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/*
* // Copyright (c) Radzivon Bartoshyk 6/2025. All rights reserved.
* //
* // Redistribution and use in source and binary forms, with or without modification,
* // are permitted provided that the following conditions are met:
* //
* // 1. Redistributions of source code must retain the above copyright notice, this
* // list of conditions and the following disclaimer.
* //
* // 2. Redistributions in binary form must reproduce the above copyright notice,
* // this list of conditions and the following disclaimer in the documentation
* // and/or other materials provided with the distribution.
* //
* // 3. Neither the name of the copyright holder nor the names of its
* // contributors may be used to endorse or promote products derived from
* // this software without specific prior written permission.
* //
* // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
use crate::bits::EXP_MASK;
use crate::common::f_fmla;
use crate::dekker::Dekker;
use crate::dyadic_float::{DyadicFloat128, DyadicSign};
use crate::sin::{LargeArgumentReduction, get_sin_k_rational, range_reduction_small};
use crate::sincos_dyadic::{r_polyeval9, range_reduction_small_f128};
#[inline]
fn tan_eval(u: Dekker) -> (Dekker, f64) {
// Evaluate tan(y) = tan(x - k * (pi/128))
// We use the degree-9 Taylor approximation:
// tan(y) ~ P(y) = y + y^3/3 + 2*y^5/15 + 17*y^7/315 + 62*y^9/2835
// Then the error is bounded by:
// |tan(y) - P(y)| < 2^-6 * |y|^11 < 2^-6 * 2^-66 = 2^-72.
// For y ~ u_hi + u_lo, fully expanding the polynomial and drop any terms
// < ulp(u_hi^3) gives us:
// P(y) = y + y^3/3 + 2*y^5/15 + 17*y^7/315 + 62*y^9/2835 = ...
// ~ u_hi + u_hi^3 * (1/3 + u_hi^2 * (2/15 + u_hi^2 * (17/315 +
// + u_hi^2 * 62/2835))) +
// + u_lo (1 + u_hi^2 * (1 + u_hi^2 * 2/3))
let u_hi_sq = u.hi * u.hi; // Error < ulp(u_hi^2) < 2^(-6 - 52) = 2^-58.
// p1 ~ 17/315 + u_hi^2 62 / 2835.
let p1 = f_fmla(
u_hi_sq,
f64::from_bits(0x3f9664f4882c10fa),
f64::from_bits(0x3faba1ba1ba1ba1c),
);
// p2 ~ 1/3 + u_hi^2 2 / 15.
let p2 = f_fmla(
u_hi_sq,
f64::from_bits(0x3fc1111111111111),
f64::from_bits(0x3fd5555555555555),
);
// q1 ~ 1 + u_hi^2 * 2/3.
let q1 = f_fmla(u_hi_sq, f64::from_bits(0x3fe5555555555555), 1.0);
let u_hi_3 = u_hi_sq * u.hi;
let u_hi_4 = u_hi_sq * u_hi_sq;
// p3 ~ 1/3 + u_hi^2 * (2/15 + u_hi^2 * (17/315 + u_hi^2 * 62/2835))
let p3 = f_fmla(u_hi_4, p1, p2);
// q2 ~ 1 + u_hi^2 * (1 + u_hi^2 * 2/3)
let q2 = f_fmla(u_hi_sq, q1, 1.0);
let tan_lo = f_fmla(u_hi_3, p3, u.lo * q2);
// Overall, |tan(y) - (u_hi + tan_lo)| < ulp(u_hi^3) <= 2^-71.
// And the relative errors is:
// |(tan(y) - (u_hi + tan_lo)) / tan(y) | <= 2*ulp(u_hi^2) < 2^-64
let err = f_fmla(
u_hi_3.abs(),
f64::from_bits(0x3cc0000000000000),
f64::from_bits(0x3990000000000000),
);
(Dekker::from_exact_add(u.hi, tan_lo), err)
}
#[inline]
fn tan_eval_rational(u: &DyadicFloat128) -> DyadicFloat128 {
let u_sq = u.quick_mul(u);
// tan(x) ~ x + x^3/3 + x^5 * 2/15 + x^7 * 17/315 + x^9 * 62/2835 +
// + x^11 * 1382/155925 + x^13 * 21844/6081075 +
// + x^15 * 929569/638512875 + x^17 * 6404582/10854718875
// Relative errors < 2^-127 for |u| < pi/256.
const TAN_COEFFS: [DyadicFloat128; 9] = [
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -127,
mantissa: 0x80000000_00000000_00000000_00000000_u128,
}, // 1
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -129,
mantissa: 0xaaaaaaaa_aaaaaaaa_aaaaaaaa_aaaaaaab_u128,
}, // 1
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -130,
mantissa: 0x88888888_88888888_88888888_88888889_u128,
}, // 2/15
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -132,
mantissa: 0xdd0dd0dd_0dd0dd0d_d0dd0dd0_dd0dd0dd_u128,
}, // 17/315
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -133,
mantissa: 0xb327a441_6087cf99_6b5dd24e_ec0b327a_u128,
}, // 62/2835
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -134,
mantissa: 0x91371aaf_3611e47a_da8e1cba_7d900eca_u128,
}, // 1382/155925
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -136,
mantissa: 0xeb69e870_abeefdaf_e606d2e4_d1e65fbc_u128,
}, // 21844/6081075
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -137,
mantissa: 0xbed1b229_5baf15b5_0ec9af45_a2619971_u128,
}, // 929569/638512875
DyadicFloat128 {
sign: DyadicSign::Pos,
exponent: -138,
mantissa: 0x9aac1240_1b3a2291_1b2ac7e3_e4627d0a_u128,
}, // 6404582/10854718875
];
u.quick_mul(&r_polyeval9(
&u_sq,
&TAN_COEFFS[0],
&TAN_COEFFS[1],
&TAN_COEFFS[2],
&TAN_COEFFS[3],
&TAN_COEFFS[4],
&TAN_COEFFS[5],
&TAN_COEFFS[6],
&TAN_COEFFS[7],
&TAN_COEFFS[8],
))
}
// Calculation a / b = a * (1/b) for Float128.
// Using the initial approximation of q ~ (1/b), then apply 2 Newton-Raphson
// iterations, before multiplying by a.
#[inline]
fn newton_raphson_div(a: &DyadicFloat128, b: &DyadicFloat128, q: f64) -> DyadicFloat128 {
let q0 = DyadicFloat128::new_from_f64(q);
const TWO: DyadicFloat128 = DyadicFloat128::new_from_f64(2.0);
let mut b = *b;
b.sign = if b.sign == DyadicSign::Pos {
DyadicSign::Neg
} else {
DyadicSign::Pos
};
let q1 = q0.quick_mul(&TWO.quick_add(&b.quick_mul(&q0)));
let q2 = q1.quick_mul(&TWO.quick_add(&b.quick_mul(&q1)));
a.quick_mul(&q2)
}
/// Tan in double precision
///
/// ULP 0.50097
#[inline]
pub fn f_tan(x: f64) -> f64 {
let x_e = (x.to_bits() >> 52) & 0x7ff;
const E_BIAS: u64 = (1u64 << (11 - 1u64)) - 1u64;
let y: Dekker;
let k;
let mut argument_reduction = LargeArgumentReduction::default();
// |x| < 2^16
if x_e < E_BIAS + 16 {
// |x| < 2^-7
if x_e < E_BIAS - 7 {
// |x| < 2^-27, |tan(x) - x| < ulp(x)/2.
if x_e < E_BIAS - 27 {
// Signed zeros.
if x == 0.0 {
return x + x;
}
return f_fmla(x, f64::from_bits(0x3c90000000000000), x);
}
// No range reduction needed.
k = 0;
y = Dekker::new(0., x);
} else {
// Small range reduction.
(y, k) = range_reduction_small(x);
}
} else {
// Inf or NaN
if x_e > 2 * E_BIAS {
if x.is_nan() {
return f64::NAN;
}
// tan(+-Inf) = NaN
return x + f64::NAN;
}
// Large range reduction.
(k, y) = argument_reduction.reduce_new(x);
}
let (tan_y, err) = tan_eval(y);
// Fast look up version, but needs 256-entry table.
// cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
let sk = crate::sin::SIN_K_PI_OVER_128[(k.wrapping_add(128) & 255) as usize];
let ck = crate::sin::SIN_K_PI_OVER_128[((k.wrapping_add(64)) & 255) as usize];
let msin_k = Dekker::new(f64::from_bits(sk.0), f64::from_bits(sk.1));
let cos_k = Dekker::new(f64::from_bits(ck.0), f64::from_bits(ck.1));
let cos_k_tan_y = Dekker::quick_mult(tan_y, cos_k);
let msin_k_tan_y = Dekker::quick_mult(tan_y, msin_k);
// num_dd = sin(k*pi/128) + tan(y) * cos(k*pi/128)
let mut num_dd = Dekker::from_full_exact_add(cos_k_tan_y.hi, -msin_k.hi);
// den_dd = cos(k*pi/128) - tan(y) * sin(k*pi/128)
let mut den_dd = Dekker::from_full_exact_add(msin_k_tan_y.hi, cos_k.hi);
num_dd.lo += cos_k_tan_y.lo - msin_k.lo;
den_dd.lo += msin_k_tan_y.lo + cos_k.lo;
let tan_x = Dekker::div(num_dd, den_dd);
// Simple error bound: |1 / den_dd| < 2^(1 + floor(-log2(den_dd)))).
let den_inv = ((E_BIAS + 1) << (52 + 1)) - (den_dd.hi.to_bits() & EXP_MASK);
// For tan_x = (num_dd + err) / (den_dd + err), the error is bounded by:
// | tan_x - num_dd / den_dd | <= err * ( 1 + | tan_x * den_dd | ).
let tan_err = err * f_fmla(f64::from_bits(den_inv), tan_x.hi.abs(), 1.0);
let err_higher = tan_x.lo + tan_err;
let err_lower = tan_x.lo - tan_err;
let tan_upper = tan_x.hi + err_higher;
let tan_lower = tan_x.hi + err_lower;
// Ziv_s rounding test.
if tan_upper == tan_lower {
return tan_upper;
}
let u_f128 = if x_e < E_BIAS + 16 {
range_reduction_small_f128(x)
} else {
argument_reduction.accurate()
};
let tan_u = tan_eval_rational(&u_f128);
// cos(k * pi/128) = sin(k * pi/128 + pi/2) = sin((k + 64) * pi/128).
let sin_k_f128 = get_sin_k_rational(k);
let cos_k_f128 = get_sin_k_rational(k.wrapping_add(64));
let msin_k_f128 = get_sin_k_rational(k.wrapping_add(128));
// num_f128 = sin(k*pi/128) + tan(y) * cos(k*pi/128)
let num_f128 = sin_k_f128 + (cos_k_f128 * tan_u);
// den_f128 = cos(k*pi/128) - tan(y) * sin(k*pi/128)
let den_f128 = cos_k_f128 + (msin_k_f128 * tan_u);
// tan(x) = (sin(k*pi/128) + tan(y) * cos(k*pi/128)) /
// / (cos(k*pi/128) - tan(y) * sin(k*pi/128))
// reused from DoubleDouble fputil::div in the fast pass.
let result = newton_raphson_div(&num_f128, &den_f128, 1.0 / den_dd.hi);
result.fast_as_f64()
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn tan_test() {
assert_eq!(f_tan(0.0), 0.0);
assert_eq!(f_tan(1.0), 1.5574077246549023);
assert_eq!(f_tan(-0.5), -0.5463024898437905);
}
}