a logarithmic number of times and split into chunks whose arctangents are
    /// summed by binary splitting, all at a working precision of about $n$; the summation is the
    /// first term, and the inversion of the $m$-bit input the second. The magnitude of the input
    /// does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_assign(5), Less);
    /// assert_eq!(x.to_string(), "0.781");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_prec_assign(20), Greater);
    /// assert_eq!(x.to_string(), "0.78539848");
    /// ```
    #[inline]
    pub fn atan_prec_assign(&mut self, prec: u64) -> Ordering {
        self.atan_prec_round_assign(prec, Nearest)
    }

    /// Computes $\arctan x$, the arctangent of a [`Float`], rounding the result with the specified
    /// rounding mode. The [`Float`] is replaced by the result, and an [`Ordering`] is returned,
    /// indicating whether the rounded arctangent is less than, equal to, or greater than the exact
    /// arctangent. Although `NaN`s are not comparable to any [`Float`], whenever this function sets
    /// a `NaN` it also returns `Equal`.
    ///
    /// The precision of the output is the precision of the input. See [`RoundingMode`] for a
    /// description of the possible rounding modes.
    ///
    /// $$
    /// x \gets \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN and $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan x|\rfloor-p+1}$, where $p$ is the precision of the input.
    /// - If $x$ is not NaN and $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan
    ///   x|\rfloor-p}$, where $p$ is the precision of the input.
    ///
    /// If the output has a precision, it is the precision of the input.
    ///
    /// See the [`Float::atan_round`] documentation for information on special cases, overflow, and
    /// underflow.
    ///
    /// If you want to specify an output precision, consider using [`Float::atan_prec_round_assign`]
    /// instead. If you know you'll be using the `Nearest` rounding mode, consider using
    /// [`Float::atan_assign`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n) = O(n (\log n)^3 \log\log n)$
    ///
    /// $M(n) = O(n \log n)$
    ///
    /// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`: an input
    /// above 1 in magnitude is first inverted, and the argument is then halved a logarithmic number
    /// of times and split into chunks whose arctangents are summed by binary splitting, all at a
    /// working precision of about $n$. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` and `self` is nonzero and not NaN, since the arctangent of a
    /// finite nonzero [`Float`] is never exactly representable and neither is $\pm\pi/2$.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_round_assign(Floor), Less);
    /// assert_eq!(x.to_string(), "0.78539816339744830961566084581983");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_round_assign(Ceiling), Greater);
    /// assert_eq!(x.to_string(), "0.78539816339744830961566084582062");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// assert_eq!(x.atan_round_assign(Nearest), Less);
    /// assert_eq!(x.to_string(), "0.78539816339744830961566084581983");
    /// ```
    #[inline]
    pub fn atan_round_assign(&mut self, rm: RoundingMode) -> Ordering {
        let prec = self.significant_bits();
        self.atan_prec_round_assign(prec, rm)
    }
}

impl Float {
    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn,
    /// rounding the result to the specified precision and with the specified rounding mode. The
    /// [`Float`] is taken by value. An [`Ordering`] is also returned, indicating whether the
    /// rounded arctangent is less than, equal to, or greater than the exact arctangent. Although
    /// `NaN`s are not comparable to any [`Float`], whenever this function returns a `NaN` it also
    /// returns `Equal`.
    ///
    /// See [`RoundingMode`] for a description of the possible rounding modes.
    ///
    /// $$
    /// f(x,u,p,m) = \arctan(x)u/(2\pi)+\varepsilon.
    /// $$
    /// - If $x$ is NaN or zero, $u = 0$, or $|x|$ is 1 or infinite, $\varepsilon$ may be ignored or
    ///   assumed to be 0.
    /// - Otherwise, if $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan(x)u/(2\pi)|\rfloor-p+1}$.
    /// - Otherwise, if $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2
    ///   |\arctan(x)u/(2\pi)|\rfloor-p}$.
    ///
    /// If the output has a precision, it is `prec`.
    ///
    /// Special cases:
    /// - $f(\text{NaN},u,p,m)=\text{NaN}$
    /// - $f(\pm\infty,u,p,m)=\pm u/4$, a quarter turn
    /// - $f(\pm0.0,u,p,m)=\pm0.0$
    /// - $f(x,0,p,m)=\pm0.0$, with the sign of $x$, so that the function stays odd
    /// - $f(\pm1,u,p,m)=\pm u/8$, an eighth of a turn
    ///
    /// The last three are the only exact cases, and the quarter and eighth turns are exact only
    /// when $p$ is large enough to hold them.
    ///
    /// Underflow:
    /// - If $0<f(x,u,p,m)<2^{-2^{30}}$, and $m$ is `Floor` or `Down`, $0.0$ is returned instead.
    /// - If $0<f(x,u,p,m)<2^{-2^{30}}$, and $m$ is `Ceiling` or `Up`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $0<f(x,u,p,m)\leq2^{-2^{30}-1}$, and $m$ is `Nearest`, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,u,p,m)<2^{-2^{30}}$, and $m$ is `Nearest`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - The negative cases mirror these, since the function is odd.
    ///
    /// Overflow is not possible, since $|f(x,u,p,m)| < u/4 < 2^{62}$. Underflow requires a tiny $x$
    /// together with a small $u$, since the result is about $xu/(2\pi)$ there.
    ///
    /// If you know you'll be using `Nearest`, consider using [`Float::atan_with_period_prec`]
    /// instead. If you know that your target precision is the precision of the input, consider
    /// using [`Float::atan_with_period_round`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `self.significant_bits()`: the arctangent is taken at a working precision of about $n$ bits,
    /// which costs the first term, and is then scaled by $u/(2\pi)$, which needs $\pi$ to that many
    /// bits; the second term covers the $m$-bit input. The magnitude of the input does not drive
    /// the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision (which is the case unless $x$ is zero or NaN, $u$ is zero, or $|x|$
    /// is 1 or infinite and $p$ is large enough to hold a quarter or an eighth of a turn).
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::{One, Two};
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::ONE.atan_with_period_prec_round(360, 10, Exact);
    /// assert_eq!(t.to_string(), "45.000");
    /// assert_eq!(o, Equal);
    ///
    /// let (t, o) = Float::TWO.atan_with_period_prec_round(360, 10, Floor);
    /// assert_eq!(t.to_string(), "63.375");
    /// assert_eq!(o, Less);
    ///
    /// let (t, o) = Float::TWO.atan_with_period_prec_round(360, 10, Ceiling);
    /// assert_eq!(t.to_string(), "63.438");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_with_period_prec_round(
        self,
        u: u64,
        prec: u64,
        rm: RoundingMode,
    ) -> (Self, Ordering) {
        self.atan_with_period_prec_round_ref(u, prec, rm)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn,
    /// rounding the result to the specified precision and with the specified rounding mode. The
    /// [`Float`] is taken by reference. An [`Ordering`] is also returned, indicating whether the
    /// rounded arctangent is less than, equal to, or greater than the exact arctangent. Although
    /// `NaN`s are not comparable to any [`Float`], whenever this function returns a `NaN` it also
    /// returns `Equal`.
    ///
    /// See [`Float::atan_with_period_prec_round`] for the error bounds, the special and closed-form
    /// cases, overflow, and the complexity; this function behaves the same way.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::{One, Two};
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = (&Float::ONE).atan_with_period_prec_round_ref(360, 10, Exact);
    /// assert_eq!(t.to_string(), "45.000");
    /// assert_eq!(o, Equal);
    ///
    /// let (t, o) = (&Float::TWO).atan_with_period_prec_round_ref(360, 10, Floor);
    /// assert_eq!(t.to_string(), "63.375");
    /// assert_eq!(o, Less);
    /// ```
    pub fn atan_with_period_prec_round_ref(
        &self,
        u: u64,
        prec: u64,
        rm: RoundingMode,
    ) -> (Self, Ordering) {
        assert_ne!(prec, 0);
        match &self.0 {
            NaN => (Self::NAN, Equal),
            // atanu(+infinity, u) = u/4, atanu(-infinity, u) = -u/4, a quarter turn
            Infinity { sign } => {
                if u == 0 {
                    (
                        if *sign {
                            Self::ZERO
                        } else {
                            Self::NEGATIVE_ZERO
                        },
                        Equal,
                    )
                } else {
                    scaled_unsigned(u, 2, *sign, prec, rm)
                }
            }
            // atanu(±0.0, u) = ±0.0, even for u = 0
            Zero { .. } => (self.clone(), Equal),
            Finite { .. } => {
                if u == 0 {
                    // atanu(x, 0) = 0 with the sign of x, which agrees with the x = 0 case and
                    // keeps the function odd
                    (
                        if *self < 0u32 {
                            Self::NEGATIVE_ZERO
                        } else {
                            Self::ZERO
                        },
                        Equal,
                    )
                } else {
                    atan_with_period_prec_round_normal_ref(self, u, prec, rm)
                }
            }
        }
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn,
    /// rounding the result to the nearest value of the specified precision. The [`Float`] is taken
    /// by value. An [`Ordering`] is also returned, indicating whether the rounded arctangent is
    /// less than, equal to, or greater than the exact arctangent. Although `NaN`s are not
    /// comparable to any [`Float`], whenever this function returns a `NaN` it also returns `Equal`.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the specified precision, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// See [`Float::atan_with_period_prec_round`] for the error bounds, the special and closed-form
    /// cases, overflow, and the complexity; this function behaves the same way with `Nearest`.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_with_period_prec_round`] instead.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::{One, Two};
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::ONE.atan_with_period_prec(360, 10);
    /// assert_eq!(t.to_string(), "45.000");
    /// assert_eq!(o, Equal);
    ///
    /// let (t, o) = Float::TWO.atan_with_period_prec(360, 10);
    /// assert_eq!(t.to_string(), "63.438");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_with_period_prec(self, u: u64, prec: u64) -> (Self, Ordering) {
        self.atan_with_period_prec_round(u, prec, Nearest)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn,
    /// rounding the result to the nearest value of the specified precision. The [`Float`] is taken
    /// by reference. An [`Ordering`] is also returned, indicating whether the rounded arctangent is
    /// less than, equal to, or greater than the exact arctangent. Although `NaN`s are not
    /// comparable to any [`Float`], whenever this function returns a `NaN` it also returns `Equal`.
    ///
    /// See [`Float::atan_with_period_prec`] and [`Float::atan_with_period_prec_round`]; this
    /// function behaves the same way.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::Two;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = (&Float::TWO).atan_with_period_prec_ref(360, 10);
    /// assert_eq!(t.to_string(), "63.438");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_with_period_prec_ref(&self, u: u64, prec: u64) -> (Self, Ordering) {
        self.atan_with_period_prec_round_ref(u, prec, Nearest)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn,
    /// rounding the result to the precision of the input and with the specified rounding mode. The
    /// [`Float`] is taken by value. An [`Ordering`] is also returned, indicating whether the
    /// rounded arctangent is less than, equal to, or greater than the exact arctangent. Although
    /// `NaN`s are not comparable to any [`Float`], whenever this function returns a `NaN` it also
    /// returns `Equal`.
    ///
    /// See [`Float::atan_with_period_prec_round`] for the error bounds, the special and closed-form
    /// cases, overflow, and the complexity; this function behaves the same way with `prec` equal to
    /// the precision of the input.
    ///
    /// If you want to specify an output precision, consider using
    /// [`Float::atan_with_period_prec_round`] instead.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` but the result cannot be represented exactly with the precision of
    /// the input.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// // the output takes the input's precision, here 10 bits
    /// let (t, o) = Float::from_unsigned_prec(2u32, 10)
    ///     .0
    ///     .atan_with_period_round(360, Floor);
    /// assert_eq!(t.to_string(), "63.375");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_with_period_round(self, u: u64, rm: RoundingMode) -> (Self, Ordering) {
        let prec = self.significant_bits();
        self.atan_with_period_prec_round(u, prec, rm)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn,
    /// rounding the result to the precision of the input and with the specified rounding mode. The
    /// [`Float`] is taken by reference. An [`Ordering`] is also returned, indicating whether the
    /// rounded arctangent is less than, equal to, or greater than the exact arctangent. Although
    /// `NaN`s are not comparable to any [`Float`], whenever this function returns a `NaN` it also
    /// returns `Equal`.
    ///
    /// See [`Float::atan_with_period_round`] and [`Float::atan_with_period_prec_round`]; this
    /// function behaves the same way.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` but the result cannot be represented exactly with the precision of
    /// the input.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// // the output takes the input's precision, here 10 bits
    /// let x = Float::from_unsigned_prec(2u32, 10).0;
    /// let (t, o) = (&x).atan_with_period_round_ref(360, Floor);
    /// assert_eq!(t.to_string(), "63.375");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_with_period_round_ref(&self, u: u64, rm: RoundingMode) -> (Self, Ordering) {
        self.atan_with_period_prec_round_ref(u, self.significant_bits(), rm)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn
    /// (so that `u = 360` is degrees), rounding the result to the precision of the input and to the
    /// nearest [`Float`]. The [`Float`] is taken by value.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the precision of the input, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// See [`Float::atan_with_period_prec_round`] for the error bounds, the special and closed-form
    /// cases, overflow, and the complexity; this function behaves the same way with `prec` equal to
    /// the precision of the input and `rm` equal to `Nearest`.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_with_period_round`] instead. If you want to specify an output precision,
    /// consider using [`Float::atan_with_period_prec`]. If you want both of these things, consider
    /// using [`Float::atan_with_period_prec_round`].
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    ///
    /// let t = Float::from_unsigned_prec(2u32, 10).0.atan_with_period(360);
    /// assert_eq!(t.to_string(), "63.438");
    /// ```
    #[inline]
    pub fn atan_with_period(self, u: u64) -> Self {
        let prec = self.significant_bits();
        self.atan_with_period_prec(u, prec).0
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn
    /// (so that `u = 360` is degrees), rounding the result to the precision of the input and to the
    /// nearest [`Float`]. The [`Float`] is taken by reference.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the precision of the input, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// See [`Float::atan_with_period_prec_round`] for the error bounds, the special and closed-form
    /// cases, overflow, and the complexity; this function behaves the same way with `prec` equal to
    /// the precision of the input and `rm` equal to `Nearest`.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_with_period_round_ref`] instead. If you want to specify an output precision,
    /// consider using [`Float::atan_with_period_prec_ref`]. If you want both of these things,
    /// consider using [`Float::atan_with_period_prec_round_ref`].
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    ///
    /// let t = (&Float::from_unsigned_prec(2u32, 10).0).atan_with_period_ref(360);
    /// assert_eq!(t.to_string(), "63.438");
    /// ```
    #[inline]
    pub fn atan_with_period_ref(&self, u: u64) -> Self {
        self.atan_with_period_prec_ref(u, self.significant_bits()).0
    }

    /// Replaces a [`Float`] measured in $u$ths of a turn with its arctangent, rounding the result
    /// to the specified precision and with the specified rounding mode. An [`Ordering`] is
    /// returned, indicating whether the rounded arctangent is less than, equal to, or greater than
    /// the exact arctangent. Although `NaN`s are not comparable to any [`Float`], whenever this
    /// function sets a `NaN` it also returns `Equal`.
    ///
    /// See [`Float::atan_with_period_prec_round`] for the error bounds, the special and closed-form
    /// cases, overflow, and the complexity; this function behaves the same way.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::Two;
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::TWO;
    /// let o = x.atan_with_period_prec_round_assign(360, 10, Floor);
    /// assert_eq!(x.to_string(), "63.375");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_with_period_prec_round_assign(
        &mut self,
        u: u64,
        prec: u64,
        rm: RoundingMode,
    ) -> Ordering {
        let (t, o) = self.atan_with_period_prec_round_ref(u, prec, rm);
        *self = t;
        o
    }

    /// Replaces a [`Float`] measured in $u$ths of a turn with its arctangent, rounding the result
    /// to the nearest value of the specified precision. An [`Ordering`] is returned, indicating
    /// whether the rounded arctangent is less than, equal to, or greater than the exact arctangent.
    /// Although `NaN`s are not comparable to any [`Float`], whenever this function sets a `NaN` it
    /// also returns `Equal`.
    ///
    /// See [`Float::atan_with_period_prec`] and [`Float::atan_with_period_prec_round`]; this
    /// function behaves the same way.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::Two;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::TWO;
    /// let o = x.atan_with_period_prec_assign(360, 10);
    /// assert_eq!(x.to_string(), "63.438");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_with_period_prec_assign(&mut self, u: u64, prec: u64) -> Ordering {
        self.atan_with_period_prec_round_assign(u, prec, Nearest)
    }

    /// Replaces a [`Float`] measured in $u$ths of a turn with its arctangent, rounding the result
    /// to the precision of the input and with the specified rounding mode. An [`Ordering`] is
    /// returned, indicating whether the rounded arctangent is less than, equal to, or greater than
    /// the exact arctangent. Although `NaN`s are not comparable to any [`Float`], whenever this
    /// function sets a `NaN` it also returns `Equal`.
    ///
    /// See [`Float::atan_with_period_round`] and [`Float::atan_with_period_prec_round`]; this
    /// function behaves the same way.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` but the result cannot be represented exactly with the precision of
    /// the input.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// // the output takes the input's precision, here 10 bits
    /// let mut x = Float::from_unsigned_prec(2u32, 10).0;
    /// let o = x.atan_with_period_round_assign(360, Floor);
    /// assert_eq!(x.to_string(), "63.375");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_with_period_round_assign(&mut self, u: u64, rm: RoundingMode) -> Ordering {
        let prec = self.significant_bits();
        self.atan_with_period_prec_round_assign(u, prec, rm)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Float`] measured in $u$ths of a turn
    /// (so that `u = 360` is degrees), rounding the result to the precision of the input and to the
    /// nearest [`Float`]. The [`Float`] is replaced by the result.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the precision of the input, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// See [`Float::atan_with_period_prec_round`] for the error bounds, the special and closed-form
    /// cases, overflow, and the complexity; this function behaves the same way with `prec` equal to
    /// the precision of the input and `rm` equal to `Nearest`.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_with_period_round_assign`] instead. If you want to specify an output
    /// precision, consider using [`Float::atan_with_period_prec_assign`]. If you want both of these
    /// things, consider using [`Float::atan_with_period_prec_round_assign`].
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    ///
    /// let mut x = Float::from_unsigned_prec(2u32, 10).0;
    /// x.atan_with_period_assign(360);
    /// assert_eq!(x.to_string(), "63.438");
    /// ```
    #[inline]
    pub fn atan_with_period_assign(&mut self, u: u64) {
        let prec = self.significant_bits();
        self.atan_with_period_prec_assign(u, prec);
    }
}

impl Float {
    /// Computes $\arctan x$, the arctangent of a [`Rational`], rounding the result to the specified
    /// precision and with the specified rounding mode and returning the result as a [`Float`]. The
    /// [`Rational`] is taken by value. An [`Ordering`] is also returned, indicating whether the
    /// rounded arctangent is less than, equal to, or greater than the exact arctangent.
    ///
    /// See [`RoundingMode`] for a description of the possible rounding modes.
    ///
    /// $$
    /// f(x,p,m) = \arctan x+\varepsilon.
    /// $$
    /// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p+1}$.
    /// - If $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$.
    ///
    /// These bounds do not apply when the result underflows; see below.
    ///
    /// The output has precision `prec`.
    ///
    /// Special cases:
    /// - $f(0,p,m)=0$.
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Floor` or `Down`, $0.0$ is returned instead.
    /// - If $0<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Ceiling` or `Up`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $0<f(x,p,m)\leq2^{-2^{30}-1}$, and $m$ is `Nearest`, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,p,m)<2^{-2^{30}}$, and $m$ is `Nearest`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is `Ceiling` or `Down`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<0$, and $m$ is `Floor` or `Up`, $-2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,p,m)<0$, and $m$ is `Nearest`, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p,m)<-2^{-2^{30}-1}$, and $m$ is `Nearest`, $-2^{-2^{30}}$ is
    ///   returned instead.
    ///
    /// Underflow requires an input of magnitude about $2^{-2^{30}}$ or less: since $|\arctan x| <
    /// |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you know you'll be using `Nearest`, consider using [`Float::atan_rational_prec`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `x.significant_bits()`: the input is rounded once to a working precision of about $n$ bits
    /// and its [`Float`] arctangent taken there, which costs the first term; the rounding of the
    /// $m$-bit input is the second. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision (which is the case for every nonzero input).
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = Float::atan_rational_prec_round(Rational::from_unsigneds(3u8, 5), 5, Floor);
    /// assert_eq!(c.to_string(), "0.531");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::atan_rational_prec_round(Rational::from_unsigneds(3u8, 5), 5, Ceiling);
    /// assert_eq!(c.to_string(), "0.562");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) = Float::atan_rational_prec_round(Rational::from_unsigneds(3u8, 5), 20, Floor);
    /// assert_eq!(c.to_string(), "0.54041862");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::atan_rational_prec_round(Rational::from_unsigneds(3u8, 5), 20, Ceiling);
    /// assert_eq!(c.to_string(), "0.54041958");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    #[allow(clippy::needless_pass_by_value)]
    pub fn atan_rational_prec_round(x: Rational, prec: u64, rm: RoundingMode) -> (Self, Ordering) {
        Self::atan_rational_prec_round_ref(&x, prec, rm)
    }

    /// Computes $\arctan x$, the arctangent of a [`Rational`], rounding the result to the specified
    /// precision and with the specified rounding mode and returning the result as a [`Float`]. The
    /// [`Rational`] is taken by reference. An [`Ordering`] is also returned, indicating whether the
    /// rounded arctangent is less than, equal to, or greater than the exact arctangent.
    ///
    /// See [`RoundingMode`] for a description of the possible rounding modes.
    ///
    /// $$
    /// f(x,p,m) = \arctan x+\varepsilon.
    /// $$
    /// - If $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p+1}$.
    /// - If $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$.
    ///
    /// These bounds do not apply when the result underflows.
    ///
    /// The output has precision `prec`.
    ///
    /// Special cases:
    /// - $f(0,p,m)=0$.
    ///
    /// See the [`Float::atan_rational_prec_round`] documentation for information on overflow and
    /// underflow.
    ///
    /// If you know you'll be using `Nearest`, consider using [`Float::atan_rational_prec_ref`]
    /// instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `x.significant_bits()`: the input is rounded once to a working precision of about $n$ bits
    /// and its [`Float`] arctangent taken there, which costs the first term; the rounding of the
    /// $m$-bit input is the second. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision (which is the case for every nonzero input).
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) =
    ///     Float::atan_rational_prec_round_ref(&Rational::from_unsigneds(3u8, 5), 5, Floor);
    /// assert_eq!(c.to_string(), "0.531");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) =
    ///     Float::atan_rational_prec_round_ref(&Rational::from_unsigneds(3u8, 5), 5, Ceiling);
    /// assert_eq!(c.to_string(), "0.562");
    /// assert_eq!(o, Greater);
    ///
    /// let (c, o) =
    ///     Float::atan_rational_prec_round_ref(&Rational::from_unsigneds(3u8, 5), 20, Floor);
    /// assert_eq!(c.to_string(), "0.54041862");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) =
    ///     Float::atan_rational_prec_round_ref(&Rational::from_unsigneds(3u8, 5), 20, Ceiling);
    /// assert_eq!(c.to_string(), "0.54041958");
    /// assert_eq!(o, Greater);
    /// ```
    pub fn atan_rational_prec_round_ref(
        x: &Rational,
        prec: u64,
        rm: RoundingMode,
    ) -> (Self, Ordering) {
        assert_ne!(prec, 0);
        if *x == 0u32 {
            // atan(0) = 0, exactly
            return (Self::ZERO, Equal);
        }
        atan_rational_helper(x, prec, rm)
    }

    /// Computes $\arctan x$, the arctangent of a [`Rational`], rounding the result to the nearest
    /// value of the specified precision and returning the result as a [`Float`]. The [`Rational`]
    /// is taken by value. An [`Ordering`] is also returned, indicating whether the rounded
    /// arctangent is less than, equal to, or greater than the exact arctangent.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the specified precision, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// $$
    /// f(x,p) = \arctan x+\varepsilon,
    /// $$
    /// where $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$ (unless the result
    /// underflows; see below).
    ///
    /// The output has precision `prec`.
    ///
    /// Special cases:
    /// - $f(0,p)=0$.
    ///
    /// Overflow and underflow:
    /// - Since $|\arctan x| < \pi/2$, the result never overflows.
    /// - If $0<f(x,p)\leq2^{-2^{30}-1}$, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,p)<2^{-2^{30}}$, $2^{-2^{30}}$ is returned instead.
    /// - If $-2^{-2^{30}-1}\leq f(x,p)<0$, $-0.0$ is returned instead.
    /// - If $-2^{-2^{30}}<f(x,p)<-2^{-2^{30}-1}$, $-2^{-2^{30}}$ is returned instead.
    ///
    /// Underflow requires an input of magnitude about $2^{-2^{30}}$ or less: since $|\arctan x| <
    /// |x|$ for nonzero $x$, no other input can reach it.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_rational_prec_round`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `x.significant_bits()`: the input is rounded once to a working precision of about $n$ bits
    /// and its [`Float`] arctangent taken there, which costs the first term; the rounding of the
    /// $m$-bit input is the second. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = Float::atan_rational_prec(Rational::from_unsigneds(3u8, 5), 5);
    /// assert_eq!(c.to_string(), "0.531");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::atan_rational_prec(Rational::from_unsigneds(3u8, 5), 20);
    /// assert_eq!(c.to_string(), "0.54041958");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    #[allow(clippy::needless_pass_by_value)]
    pub fn atan_rational_prec(x: Rational, prec: u64) -> (Self, Ordering) {
        Self::atan_rational_prec_round_ref(&x, prec, Nearest)
    }

    /// Computes $\arctan x$, the arctangent of a [`Rational`], rounding the result to the nearest
    /// value of the specified precision and returning the result as a [`Float`]. The [`Rational`]
    /// is taken by reference. An [`Ordering`] is also returned, indicating whether the rounded
    /// arctangent is less than, equal to, or greater than the exact arctangent.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the specified precision, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// $$
    /// f(x,p) = \arctan x+\varepsilon,
    /// $$
    /// where $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$ (unless the result
    /// underflows).
    ///
    /// The output has precision `prec`.
    ///
    /// Special cases:
    /// - $f(0,p)=0$.
    ///
    /// See the [`Float::atan_rational_prec`] documentation for information on overflow and
    /// underflow.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_rational_prec_round_ref`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `x.significant_bits()`: the input is rounded once to a working precision of about $n$ bits
    /// and its [`Float`] arctangent taken there, which costs the first term; the rounding of the
    /// $m$-bit input is the second. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (c, o) = Float::atan_rational_prec_ref(&Rational::from_unsigneds(3u8, 5), 5);
    /// assert_eq!(c.to_string(), "0.531");
    /// assert_eq!(o, Less);
    ///
    /// let (c, o) = Float::atan_rational_prec_ref(&Rational::from_unsigneds(3u8, 5), 20);
    /// assert_eq!(c.to_string(), "0.54041958");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_rational_prec_ref(x: &Rational, prec: u64) -> (Self, Ordering) {
        Self::atan_rational_prec_round_ref(x, prec, Nearest)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Rational`] measured in $u$ths of a
    /// turn, rounding the result to the specified precision and with the specified rounding mode
    /// and returning the result as a [`Float`]. The [`Rational`] is taken by value. An [`Ordering`]
    /// is also returned, indicating whether the rounded arctangent is less than, equal to, or
    /// greater than the exact arctangent.
    ///
    /// See [`RoundingMode`] for a description of the possible rounding modes.
    ///
    /// $$
    /// f(x,u,p,m) = \arctan(x)u/(2\pi)+\varepsilon.
    /// $$
    /// - If $x$ is zero, $u = 0$, or $|x|$ is 1, $\varepsilon$ may be ignored or assumed to be 0.
    /// - Otherwise, if $m$ is not `Nearest`, then $|\varepsilon| < 2^{\lfloor\log_2
    ///   |\arctan(x)u/(2\pi)|\rfloor-p+1}$.
    /// - Otherwise, if $m$ is `Nearest`, then $|\varepsilon| \leq 2^{\lfloor\log_2
    ///   |\arctan(x)u/(2\pi)|\rfloor-p}$.
    ///
    /// The output has precision `prec`.
    ///
    /// Special cases:
    /// - $f(0,u,p,m)=0.0$
    /// - $f(x,0,p,m)=\pm0.0$, with the sign of $x$, so that the function stays odd
    /// - $f(\pm1,u,p,m)=\pm u/8$, an eighth of a turn
    ///
    /// These are the only exact cases, and the eighth turn is exact only when $p$ is large enough
    /// to hold it.
    ///
    /// Underflow:
    /// - If $0<f(x,u,p,m)<2^{-2^{30}}$, and $m$ is `Floor` or `Down`, $0.0$ is returned instead.
    /// - If $0<f(x,u,p,m)<2^{-2^{30}}$, and $m$ is `Ceiling` or `Up`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - If $0<f(x,u,p,m)\leq2^{-2^{30}-1}$, and $m$ is `Nearest`, $0.0$ is returned instead.
    /// - If $2^{-2^{30}-1}<f(x,u,p,m)<2^{-2^{30}}$, and $m$ is `Nearest`, $2^{-2^{30}}$ is returned
    ///   instead.
    /// - The negative cases mirror these, since the function is odd.
    ///
    /// Overflow is not possible, since $|f(x,u,p,m)| < u/4 < 2^{62}$. Underflow requires a tiny $x$
    /// together with a small $u$, since the result is about $xu/(2\pi)$ there. Unlike the [`Float`]
    /// case, $x$ itself may be far below the bottom of the exponent range.
    ///
    /// If you know you'll be using `Nearest`, consider using
    /// [`Float::atan_with_period_rational_prec`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `x.significant_bits()`: the arctangent is taken at a working precision of about $n$ bits and
    /// scaled by $u/(2\pi)$, which needs $\pi$ to that many bits, and both cost the first term; the
    /// second covers the $m$-bit input. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision (which is the case unless $x$ is zero, $u$ is zero, or $|x|$ is 1
    /// and $p$ is large enough to hold an eighth of a turn).
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::One;
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::atan_with_period_rational_prec_round(Rational::ONE, 360, 10, Exact);
    /// assert_eq!(t.to_string(), "45.000");
    /// assert_eq!(o, Equal);
    ///
    /// let (t, o) = Float::atan_with_period_rational_prec_round(
    ///     Rational::from_unsigneds(3u8, 5),
    ///     360,
    ///     10,
    ///     Floor,
    /// );
    /// assert_eq!(t.to_string(), "30.938");
    /// assert_eq!(o, Less);
    ///
    /// let (t, o) = Float::atan_with_period_rational_prec_round(
    ///     Rational::from_unsigneds(3u8, 5),
    ///     360,
    ///     10,
    ///     Ceiling,
    /// );
    /// assert_eq!(t.to_string(), "30.969");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    #[allow(clippy::needless_pass_by_value)]
    pub fn atan_with_period_rational_prec_round(
        x: Rational,
        u: u64,
        prec: u64,
        rm: RoundingMode,
    ) -> (Self, Ordering) {
        Self::atan_with_period_rational_prec_round_ref(&x, u, prec, rm)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Rational`] measured in $u$ths of a
    /// turn, rounding the result to the specified precision and with the specified rounding mode
    /// and returning the result as a [`Float`]. The [`Rational`] is taken by reference. An
    /// [`Ordering`] is also returned, indicating whether the rounded arctangent is less than, equal
    /// to, or greater than the exact arctangent.
    ///
    /// See [`Float::atan_with_period_rational_prec_round`] for the error bounds, the special cases,
    /// underflow, and the complexity; this function behaves the same way.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::One;
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) =
    ///     Float::atan_with_period_rational_prec_round_ref(&Rational::ONE, 360, 10, Exact);
    /// assert_eq!(t.to_string(), "45.000");
    /// assert_eq!(o, Equal);
    ///
    /// let (t, o) = Float::atan_with_period_rational_prec_round_ref(
    ///     &Rational::from_unsigneds(3u8, 5),
    ///     360,
    ///     10,
    ///     Floor,
    /// );
    /// assert_eq!(t.to_string(), "30.938");
    /// assert_eq!(o, Less);
    /// ```
    pub fn atan_with_period_rational_prec_round_ref(
        x: &Rational,
        u: u64,
        prec: u64,
        rm: RoundingMode,
    ) -> (Self, Ordering) {
        assert_ne!(prec, 0);
        // atanu(0, u) = 0, and atanu(x, 0) = 0 with the sign of x, so that the function stays odd;
        // a `Rational` zero has no sign, so the first case gives a positive zero
        if *x == 0u32 || u == 0 {
            return (
                if *x < 0u32 {
                    Self::NEGATIVE_ZERO
                } else {
                    Self::ZERO
                },
                Equal,
            );
        }
        atan_with_period_rational_helper(x, u, prec, rm)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Rational`] measured in $u$ths of a
    /// turn, rounding the result to the nearest value of the specified precision and returning the
    /// result as a [`Float`]. The [`Rational`] is taken by value. An [`Ordering`] is also returned,
    /// indicating whether the rounded arctangent is less than, equal to, or greater than the exact
    /// arctangent.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the specified precision, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// $$
    /// f(x,u,p) = \arctan(x)u/(2\pi)+\varepsilon,
    /// $$
    /// where $|\varepsilon| \leq 2^{\lfloor\log_2 |\arctan(x)u/(2\pi)|\rfloor-p}$ (unless the
    /// result underflows, or is one of the exact cases below).
    ///
    /// The output has precision `prec`.
    ///
    /// Special cases:
    /// - $f(0,u,p)=0.0$
    /// - $f(x,0,p)=\pm0.0$, with the sign of $x$, so that the function stays odd
    /// - $f(\pm1,u,p)=\pm u/8$, an eighth of a turn
    ///
    /// See the [`Float::atan_with_period_rational_prec_round`] documentation for information on
    /// overflow and underflow.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_with_period_rational_prec_round`] instead.
    ///
    /// # Worst-case complexity
    /// $T(n, m) = O(n (\log n)^3 \log\log n + (n+m) (\log (n+m))^2 \log\log (n+m))$
    ///
    /// $M(n, m) = O((n+m) \log (n+m))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `prec`, and $m$ is
    /// `x.significant_bits()`: the arctangent is taken at a working precision of about $n$ bits and
    /// scaled by $u/(2\pi)$, which needs $\pi$ to that many bits, and both cost the first term; the
    /// second covers the $m$-bit input. The magnitude of the input does not drive the cost.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) =
    ///     Float::atan_with_period_rational_prec(Rational::from_unsigneds(3u8, 5), 360, 10);
    /// assert_eq!(t.to_string(), "30.969");
    /// assert_eq!(o, Greater);
    ///
    /// let (t, o) =
    ///     Float::atan_with_period_rational_prec(Rational::from_unsigneds(3u8, 5), 360, 20);
    /// assert_eq!(t.to_string(), "30.963745");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    #[allow(clippy::needless_pass_by_value)]
    pub fn atan_with_period_rational_prec(x: Rational, u: u64, prec: u64) -> (Self, Ordering) {
        Self::atan_with_period_rational_prec_round_ref(&x, u, prec, Nearest)
    }

    /// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Rational`] measured in $u$ths of a
    /// turn, rounding the result to the nearest value of the specified precision and returning the
    /// result as a [`Float`]. The [`Rational`] is taken by reference. An [`Ordering`] is also
    /// returned, indicating whether the rounded arctangent is less than, equal to, or greater than
    /// the exact arctangent.
    ///
    /// See [`Float::atan_with_period_rational_prec`] for the error bounds, the special cases,
    /// underflow, and the complexity; this function behaves the same way.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) =
    ///     Float::atan_with_period_rational_prec_ref(&Rational::from_unsigneds(3u8, 5), 360, 10);
    /// assert_eq!(t.to_string(), "30.969");
    /// assert_eq!(o, Greater);
    ///
    /// let (t, o) =
    ///     Float::atan_with_period_rational_prec_ref(&Rational::from_unsigneds(3u8, 5), 360, 20);
    /// assert_eq!(t.to_string(), "30.963745");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_with_period_rational_prec_ref(x: &Rational, u: u64, prec: u64) -> (Self, Ordering) {
        Self::atan_with_period_rational_prec_round_ref(x, u, prec, Nearest)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the specified precision and with the specified rounding mode. The [`Float`] is
    /// taken by value. An [`Ordering`] is also returned, indicating whether the rounded arctangent
    /// is less than, equal to, or greater than the exact arctangent. Although `NaN`s are not
    /// comparable to any [`Float`], whenever this function returns a `NaN` it also returns `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period_prec_round`]
    /// for the error bounds, the special cases, underflow, and the complexity, with $u = 2$. An
    /// infinite input gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every
    /// precision, since a half and a quarter need only one bit; a zero input gives $\pm0.0$. Those
    /// are the only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::One;
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::from(0.1f64).atan_pi_prec_round(10, Floor);
    /// assert_eq!(t.to_string(), "0.031677");
    /// assert_eq!(o, Less);
    ///
    /// let (t, o) = Float::from(0.1f64).atan_pi_prec_round(10, Ceiling);
    /// assert_eq!(t.to_string(), "0.031738");
    /// assert_eq!(o, Greater);
    ///
    /// // an input of 1 gives an eighth of a turn, which is a quarter of a half-turn, exactly
    /// let (t, o) = Float::ONE.atan_pi_prec_round(10, Exact);
    /// assert_eq!(t.to_string(), "0.25000");
    /// assert_eq!(o, Equal);
    /// ```
    #[inline]
    pub fn atan_pi_prec_round(self, prec: u64, rm: RoundingMode) -> (Self, Ordering) {
        self.atan_with_period_prec_round(2, prec, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the specified precision and with the specified rounding mode. The [`Float`] is
    /// taken by reference. An [`Ordering`] is also returned, indicating whether the rounded
    /// arctangent is less than, equal to, or greater than the exact arctangent. Although `NaN`s are
    /// not comparable to any [`Float`], whenever this function returns a `NaN` it also returns
    /// `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see
    /// [`Float::atan_with_period_prec_round_ref`] for the error bounds, the special cases,
    /// underflow, and the complexity, with $u = 2$. An infinite input gives $\pm1/2$ and an input
    /// of $\pm1$ gives $\pm1/4$, both exact at every precision, since a half and a quarter need
    /// only one bit; a zero input gives $\pm0.0$. Those are the only exact cases. Overflow is not
    /// possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::One;
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = (Float::from(0.1f64)).atan_pi_prec_round_ref(10, Floor);
    /// assert_eq!(t.to_string(), "0.031677");
    /// assert_eq!(o, Less);
    ///
    /// let (t, o) = (Float::from(0.1f64)).atan_pi_prec_round_ref(10, Ceiling);
    /// assert_eq!(t.to_string(), "0.031738");
    /// assert_eq!(o, Greater);
    ///
    /// // an input of 1 gives an eighth of a turn, which is a quarter of a half-turn, exactly
    /// let (t, o) = (&Float::ONE).atan_pi_prec_round_ref(10, Exact);
    /// assert_eq!(t.to_string(), "0.25000");
    /// assert_eq!(o, Equal);
    /// ```
    #[inline]
    pub fn atan_pi_prec_round_ref(&self, prec: u64, rm: RoundingMode) -> (Self, Ordering) {
        self.atan_with_period_prec_round_ref(2, prec, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the nearest value of the specified precision. The [`Float`] is taken by value.
    /// An [`Ordering`] is also returned, indicating whether the rounded arctangent is less than,
    /// equal to, or greater than the exact arctangent. Although `NaN`s are not comparable to any
    /// [`Float`], whenever this function returns a `NaN` it also returns `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period_prec`] for the
    /// error bounds, the special cases, underflow, and the complexity, with $u = 2$. An infinite
    /// input gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every precision,
    /// since a half and a quarter need only one bit; a zero input gives $\pm0.0$. Those are the
    /// only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::from(0.1f64).atan_pi_prec(10);
    /// assert_eq!(t.to_string(), "0.031738");
    /// assert_eq!(o, Greater);
    ///
    /// let (t, o) = Float::from(0.1f64).atan_pi_prec(53);
    /// assert_eq!(t.to_string(), "0.031725517430553574");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_pi_prec(self, prec: u64) -> (Self, Ordering) {
        self.atan_with_period_prec(2, prec)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the nearest value of the specified precision. The [`Float`] is taken by
    /// reference. An [`Ordering`] is also returned, indicating whether the rounded arctangent is
    /// less than, equal to, or greater than the exact arctangent. Although `NaN`s are not
    /// comparable to any [`Float`], whenever this function returns a `NaN` it also returns `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period_prec_ref`] for
    /// the error bounds, the special cases, underflow, and the complexity, with $u = 2$. An
    /// infinite input gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every
    /// precision, since a half and a quarter need only one bit; a zero input gives $\pm0.0$. Those
    /// are the only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = (Float::from(0.1f64)).atan_pi_prec_ref(10);
    /// assert_eq!(t.to_string(), "0.031738");
    /// assert_eq!(o, Greater);
    ///
    /// let (t, o) = (Float::from(0.1f64)).atan_pi_prec_ref(53);
    /// assert_eq!(t.to_string(), "0.031725517430553574");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_pi_prec_ref(&self, prec: u64) -> (Self, Ordering) {
        self.atan_with_period_prec_ref(2, prec)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result with the specified rounding mode. The precision of the output is the precision of
    /// the input. The [`Float`] is taken by value. An [`Ordering`] is also returned, indicating
    /// whether the rounded arctangent is less than, equal to, or greater than the exact arctangent.
    /// Although `NaN`s are not comparable to any [`Float`], whenever this function returns a `NaN`
    /// it also returns `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period_round`] for the
    /// error bounds, the special cases, underflow, and the complexity, with $u = 2$. An infinite
    /// input gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every precision,
    /// since a half and a quarter need only one bit; a zero input gives $\pm0.0$. Those are the
    /// only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` but the result cannot be represented exactly with the input
    /// precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::from(0.1f64).atan_pi_round(Floor);
    /// assert_eq!(t.to_string(), "0.031725517430553560");
    /// assert_eq!(o, Less);
    ///
    /// let (t, o) = Float::from(0.1f64).atan_pi_round(Nearest);
    /// assert_eq!(t.to_string(), "0.031725517430553574");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_pi_round(self, rm: RoundingMode) -> (Self, Ordering) {
        self.atan_with_period_round(2, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result with the specified rounding mode. The precision of the output is the precision of
    /// the input. The [`Float`] is taken by reference. An [`Ordering`] is also returned, indicating
    /// whether the rounded arctangent is less than, equal to, or greater than the exact arctangent.
    /// Although `NaN`s are not comparable to any [`Float`], whenever this function returns a `NaN`
    /// it also returns `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period_round_ref`] for
    /// the error bounds, the special cases, underflow, and the complexity, with $u = 2$. An
    /// infinite input gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every
    /// precision, since a half and a quarter need only one bit; a zero input gives $\pm0.0$. Those
    /// are the only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` but the result cannot be represented exactly with the input
    /// precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = (Float::from(0.1f64)).atan_pi_round_ref(Floor);
    /// assert_eq!(t.to_string(), "0.031725517430553560");
    /// assert_eq!(o, Less);
    ///
    /// let (t, o) = (Float::from(0.1f64)).atan_pi_round_ref(Nearest);
    /// assert_eq!(t.to_string(), "0.031725517430553574");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_pi_round_ref(&self, rm: RoundingMode) -> (Self, Ordering) {
        self.atan_with_period_round_ref(2, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the precision of the input and to the nearest [`Float`]. The [`Float`] is
    /// taken by value.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the precision of the input, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period`] for the error
    /// bounds, the special cases, underflow, and the complexity, with $u = 2$. An infinite input
    /// gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every precision, since a
    /// half and a quarter need only one bit; a zero input gives $\pm0.0$. Those are the only exact
    /// cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_pi_round`] instead. If you want to specify an output precision, consider using
    /// [`Float::atan_pi_prec`]. If you want both of these things, consider using
    /// [`Float::atan_pi_prec_round`].
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    ///
    /// let t = Float::from(0.1f64).atan_pi();
    /// assert_eq!(t.to_string(), "0.031725517430553574");
    ///
    /// assert_eq!(Float::from(0.5f64).atan_pi().to_string(), "0.12");
    /// ```
    #[inline]
    pub fn atan_pi(self) -> Self {
        let prec = self.significant_bits();
        self.atan_pi_prec(prec).0
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the precision of the input and to the nearest [`Float`]. The [`Float`] is
    /// taken by reference.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the precision of the input, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period`] for the error
    /// bounds, the special cases, underflow, and the complexity, with $u = 2$. An infinite input
    /// gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every precision, since a
    /// half and a quarter need only one bit; a zero input gives $\pm0.0$. Those are the only exact
    /// cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_pi_round_ref`] instead. If you want to specify an output precision, consider
    /// using [`Float::atan_pi_prec_ref`]. If you want both of these things, consider using
    /// [`Float::atan_pi_prec_round_ref`].
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    ///
    /// let t = (&Float::from(0.1f64)).atan_pi_ref();
    /// assert_eq!(t.to_string(), "0.031725517430553574");
    /// ```
    #[inline]
    pub fn atan_pi_ref(&self) -> Self {
        self.atan_pi_prec_ref(self.significant_bits()).0
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the specified precision and with the specified rounding mode. The [`Float`] is
    /// replaced by the result, and an [`Ordering`] is returned, indicating whether the rounded
    /// arctangent is less than, equal to, or greater than the exact arctangent. Although `NaN`s are
    /// not comparable to any [`Float`], whenever this function sets a `NaN` it also returns
    /// `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see
    /// [`Float::atan_with_period_prec_round_assign`] for the error bounds, the special cases,
    /// underflow, and the complexity, with $u = 2$. An infinite input gives $\pm1/2$ and an input
    /// of $\pm1$ gives $\pm1/4$, both exact at every precision, since a half and a quarter need
    /// only one bit; a zero input gives $\pm0.0$. Those are the only exact cases. Overflow is not
    /// possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::from(0.1f64);
    /// assert_eq!(x.atan_pi_prec_round_assign(10, Floor), Less);
    /// assert_eq!(x.to_string(), "0.031677");
    ///
    /// let mut x = Float::from(0.1f64);
    /// assert_eq!(x.atan_pi_prec_round_assign(10, Ceiling), Greater);
    /// assert_eq!(x.to_string(), "0.031738");
    /// ```
    #[inline]
    pub fn atan_pi_prec_round_assign(&mut self, prec: u64, rm: RoundingMode) -> Ordering {
        self.atan_with_period_prec_round_assign(2, prec, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the nearest value of the specified precision. The [`Float`] is replaced by the
    /// result, and an [`Ordering`] is returned, indicating whether the rounded arctangent is less
    /// than, equal to, or greater than the exact arctangent. Although `NaN`s are not comparable to
    /// any [`Float`], whenever this function sets a `NaN` it also returns `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period_prec_assign`]
    /// for the error bounds, the special cases, underflow, and the complexity, with $u = 2$. An
    /// infinite input gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every
    /// precision, since a half and a quarter need only one bit; a zero input gives $\pm0.0$. Those
    /// are the only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::from(0.1f64);
    /// assert_eq!(x.atan_pi_prec_assign(10), Greater);
    /// assert_eq!(x.to_string(), "0.031738");
    /// ```
    #[inline]
    pub fn atan_pi_prec_assign(&mut self, prec: u64) -> Ordering {
        self.atan_with_period_prec_assign(2, prec)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result with the specified rounding mode. The precision of the output is the precision of
    /// the input. The [`Float`] is replaced by the result, and an [`Ordering`] is returned,
    /// indicating whether the rounded arctangent is less than, equal to, or greater than the exact
    /// arctangent. Although `NaN`s are not comparable to any [`Float`], whenever this function sets
    /// a `NaN` it also returns `Equal`.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period_round_assign`]
    /// for the error bounds, the special cases, underflow, and the complexity, with $u = 2$. An
    /// infinite input gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every
    /// precision, since a half and a quarter need only one bit; a zero input gives $\pm0.0$. Those
    /// are the only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `rm` is `Exact` but the result cannot be represented exactly with the input
    /// precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use std::cmp::Ordering::*;
    ///
    /// let mut x = Float::from(0.1f64);
    /// assert_eq!(x.atan_pi_round_assign(Floor), Less);
    /// assert_eq!(x.to_string(), "0.031725517430553560");
    /// ```
    #[inline]
    pub fn atan_pi_round_assign(&mut self, rm: RoundingMode) -> Ordering {
        self.atan_with_period_round_assign(2, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Float`] measured in half-turns, rounding
    /// the result to the precision of the input and to the nearest [`Float`]. The [`Float`] is
    /// replaced by the result.
    ///
    /// If the arctangent is equidistant from two [`Float`]s with the precision of the input, the
    /// [`Float`] with fewer 1s in its binary expansion is chosen. See [`RoundingMode`] for a
    /// description of the `Nearest` rounding mode.
    ///
    /// This is `atan_with_period` with a period of 2: see [`Float::atan_with_period`] for the error
    /// bounds, the special cases, underflow, and the complexity, with $u = 2$. An infinite input
    /// gives $\pm1/2$ and an input of $\pm1$ gives $\pm1/4$, both exact at every precision, since a
    /// half and a quarter need only one bit; a zero input gives $\pm0.0$. Those are the only exact
    /// cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_pi_round_assign`] instead. If you want to specify an output precision,
    /// consider using [`Float::atan_pi_prec_assign`]. If you want both of these things, consider
    /// using [`Float::atan_pi_prec_round_assign`].
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    ///
    /// let mut x = Float::from(0.1f64);
    /// x.atan_pi_assign();
    /// assert_eq!(x.to_string(), "0.031725517430553574");
    /// ```
    #[inline]
    pub fn atan_pi_assign(&mut self) {
        let prec = self.significant_bits();
        self.atan_pi_prec_assign(prec);
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Rational`] measured in half-turns, rounding
    /// the result to the specified precision and with the specified rounding mode and returning the
    /// result as a [`Float`]. The [`Rational`] is taken by value. An [`Ordering`] is also returned,
    /// indicating whether the rounded arctangent is less than, equal to, or greater than the exact
    /// arctangent.
    ///
    /// This is `atan_with_period_rational` with a period of 2: see
    /// [`Float::atan_with_period_rational_prec_round`] for the error bounds, the special cases,
    /// underflow, and the complexity, with $u = 2$. An input of $\pm1$ gives $\pm1/4$ and a zero
    /// input gives $0.0$, both exact at every precision; those are the only exact cases. Overflow
    /// is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::basic::traits::One;
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) =
    ///     Float::atan_pi_rational_prec_round(Rational::from_unsigneds(1u8, 7), 10, Floor);
    /// assert_eq!(t.to_string(), "0.045166");
    /// assert_eq!(o, Less);
    ///
    /// // an input of 1 gives a quarter of a half-turn, exactly
    /// let (t, o) = Float::atan_pi_rational_prec_round(Rational::ONE, 10, Exact);
    /// assert_eq!(t.to_string(), "0.25000");
    /// assert_eq!(o, Equal);
    /// ```
    #[inline]
    #[allow(clippy::needless_pass_by_value)]
    pub fn atan_pi_rational_prec_round(
        x: Rational,
        prec: u64,
        rm: RoundingMode,
    ) -> (Self, Ordering) {
        Self::atan_with_period_rational_prec_round_ref(&x, 2, prec, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Rational`] measured in half-turns, rounding
    /// the result to the specified precision and with the specified rounding mode and returning the
    /// result as a [`Float`]. The [`Rational`] is taken by reference. An [`Ordering`] is also
    /// returned, indicating whether the rounded arctangent is less than, equal to, or greater than
    /// the exact arctangent.
    ///
    /// This is `atan_with_period_rational` with a period of 2: see
    /// [`Float::atan_with_period_rational_prec_round_ref`] for the error bounds, the special cases,
    /// underflow, and the complexity, with $u = 2$. An input of $\pm1$ gives $\pm1/4$ and a zero
    /// input gives $0.0$, both exact at every precision; those are the only exact cases. Overflow
    /// is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero, or if `rm` is `Exact` but the result cannot be represented exactly
    /// with the given precision.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::rounding_modes::RoundingMode::*;
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) =
    ///     Float::atan_pi_rational_prec_round_ref(&Rational::from_unsigneds(1u8, 7), 10, Ceiling);
    /// assert_eq!(t.to_string(), "0.045227");
    /// assert_eq!(o, Greater);
    /// ```
    #[inline]
    pub fn atan_pi_rational_prec_round_ref(
        x: &Rational,
        prec: u64,
        rm: RoundingMode,
    ) -> (Self, Ordering) {
        Self::atan_with_period_rational_prec_round_ref(x, 2, prec, rm)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Rational`] measured in half-turns, rounding
    /// the result to the nearest value of the specified precision and returning the result as a
    /// [`Float`]. The [`Rational`] is taken by value. An [`Ordering`] is also returned, indicating
    /// whether the rounded arctangent is less than, equal to, or greater than the exact arctangent.
    ///
    /// This is `atan_with_period_rational` with a period of 2: see
    /// [`Float::atan_with_period_rational_prec`] for the error bounds, the special cases,
    /// underflow, and the complexity, with $u = 2$. An input of $\pm1$ gives $\pm1/4$ and a zero
    /// input gives $0.0$, both exact at every precision; those are the only exact cases. Overflow
    /// is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::atan_pi_rational_prec(Rational::from_unsigneds(1u8, 7), 53);
    /// assert_eq!(t.to_string(), "0.045167235300866547");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    #[allow(clippy::needless_pass_by_value)]
    pub fn atan_pi_rational_prec(x: Rational, prec: u64) -> (Self, Ordering) {
        Self::atan_with_period_rational_prec_ref(&x, 2, prec)
    }

    /// Computes $\arctan(x)/\pi$, the arctangent of a [`Rational`] measured in half-turns, rounding
    /// the result to the nearest value of the specified precision and returning the result as a
    /// [`Float`]. The [`Rational`] is taken by reference. An [`Ordering`] is also returned,
    /// indicating whether the rounded arctangent is less than, equal to, or greater than the exact
    /// arctangent.
    ///
    /// This is `atan_with_period_rational` with a period of 2: see
    /// [`Float::atan_with_period_rational_prec_ref`] for the error bounds, the special cases,
    /// underflow, and the complexity, with $u = 2$. An input of $\pm1$ gives $\pm1/4$ and a zero
    /// input gives $0.0$, both exact at every precision; those are the only exact cases. Overflow
    /// is not possible, since $|\arctan(x)/\pi| < 1/2$.
    ///
    /// # Panics
    /// Panics if `prec` is zero.
    ///
    /// # Examples
    /// ```
    /// use malachite_float::Float;
    /// use malachite_q::Rational;
    /// use std::cmp::Ordering::*;
    ///
    /// let (t, o) = Float::atan_pi_rational_prec_ref(&Rational::from_unsigneds(1u8, 7), 53);
    /// assert_eq!(t.to_string(), "0.045167235300866547");
    /// assert_eq!(o, Less);
    /// ```
    #[inline]
    pub fn atan_pi_rational_prec_ref(x: &Rational, prec: u64) -> (Self, Ordering) {
        Self::atan_with_period_rational_prec_ref(x, 2, prec)
    }
}

impl Atan for Float {
    type Output = Self;

    /// Computes $\arctan x$, the arctangent of a [`Float`], taking it by value.
    ///
    /// If the output has a precision, it is the precision of the input. If the arctangent is
    /// equidistant from two [`Float`]s with the specified precision, the [`Float`] with fewer 1s in
    /// its binary expansion is chosen. See [`RoundingMode`] for a description of the `Nearest`
    /// rounding mode.
    ///
    /// $$
    /// f(x) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$, where
    ///   $p$ is the precision of the input.
    ///
    /// Special cases:
    /// - $f(\text{NaN})=\text{NaN}$
    /// - $f(\pm\infty)=\pm\pi/2$, rounded
    /// - $f(\pm0.0)=\pm0.0$
    ///
    /// See the [`Float::atan_round`] documentation for information on overflow and underflow.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_round`] instead. If you want to specify the output precision, consider using
    /// [`Float::atan_prec`]. If you want both of these things, consider using
    /// [`Float::atan_prec_round`].
    ///
    /// # Worst-case complexity
    /// $T(n, e) = O(n (\log n)^3 \log\log n + (n+e) (\log (n+e))^2 \log\log (n+e))$
    ///
    /// $M(n, e) = O((n+e) \log (n+e))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `self.significant_bits()`, and $e$ is
    /// the exponent of `self` (0 if `self` has no exponent or a negative one): the Taylor series at
    /// working precision $n$, summed by binary splitting for large $n$, costs the first term, and
    /// for $|x| \geq 4$ the argument is reduced modulo $2\pi$, which requires $\pi$ to about $n +
    /// e$ bits. Unlike most functions, `atan` therefore gets slower as the magnitude of its input
    /// grows, not just as the precision does.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::arithmetic::traits::Atan;
    /// use malachite_base::num::basic::traits::*;
    /// use malachite_float::Float;
    ///
    /// assert!(Float::NAN.atan().is_nan());
    /// // an infinity has a precision of 1, so pi/2 rounds to 2
    /// assert_eq!(Float::INFINITY.atan().to_string(), "2.0");
    /// assert_eq!(Float::NEGATIVE_INFINITY.atan().to_string(), "-2.0");
    /// assert_eq!(Float::ZERO.atan().to_string(), "0.0");
    /// assert_eq!(Float::NEGATIVE_ZERO.atan().to_string(), "-0.0");
    /// assert_eq!(
    ///     Float::from_unsigned_prec(1u32, 100).0.atan().to_string(),
    ///     "0.78539816339744830961566084581983"
    /// );
    /// assert_eq!(
    ///     Float::from_unsigned_prec(100u32, 100).0.atan().to_string(),
    ///     "1.5607966601082313810249815754304"
    /// );
    /// ```
    #[inline]
    fn atan(self) -> Self {
        let prec = self.significant_bits();
        self.atan_prec_round(prec, Nearest).0
    }
}

impl Atan for &Float {
    type Output = Float;

    /// Computes $\arctan x$, the arctangent of a [`Float`], taking it by reference.
    ///
    /// If the output has a precision, it is the precision of the input. If the arctangent is
    /// equidistant from two [`Float`]s with the specified precision, the [`Float`] with fewer 1s in
    /// its binary expansion is chosen. See [`RoundingMode`] for a description of the `Nearest`
    /// rounding mode.
    ///
    /// $$
    /// f(x) = \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$, where
    ///   $p$ is the precision of the input.
    ///
    /// Special cases:
    /// - $f(\text{NaN})=\text{NaN}$
    /// - $f(\pm\infty)=\pm\pi/2$, rounded
    /// - $f(\pm0.0)=\pm0.0$
    ///
    /// See the [`Float::atan_round`] documentation for information on overflow and underflow.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_round_ref`] instead. If you want to specify the output precision, consider
    /// using [`Float::atan_prec_ref`]. If you want both of these things, consider using
    /// [`Float::atan_prec_round_ref`].
    ///
    /// # Worst-case complexity
    /// $T(n, e) = O(n (\log n)^3 \log\log n + (n+e) (\log (n+e))^2 \log\log (n+e))$
    ///
    /// $M(n, e) = O((n+e) \log (n+e))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `self.significant_bits()`, and $e$ is
    /// the exponent of `self` (0 if `self` has no exponent or a negative one): the Taylor series at
    /// working precision $n$, summed by binary splitting for large $n$, costs the first term, and
    /// for $|x| \geq 4$ the argument is reduced modulo $2\pi$, which requires $\pi$ to about $n +
    /// e$ bits. Unlike most functions, `atan` therefore gets slower as the magnitude of its input
    /// grows, not just as the precision does.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::arithmetic::traits::Atan;
    /// use malachite_base::num::basic::traits::*;
    /// use malachite_float::Float;
    ///
    /// assert!(Float::NAN.atan().is_nan());
    /// // an infinity has a precision of 1, so pi/2 rounds to 2
    /// assert_eq!(Float::INFINITY.atan().to_string(), "2.0");
    /// assert_eq!(Float::NEGATIVE_INFINITY.atan().to_string(), "-2.0");
    /// assert_eq!(Float::ZERO.atan().to_string(), "0.0");
    /// assert_eq!(Float::NEGATIVE_ZERO.atan().to_string(), "-0.0");
    /// assert_eq!(
    ///     (&Float::from_unsigned_prec(1u32, 100).0).atan().to_string(),
    ///     "0.78539816339744830961566084581983"
    /// );
    /// assert_eq!(
    ///     (&Float::from_unsigned_prec(100u32, 100).0)
    ///         .atan()
    ///         .to_string(),
    ///     "1.5607966601082313810249815754304"
    /// );
    /// ```
    #[inline]
    fn atan(self) -> Float {
        self.atan_prec_round_ref(self.significant_bits(), Nearest).0
    }
}

impl AtanAssign for Float {
    /// Computes $\arctan x$, the arctangent of a [`Float`], in place.
    ///
    /// If the output has a precision, it is the precision of the input. If the arctangent is
    /// equidistant from two [`Float`]s with the specified precision, the [`Float`] with fewer 1s in
    /// its binary expansion is chosen. See [`RoundingMode`] for a description of the `Nearest`
    /// rounding mode.
    ///
    /// $$
    /// x \gets \arctan x+\varepsilon.
    /// $$
    /// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
    /// - If $x$ is not NaN, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$, where
    ///   $p$ is the precision of the input.
    ///
    /// See the [`Float::atan`] documentation for information on special cases, overflow, and
    /// underflow.
    ///
    /// If you want to use a rounding mode other than `Nearest`, consider using
    /// [`Float::atan_round_assign`] instead. If you want to specify the output precision, consider
    /// using [`Float::atan_prec_assign`]. If you want both of these things, consider using
    /// [`Float::atan_prec_round_assign`].
    ///
    /// # Worst-case complexity
    /// $T(n, e) = O(n (\log n)^3 \log\log n + (n+e) (\log (n+e))^2 \log\log (n+e))$
    ///
    /// $M(n, e) = O((n+e) \log (n+e))$
    ///
    /// where $T$ is time, $M$ is additional memory, $n$ is `self.significant_bits()`, and $e$ is
    /// the exponent of `self` (0 if `self` has no exponent or a negative one): the Taylor series at
    /// working precision $n$, summed by binary splitting for large $n$, costs the first term, and
    /// for $|x| \geq 4$ the argument is reduced modulo $2\pi$, which requires $\pi$ to about $n +
    /// e$ bits. Unlike most functions, `atan` therefore gets slower as the magnitude of its input
    /// grows, not just as the precision does.
    ///
    /// # Examples
    /// ```
    /// use malachite_base::num::arithmetic::traits::AtanAssign;
    /// use malachite_base::num::basic::traits::*;
    /// use malachite_float::Float;
    ///
    /// let mut x = Float::NAN;
    /// x.atan_assign();
    /// assert!(x.is_nan());
    ///
    /// let mut x = Float::INFINITY;
    /// x.atan_assign();
    /// assert_eq!(x.to_string(), "2.0");
    ///
    /// let mut x = Float::NEGATIVE_INFINITY;
    /// x.atan_assign();
    /// assert_eq!(x.to_string(), "-2.0");
    ///
    /// let mut x = Float::ZERO;
    /// x.atan_assign();
    /// assert_eq!(x.to_string(), "0.0");
    ///
    /// let mut x = Float::NEGATIVE_ZERO;
    /// x.atan_assign();
    /// assert_eq!(x.to_string(), "-0.0");
    ///
    /// let mut x = Float::from_unsigned_prec(1u32, 100).0;
    /// x.atan_assign();
    /// assert_eq!(x.to_string(), "0.78539816339744830961566084581983");
    ///
    /// let mut x = Float::from_unsigned_prec(100u32, 100).0;
    /// x.atan_assign();
    /// assert_eq!(x.to_string(), "1.5607966601082313810249815754304");
    /// ```
    #[inline]
    fn atan_assign(&mut self) {
        let prec = self.significant_bits();
        self.atan_prec_round_assign(prec, Nearest);
    }
}

/// Computes $\arctan x$, the arctangent of a primitive float. Using this function is more accurate
/// than using the default `atan` function or the one provided by `libm`.
///
/// $$
/// f(x) = \arctan x+\varepsilon.
/// $$
/// - If $x$ is NaN, $\varepsilon$ may be ignored or assumed to be 0.
/// - If $x$ is not NaN, then $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$, where $p$ is
///   the precision of the output (24 if `T` is a [`f32`] and 53 if `T` is a [`f64`]).
///
/// Special cases:
/// - $f(\text{NaN})=\text{NaN}$
/// - $f(\pm\infty)=\pm\pi/2$, rounded
/// - $f(\pm0.0)=\pm0.0$
///
/// Overflow is not possible, since the result lies in $(-\pi/2, \pi/2)$. The result is subnormal
/// only when $x$ is, and then it is $x$ itself, since $|\arctan x - x| < |x|^3/3$.
///
/// # Worst-case complexity
/// Constant time and additional memory.
///
/// # Examples
/// ```
/// use malachite_base::num::basic::traits::NegativeInfinity;
/// use malachite_base::num::float::NiceFloat;
/// use malachite_float::float::arithmetic::atan::primitive_float_atan;
///
/// assert!(primitive_float_atan(f32::NAN).is_nan());
/// assert_eq!(
///     NiceFloat(primitive_float_atan(f32::INFINITY)),
///     NiceFloat(core::f32::consts::FRAC_PI_2)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan(f32::NEGATIVE_INFINITY)),
///     NiceFloat(-core::f32::consts::FRAC_PI_2)
/// );
/// assert_eq!(NiceFloat(primitive_float_atan(0.0f32)), NiceFloat(0.0));
/// assert_eq!(NiceFloat(primitive_float_atan(-0.0f32)), NiceFloat(-0.0));
/// assert_eq!(
///     NiceFloat(primitive_float_atan(1.0f32)),
///     NiceFloat(core::f32::consts::FRAC_PI_4)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan(1.0f64)),
///     NiceFloat(core::f64::consts::FRAC_PI_4)
/// );
/// ```
#[inline]
#[allow(clippy::type_repetition_in_bounds)]
pub fn primitive_float_atan<T: PrimitiveFloat>(x: T) -> T
where
    Float: From<T> + PartialOrd<T>,
    for<'a> T: ExactFrom<&'a Float> + RoundingFrom<&'a Float>,
{
    emulate_float_to_float_fn(Float::atan_prec, x)
}

/// Computes $\arctan x$, the arctangent of a [`Rational`], returning the result as a primitive
/// float.
///
/// $$
/// f(x) = \arctan x+\varepsilon,
/// $$
/// where $|\varepsilon| < 2^{\lfloor\log_2 |\arctan x|\rfloor-p}$, and $p$ is the precision of the
/// output (24 if `T` is a [`f32`] and 53 if `T` is a [`f64`]).
///
/// Special cases:
/// - $f(0)=0$
///
/// Overflow is not possible, since the result lies in $(-\pi/2, \pi/2)$. The result underflows, to
/// a subnormal or to zero, only when $x$ is tiny, since $|\arctan x| < |x|$ and $\arctan x$ is very
/// close to $x$ there.
///
/// # Worst-case complexity
/// $T(m) = O(m (\log m)^2 \log\log m)$
///
/// $M(m) = O(m \log m)$
///
/// where $T$ is time, $M$ is additional memory, and $m$ is `x.significant_bits()`.
///
/// # Examples
/// ```
/// use malachite_base::num::basic::traits::Zero;
/// use malachite_base::num::float::NiceFloat;
/// use malachite_float::float::arithmetic::atan::primitive_float_atan_rational;
/// use malachite_q::Rational;
///
/// assert_eq!(
///     NiceFloat(primitive_float_atan_rational::<f64>(&Rational::ZERO)),
///     NiceFloat(0.0)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_rational::<f64>(
///         &Rational::from_unsigneds(1u8, 3)
///     )),
///     NiceFloat(0.3217505543966422)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_rational::<f32>(
///         &Rational::from_unsigneds(1u8, 3)
///     )),
///     NiceFloat(0.32175055)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_rational::<f64>(&Rational::from(10000))),
///     NiceFloat(1.5706963267952299)
/// );
/// ```
#[inline]
#[allow(clippy::type_repetition_in_bounds)]
pub fn primitive_float_atan_rational<T: PrimitiveFloat>(x: &Rational) -> T
where
    Float: PartialOrd<T>,
    for<'a> T: ExactFrom<&'a Float> + RoundingFrom<&'a Float>,
{
    emulate_rational_to_float_fn(Float::atan_rational_prec_ref, x)
}

/// Computes $\arctan(x)u/(2\pi)$, the arctangent of a primitive float measured in $u$ths of a turn
/// (so that `u = 360` gives degrees).
///
/// $$
/// f(x,u) = \arctan(x)u/(2\pi)+\varepsilon.
/// $$
/// - If $x$ is NaN or zero, $u = 0$, or $|x|$ is 1 or infinite, $\varepsilon$ may be ignored or
///   assumed to be 0.
/// - Otherwise, $|\varepsilon| < 2^{\lfloor\log_2 |\arctan(x)u/(2\pi)|\rfloor-p}$, where $p$ is the
///   precision of the output (24 if `T` is a [`f32`] and 53 if `T` is a [`f64`]).
///
/// Special cases:
/// - $f(\text{NaN},u)=\text{NaN}$
/// - $f(\pm\infty,u)=\pm u/4$, a quarter turn
/// - $f(\pm0.0,u)=\pm0.0$
/// - $f(x,0)=\pm0.0$, with the sign of $x$, so that the function stays odd
/// - $f(\pm1,u)=\pm u/8$, an eighth of a turn
///
/// Overflow is not possible, since $|f(x,u)| < u/4 < 2^{62}$. The result is subnormal, or zero,
/// only when $x$ is tiny and $u$ is small, since the result is about $xu/(2\pi)$ there.
///
/// # Worst-case complexity
/// Constant time and additional memory.
///
/// # Examples
/// ```
/// use malachite_base::num::basic::traits::NegativeInfinity;
/// use malachite_base::num::float::NiceFloat;
/// use malachite_float::float::arithmetic::atan::primitive_float_atan_with_period;
///
/// assert!(primitive_float_atan_with_period(f32::NAN, 360).is_nan());
/// // an infinite input is a quarter turn
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period(f32::INFINITY, 360)),
///     NiceFloat(90.0)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period(
///         f32::NEGATIVE_INFINITY,
///         360
///     )),
///     NiceFloat(-90.0)
/// );
/// // an input of 1 is an eighth of a turn
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period(1.0f32, 360)),
///     NiceFloat(45.0)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period(2.0f32, 360)),
///     NiceFloat(63.434948)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period(2.0f64, 360)),
///     NiceFloat(63.43494882292201)
/// );
/// ```
#[inline]
#[allow(clippy::type_repetition_in_bounds)]
pub fn primitive_float_atan_with_period<T: PrimitiveFloat>(x: T, u: u64) -> T
where
    Float: From<T> + PartialOrd<T>,
    for<'a> T: ExactFrom<&'a Float> + RoundingFrom<&'a Float>,
{
    emulate_float_to_float_fn(|x, prec| Float::atan_with_period_prec(x, u, prec), x)
}

/// Computes $\arctan(x)u/(2\pi)$, the arctangent of a [`Rational`] measured in $u$ths of a turn (so
/// that `u = 360` gives degrees), returning the result as a primitive float.
///
/// $$
/// f(x,u) = \arctan(x)u/(2\pi)+\varepsilon.
/// $$
/// - If $x$ is zero, $u = 0$, or $|x|$ is 1, $\varepsilon$ may be ignored or assumed to be 0.
/// - Otherwise, $|\varepsilon| < 2^{\lfloor\log_2 |\arctan(x)u/(2\pi)|\rfloor-p}$, where $p$ is the
///   precision of the output (24 if `T` is a [`f32`] and 53 if `T` is a [`f64`]).
///
/// Special cases:
/// - $f(0,u)=0.0$
/// - $f(x,0)=\pm0.0$, with the sign of $x$, so that the function stays odd
/// - $f(\pm1,u)=\pm u/8$, an eighth of a turn
///
/// Overflow is not possible, since $|f(x,u)| < u/4 < 2^{62}$. The result is subnormal, or zero,
/// only when $x$ is tiny and $u$ is small, since the result is about $xu/(2\pi)$ there.
///
/// # Worst-case complexity
/// $T(m) = O(m \log m \log\log m)$
///
/// $M(m) = O(m \log m)$
///
/// where $T$ is time, $M$ is additional memory, and $m$ is `x.significant_bits()`.
///
/// # Examples
/// ```
/// use malachite_base::num::basic::traits::{One, Zero};
/// use malachite_base::num::float::NiceFloat;
/// use malachite_float::float::arithmetic::atan::primitive_float_atan_with_period_rational;
/// use malachite_q::Rational;
///
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period_rational::<f64>(
///         &Rational::ZERO,
///         360
///     )),
///     NiceFloat(0.0)
/// );
/// // an input of 1 is an eighth of a turn
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period_rational::<f64>(
///         &Rational::ONE,
///         360
///     )),
///     NiceFloat(45.0)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period_rational::<f64>(
///         &Rational::from_unsigneds(1u8, 3),
///         360
///     )),
///     NiceFloat(18.43494882292201)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_with_period_rational::<f32>(
///         &Rational::from_unsigneds(1u8, 3),
///         360
///     )),
///     NiceFloat(18.434948)
/// );
/// ```
#[inline]
#[allow(clippy::type_repetition_in_bounds)]
pub fn primitive_float_atan_with_period_rational<T: PrimitiveFloat>(x: &Rational, u: u64) -> T
where
    Float: PartialOrd<T>,
    for<'a> T: ExactFrom<&'a Float> + RoundingFrom<&'a Float>,
{
    emulate_rational_to_float_fn(
        |x, prec| Float::atan_with_period_rational_prec_ref(x, u, prec),
        x,
    )
}

/// Computes $\arctan(x)/\pi$, the arctangent of a primitive float measured in half-turns.
///
/// This is `primitive_float_atan_with_period` with a period of 2: see
/// [`primitive_float_atan_with_period`] for the error bound and the special cases, with $u = 2$. An
/// infinite input gives exactly $\pm1/2$, an input of $\pm1$ exactly $\pm1/4$, and a zero input
/// exactly $\pm0.0$; those are the only exact cases. Overflow is not possible, since
/// $|\arctan(x)/\pi| < 1/2$, and the result is subnormal, or zero, only for a subnormal input.
///
/// # Worst-case complexity
/// Constant time and additional memory.
///
/// # Examples
/// ```
/// use malachite_base::num::basic::traits::NegativeInfinity;
/// use malachite_base::num::float::NiceFloat;
/// use malachite_float::float::arithmetic::atan::primitive_float_atan_pi;
///
/// assert!(primitive_float_atan_pi(f32::NAN).is_nan());
/// // an infinite input is half a turn
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi(f32::INFINITY)),
///     NiceFloat(0.5)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi(f32::NEGATIVE_INFINITY)),
///     NiceFloat(-0.5)
/// );
/// // an input of 1 is a quarter of a half-turn
/// assert_eq!(NiceFloat(primitive_float_atan_pi(1.0f32)), NiceFloat(0.25));
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi(0.1f32)),
///     NiceFloat(0.03172552)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi(0.1f64)),
///     NiceFloat(0.031725517430553574)
/// );
/// ```
#[inline]
#[allow(clippy::type_repetition_in_bounds)]
pub fn primitive_float_atan_pi<T: PrimitiveFloat>(x: T) -> T
where
    Float: From<T> + PartialOrd<T>,
    for<'a> T: ExactFrom<&'a Float> + RoundingFrom<&'a Float>,
{
    primitive_float_atan_with_period(x, 2)
}

/// Computes $\arctan(x)/\pi$, the arctangent of a [`Rational`] measured in half-turns, returning
/// the result as a primitive float.
///
/// This is `primitive_float_atan_with_period_rational` with a period of 2: see
/// [`primitive_float_atan_with_period_rational`] for the error bound and the special cases, with $u
/// = 2$. An input of $\pm1$ gives exactly $\pm1/4$ and a zero input exactly $0.0$; those are the
/// only exact cases. Overflow is not possible, since $|\arctan(x)/\pi| < 1/2$.
///
/// # Worst-case complexity
/// $T(m) = O(m \log m \log\log m)$
///
/// $M(m) = O(m \log m)$
///
/// where $T$ is time, $M$ is additional memory, and $m$ is `x.significant_bits()`.
///
/// # Examples
/// ```
/// use malachite_base::num::basic::traits::{One, Zero};
/// use malachite_base::num::float::NiceFloat;
/// use malachite_float::float::arithmetic::atan::primitive_float_atan_pi_rational;
/// use malachite_q::Rational;
///
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi_rational::<f64>(&Rational::ZERO)),
///     NiceFloat(0.0)
/// );
/// // an input of 1 is a quarter of a half-turn
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi_rational::<f64>(&Rational::ONE)),
///     NiceFloat(0.25)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi_rational::<f64>(
///         &Rational::from_unsigneds(1u8, 3)
///     )),
///     NiceFloat(0.10241638234956672)
/// );
/// assert_eq!(
///     NiceFloat(primitive_float_atan_pi_rational::<f32>(
///         &Rational::from_unsigneds(1u8, 3)
///     )),
///     NiceFloat(0.10241638)
/// );
/// ```
#[inline]
#[allow(clippy::type_repetition_in_bounds)]
pub fn primitive_float_atan_pi_rational<T: PrimitiveFloat>(x: &Rational) -> T
where
    Float: PartialOrd<T>,
    for<'a> T: ExactFrom<&'a Float> + RoundingFrom<&'a Float>,
{
    primitive_float_atan_with_period_rational(x, 2)
}