feldera_fxp/

fixed.rs

use std::{
    cmp::Ordering,
    fmt::{Debug, Display},
    ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign},
    str::FromStr,
};

use num_traits::{CheckedAdd, CheckedDiv, CheckedMul, CheckedSub, One, Zero, cast};

use crate::{
    Halfway, OutOfRange, ParseDecimalError, checked_pow10, debug_decimal, display_decimal,
    div_ceil, div_floor, i128_mul_pow10_round_even, parse_decimal, pow10, round_inner, u256::I256,
};

/// Decimal real number with fixed precision and scale.
///
/// `Fixed<P, S>`, where `P` in `1..=38` is the "precision" and `S` in `0..=P`
/// is the "scale", represents a signed decimal number in which `S - P` digits
/// precede the decimal point and `S` digits follow it.  The table below shows
/// the maximum values for a few combinations of `P` and `S`.  For each type,
/// the minimum value is the negation of the maximum:
///
/// |          Type |                                              Maximum Value |
/// |:--------------|-----------------------------------------------------------:|
/// | `Fixed<5,5>`  | `                                                 0.99999` |
/// | `Fixed<5,4>`  | `                                                 9.9999 ` |
/// | `Fixed<5,3>`  | `                                                99.999  ` |
/// | `Fixed<5,2>`  | `                                               999.99   ` |
/// | `Fixed<5,1>`  | `                                             9,999.9    ` |
/// | `Fixed<5,0>`  | `                                            99,999      ` |
/// | `Fixed<38,0>` | `99,999,999,999,999,999,999,999,999,999,999,999,999      ` |
/// | `Fixed<38,5>` | `       999,999,999,999,999,999,999,999,999,999,999.99999` |
///
/// # Implementation
///
/// `Fixed<P, S>` internally contains a single `i128` that represents a value
/// `x` as `x * 10**P`.  This limits `S` to 38 because `10**38 ≤ 2**127 - 1 <
/// 10**39`.  A single `i64` would be sufficient for `S ≤ 18`, and a single
/// `i32` for `S ≤ 9`, but the implementation does not optimize for those cases.
#[derive(Copy, Clone, Default, Eq, Ord, Hash)]
#[cfg_attr(feature = "size_of", derive(size_of::SizeOf))]
pub struct Fixed<const P: usize, const S: usize>(pub(super) i128);

impl<const P0: usize, const S0: usize, const P1: usize, const S1: usize> PartialEq<Fixed<P1, S1>>
    for Fixed<P0, S0>
{
    fn eq(&self, other: &Fixed<P1, S1>) -> bool {
        match S0.cmp(&S1) {
            Ordering::Less => I256::from_product(self.0, pow10(S1 - S0)) == I256::from(other.0),
            Ordering::Equal => self.0 == other.0,
            Ordering::Greater => I256::from(self.0) == I256::from_product(other.0, pow10(S0 - S1)),
        }
    }
}

macro_rules! partial_eq_int {
    ($type_name:ty) => {
        impl<const P0: usize, const S0: usize> PartialEq<$type_name> for Fixed<P0, S0> {
            fn eq(&self, other: &$type_name) -> bool {
                self.0 % Self::scale() == 0 && self.0 / Self::scale() == *other as i128
            }
        }
    };
}
partial_eq_int!(i8);
partial_eq_int!(i16);
partial_eq_int!(i32);
partial_eq_int!(i64);
partial_eq_int!(i128);
partial_eq_int!(isize);
partial_eq_int!(u8);
partial_eq_int!(u16);
partial_eq_int!(u32);
partial_eq_int!(u64);

impl<const P0: usize, const S0: usize> PartialEq<u128> for Fixed<P0, S0> {
    fn eq(&self, other: &u128) -> bool {
        self.0 >= 0
            && self.0 % Self::scale() == 0
            && (self.0 / Self::scale()).cast_unsigned() == *other
    }
}

impl<const P0: usize, const S0: usize> PartialEq<usize> for Fixed<P0, S0> {
    fn eq(&self, other: &usize) -> bool {
        self.0 >= 0
            && self.0 % Self::scale() == 0
            && (self.0 / Self::scale()).cast_unsigned() == *other as u128
    }
}

impl<const P0: usize, const S0: usize, const P1: usize, const S1: usize> PartialOrd<Fixed<P1, S1>>
    for Fixed<P0, S0>
{
    fn partial_cmp(&self, other: &Fixed<P1, S1>) -> Option<Ordering> {
        match S0.cmp(&S1) {
            Ordering::Less => {
                I256::from_product(self.0, pow10(S1 - S0)).partial_cmp(&I256::from(other.0))
            }
            Ordering::Equal => self.0.partial_cmp(&other.0),
            Ordering::Greater => {
                I256::from(self.0).partial_cmp(&I256::from_product(other.0, pow10(S0 - S1)))
            }
        }
    }
}

impl<const P: usize, const S: usize> Fixed<P, S> {
    /// Largest value for this type, e.g. 999.99 for `Fixed<5,2>`.
    pub const MAX: Self = Self(pow10(P) - 1);

    /// Smallest value for this type, e.g. -999.99 for `Fixed<5,2>`.
    ///
    /// `MIN` is always `-MAX`.
    pub const MIN: Self = Self(-Self::MAX.0);

    /// Zero in this type.
    pub const ZERO: Self = Self(0);

    /// 1 in this type.
    ///
    /// # Panic
    ///
    /// If `S == P`, this is undefined because 1 is not a value in this type,
    /// and referring to it yields a compile-time error.
    pub const ONE: Self = {
        if S < P {
            Self(pow10(S))
        } else {
            panic!("all values of Fixed::<S,P>::one() for S >= P have magnitude less than one");
        }
    };

    /// Returns the mantissa of this fixed-point decimal number.
    ///
    /// The mantissa is the unscaled integer representation of the decimal value.
    /// For a decimal number `d` with a given scale `s`, the mantissa `m` satisfies:
    /// `d = m × 10^(-s)`, or equivalently, `m = d × 10^s`.
    pub fn mantissa(&self) -> i128 {
        self.0
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `i64` value.
    pub const fn for_i64(value: i64) -> Self {
        assert!(P.saturating_sub(S) >= 19);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `u64` value.
    pub const fn for_u64(value: u64) -> Self {
        assert!(P.saturating_sub(S) >= 19);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `i32` value.
    pub const fn for_i32(value: i32) -> Self {
        assert!(P.saturating_sub(S) >= 10);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `u32` value.
    pub const fn for_u32(value: u32) -> Self {
        assert!(P.saturating_sub(S) >= 10);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `i16` value.
    pub const fn for_i16(value: i16) -> Self {
        assert!(P.saturating_sub(S) >= 5);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `u16` value.
    pub const fn for_u16(value: u16) -> Self {
        assert!(P.saturating_sub(S) >= 5);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `i8` value.
    pub const fn for_i8(value: i8) -> Self {
        assert!(P.saturating_sub(S) >= 3);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `u8` value.
    pub const fn for_u8(value: u8) -> Self {
        assert!(P.saturating_sub(S) >= 3);
        Self(value as i128 * Self::scale())
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `isize` value.
    pub const fn for_isize(value: isize) -> Self {
        match isize::BITS {
            64 => Self::for_i64(value as i64),
            32 => Self::for_i32(value as i32),
            16 => Self::for_i16(value as i16),
            _ => panic!(),
        }
    }

    /// Returns `value` in this type.
    ///
    /// # Panic
    ///
    /// Panics if this type cannot hold every `usize` value.
    pub const fn for_usize(value: usize) -> Self {
        match usize::BITS {
            64 => Self::for_u64(value as u64),
            32 => Self::for_u32(value as u32),
            16 => Self::for_u16(value as u16),
            _ => panic!(),
        }
    }

    /// Returns the value that represents `value * 10**-scale`, rounding
    /// toward zero if necessary, or `None` if the rounded value is out of range
    /// for this type.
    ///
    /// # Examples
    ///
    /// ```
    /// # use feldera_fxp::Fixed;
    ///
    /// assert_eq!(Fixed::<11, 1>::new(435, 0).unwrap().to_string(), "435");
    /// assert_eq!(Fixed::<11, 1>::new(435, 1).unwrap().to_string(), "43.5");
    /// assert_eq!(Fixed::<11, 1>::new(435, 2).unwrap().to_string(), "4.3");
    /// assert_eq!(Fixed::<11, 1>::new(435, 3).unwrap().to_string(), "0.4");
    /// ```
    pub fn new(value: i128, scale: i32) -> Option<Self> {
        Self::try_new_with_exponent(value, (S as i32).saturating_sub(scale))
    }

    /// Returns the value that represents `value * 10**-scale`, rounding to
    /// nearest if necessary, with ties rounded to even, or `None` if the
    /// rounded value is out of range for this type.
    ///
    /// # Examples
    ///
    /// ```
    /// # use feldera_fxp::Fixed;
    ///
    /// assert_eq!(Fixed::<11, 1>::new_round_even(435, 0).unwrap().to_string(), "435");
    /// assert_eq!(Fixed::<11, 1>::new_round_even(435, 1).unwrap().to_string(), "43.5");
    /// assert_eq!(Fixed::<11, 1>::new_round_even(435, 2).unwrap().to_string(), "4.4");
    /// assert_eq!(Fixed::<11, 1>::new_round_even(435, 3).unwrap().to_string(), "0.4");
    /// ```
    pub fn new_round_even(value: i128, scale: i32) -> Option<Self> {
        Self::try_new_with_exponent_round_even(value, (S as i32).saturating_sub(scale))
    }

    /// Returns `Self(value)`, if `value` is in the correct range for this type.
    fn try_new(value: i128) -> Option<Self> {
        const { Self::check_constraints() };
        (Self::MIN.0..=Self::MAX.0)
            .contains(&value)
            .then_some(Self(value))
    }

    /// Returns the internal representation of this Fixed value
    /// This is an unstable API, use at your own risk!
    pub fn internal_representation(self) -> i128 {
        self.0
    }

    /// Returns `Self(value * 10**exponent)`, rounding to even if `exponent` is
    /// negative, if the computed value is in the correct range for the type.
    pub(super) fn try_new_with_exponent_round_even(value: i128, exponent: i32) -> Option<Self> {
        i128_mul_pow10_round_even(value, exponent).and_then(Self::try_new)
    }

    /// Returns `Self(value * 10**exponent)`, rounding toward zero, if the
    /// computed value is in the correct range for the type.
    pub(super) fn try_new_with_exponent(value: i128, exponent: i32) -> Option<Self> {
        // Non-generic inner function to reduce monomorphization cost.
        fn inner(value: i128, exponent: i32) -> Option<i128> {
            Some(match exponent.cmp(&0) {
                Ordering::Less => {
                    // Divide by a negative exponent.
                    if let Some(divisor) = checked_pow10(exponent.unsigned_abs()) {
                        value / divisor
                    } else {
                        // `10**-exponent` is greater than `i128::MAX`.  The result
                        // must be zero.
                        0
                    }
                }
                Ordering::Equal => value,
                Ordering::Greater => {
                    // Multiply by a positive exponent.
                    value.checked_mul(checked_pow10(exponent.cast_unsigned())?)?
                }
            })
        }
        inner(value, exponent).and_then(Self::try_new)
    }

    /// Validates the constraints on `S` and `P`.
    const fn check_constraints() {
        assert!(P >= 1 && P <= 38, "Fixed<S,P> must have 1 <= S <= 38");
        assert!(S <= P, "Fixed<S,P> must have S <= P");
    }

    /// Returns `pow10(S)`.
    const fn scale() -> i128 {
        Self::check_constraints();
        pow10(S)
    }

    /// Integer division, as defined for `divide-integer` in [General Decimal
    /// Arithmetic].  Returns `None` if `other` is zero or the result is greater
    /// than `i128::MAX`.
    ///
    /// [General Decimal Arithmetic]: https://speleotrove.com/decimal/decarith.pdf
    pub const fn checked_div_integer(self, other: Self) -> Option<i128> {
        self.0.checked_div(other.0)
    }

    /// Integer division like [checked_div_integer](Self::checked_div_integer),
    /// but panic on error.
    ///
    /// # Panic
    ///
    /// Panics if `other` is zero or the result is greater than `i128::MAX`.
    pub const fn strict_div_integer(self, other: Self) -> i128 {
        self.checked_div_integer(other).unwrap()
    }

    /// Remainder, as defined for `remainder` in [General Decimal Arithmetic].
    /// Returns `None` if `other` is zero.
    ///
    /// [General Decimal Arithmetic]: https://speleotrove.com/decimal/decarith.pdf
    pub fn checked_rem<const P0: usize, const S0: usize, const P1: usize, const S1: usize>(
        self,
        other: Fixed<P0, S0>,
    ) -> Option<Fixed<P1, S1>> {
        let neg = self.is_negative();
        let left = self.abs();
        let right = other.abs();
        let div: Self = left.checked_div_generic(right)?;
        let trunc: Self = div.trunc();
        let mul: Self = right.checked_mul_generic(trunc)?;
        let rem: Fixed<P1, S1> = left.checked_sub_generic(mul)?;
        Some(if neg { rem.neg() } else { rem })
    }

    /// Returns the absolute value.  This is an exact calculation that cannot
    /// overflow.
    pub const fn abs(self) -> Self {
        Self(self.0.abs())
    }

    /// Returns true if this value is negative, false if it is zero or positive.
    pub const fn is_negative(self) -> bool {
        self.0.is_negative()
    }

    /// Returns the square root of this value, rounded down, or `None` if this
    /// value is negative.
    ///
    /// It probably makes more sense to convert to `f64` and take the
    /// floating-point square root.
    pub fn checked_sqrt(self) -> Option<Self> {
        Some(Self(
            I256::from_product(self.0, Self::scale()).checked_isqrt()?,
        ))
    }

    /// Returns the square root of this value, rounded down.
    ///
    /// It probably makes more sense to convert to `f64` and take the
    /// floating-point square root.
    ///
    /// # Panic
    ///
    /// Panics if this value is negative.
    pub fn sqrt(self) -> Self {
        self.checked_sqrt().unwrap()
    }

    /// Returns this value rounded to `n` digits after the decimal point, or
    /// `None` if rounding caused overflow, `n` may be negative.
    ///
    /// If the value is halfway between two integers, rounds away from zero.
    pub fn checked_round(&self, n: i32) -> Option<Self> {
        round_inner(self.0, S as i32, n, Halfway::AwayFromZero).and_then(Self::try_new)
    }

    /// Rounds to `n` digits after the decimal point, like [checked_round].
    /// If the value is halfway between two integers, rounds away from zero.
    ///
    /// # Panic
    ///
    /// Panics if rounding causes overflow.
    ///
    /// [checked_round]: Self::checked_round
    pub fn round(&self, n: i32) -> Self {
        self.checked_round(n)
            .unwrap_or_else(|| panic!("Could not round value {} to {} digits", self, n))
    }

    /// Returns this value rounded to `n` digits after the decimal point, or
    /// `None` if rounding caused overflow, `n` may be negative.  If the value
    /// is halfway between two integers, rounds toward an even least significant
    /// digit.
    pub fn checked_round_ties_even(&self, n: i32) -> Option<Self> {
        round_inner(self.0, S as i32, n, Halfway::Even).and_then(Self::try_new)
    }

    /// Rounds to `n` digits after the decimal point, like
    /// [checked_round_ties_even].  If the value is halfway between two
    /// integers, rounds toward an even least significant digit.
    ///
    /// # Panic
    ///
    /// Panics if rounding causes overflow.
    ///
    /// [checked_round_ties_even]: Self::checked_round_ties_even
    pub fn round_ties_even(&self, n: i32) -> Self {
        self.checked_round_ties_even(n).unwrap()
    }

    /// Returns this value rounded down to the nearest integer, or `None` if
    /// rounding caused overflow.
    pub fn checked_floor(&self) -> Option<Self> {
        if S > 0 {
            Self::try_new(div_floor(self.0, Self::scale()) * Self::scale())
        } else {
            Some(*self)
        }
    }

    /// Rounds down to the nearest integer, like [checked_floor].
    ///
    /// # Panic
    ///
    /// Panics if rounding causes overflow.
    ///
    /// [checked_floor]: Self::checked_floor
    pub fn floor(&self) -> Self {
        self.checked_floor().unwrap()
    }

    /// Returns this value rounded down to the nearest integer, with type Fixed<P, 0>
    pub fn int_floor(&self) -> Fixed<P, 0> {
        if S > 0 {
            Fixed::<P, 0>::new(div_floor(self.0, Self::scale()), 0i32).unwrap()
        } else {
            Fixed::<P, 0>::new(self.0, S as i32).unwrap()
        }
    }

    /// Returns the integer part of this value, truncating non-integers toward
    /// zero.  This is an exact calculation that cannot overflow.
    ///
    /// `trunc()` is equivalent to `trunc_digits(0)`, but it might be more
    /// efficient due to constant folding.
    pub fn trunc(&self) -> Self {
        Self(self.0 / Self::scale() * Self::scale())
    }

    /// Returns this value, truncated toward zero at `digits` after the decimal
    /// point. This is an exact calculation that cannot overflow.
    pub fn trunc_digits(&self, digits: i32) -> Self {
        let exponent = (S as i32).saturating_sub(digits);
        if exponent <= 0 {
            *self
        } else if let Some(divisor) = checked_pow10(exponent.cast_unsigned()) {
            Self(self.0 / divisor * divisor)
        } else {
            Self::ZERO
        }
    }

    /// Returns this value rounded up to the nearest integer, or `None` if
    /// rounding caused overflow.
    pub fn checked_ceil(&self) -> Option<Self> {
        if S > 0 {
            Self::try_new(div_ceil(self.0, Self::scale()) * Self::scale())
        } else {
            Some(*self)
        }
    }

    /// Rounds up to the nearest integer, like [checked_ceil].
    ///
    /// # Panic
    ///
    /// Panics if rounding causes overflow.
    ///
    /// [checked_ceil]: Self::checked_ceil
    pub fn ceil(&self) -> Self {
        self.checked_ceil().unwrap()
    }

    /// Returns this value rounded up to the nearest integer, or `None` if
    /// rounding caused overflow.
    pub fn int_ceil(&self) -> Fixed<P, 0> {
        if S > 0 {
            Fixed::<P, 0>::new(div_ceil(self.0, Self::scale()), 0i32).unwrap()
        } else {
            Fixed::<P, 0>::new(self.0, S as i32).unwrap()
        }
    }

    /// Returns -1 if this value is less than zero, 0 if this value is zero, and
    /// 1 if this value is greater than zero, as `Fixed<1,0>`.
    pub fn sign(&self) -> Fixed<1, 0> {
        self.checked_sign_generic().unwrap()
    }

    /// Returns the reciprocal (inverse) of this value, `1/x`, or `None` if `x`
    /// is zero or `1/x` is out of range.
    pub fn checked_recip(&self) -> Option<Self> {
        if S < P {
            Self(Self::scale()).checked_div(self)
        } else {
            // `1` is out of range for this type, therefore `abs(self) < 1`,
            // therefore `abs(1/self) > 1`, therefore the result is out of
            // range.
            None
        }
    }

    /// Returns the reciprocal (inverse) of this value, `1/x`.  This works even
    /// if `1` is out of range for this type, as long as `1/x` is in range.
    ///
    /// # Panic
    ///
    /// Panics if `x` is zero or `1/x` is out of range.
    pub fn recip(&self) -> Self {
        self.checked_recip().unwrap()
    }

    /// Returns this value raised to `exp` power, rounding toward zero, or
    /// `None` if the result is out of range or if this value is 0 and `exp` is
    /// nonpositive.
    ///
    /// # Accuracy
    ///
    /// For `exp > 0`, this computes intermediate results with more than `S`
    /// digits of precision, if possible, to allow to better accuracy in the
    /// result.  For `exp < 0`, this isn't implemented yet.
    pub fn checked_powi(&self, exp: i32) -> Option<Self> {
        if self.is_zero() {
            (exp > 0).then_some(Self::ZERO)
        } else if exp == 0 {
            if S < P {
                Some(Self::ONE)
            } else {
                // 1 is not representable.
                None
            }
        } else if exp > 0 {
            let mut exp = exp.unsigned_abs();
            let mut base = self.0;
            let mut base_scale = S as i32;
            let mut acc = None;
            loop {
                if (exp & 1) == 1 {
                    acc = if let Some((acc, acc_scale)) = acc {
                        let (acc, shift) = I256::from_product(acc, base).reduce_to_i128();
                        Some((acc, (acc_scale + base_scale) - shift as i32))
                    } else {
                        Some((base, base_scale))
                    };
                }
                exp /= 2;
                if exp == 0 {
                    let (acc, acc_scale) = acc.unwrap();
                    return Self::try_new_with_exponent(acc, S as i32 - acc_scale);
                }

                let (next_base, shift) = I256::from_product(base, base).reduce_to_i128();
                base = next_base;
                base_scale = base_scale * 2 - shift as i32;
            }
        } else {
            let mut exp = exp.unsigned_abs();
            let mut base = *self;
            let mut acc: Option<Fixed<P, S>> = None;
            loop {
                if (exp & 1) == 1 {
                    acc = Some(if let Some(acc) = acc {
                        acc.checked_div(&base)
                    } else {
                        base.checked_recip()
                    }?)
                }
                exp /= 2;
                if exp == 0 {
                    return acc;
                }
                base *= base;
            }
        }
    }

    /// Returns this value raised to `exp` power, rounding toward zero.
    ///
    /// For `exp > 0`, this computes intermediate results with more than `S`
    /// digits of precision, if possible, to allow to better accuracy in the
    /// result.  For `exp < 0`, this isn't implemented yet.
    ///
    /// # Panics
    ///
    /// Panics if the result is out of range or if this value is 0 and `exp` is
    /// nonpositive.
    pub fn powi(&self, exp: i32) -> Self {
        self.checked_powi(exp).unwrap()
    }

    /// Returns the least number greater than `self`, or `None` if this is
    /// `Self::MAX`.
    pub fn next_up(&self) -> Option<Self> {
        if *self < Self::MAX {
            Some(Self(self.0 + 1))
        } else {
            None
        }
    }

    /// Returns the greatest number less than `self`, or `None` if this is
    /// `Self::MAX`.
    pub fn next_down(&self) -> Option<Self> {
        if *self > Self::MIN {
            Some(Self(self.0 - 1))
        } else {
            None
        }
    }
}

impl<const P0: usize, const S0: usize> Fixed<P0, S0> {
    /// Returns this value converted into another type `Fixed<P1, S1>`, or
    /// `None` if this value is outside the range of the target type.  If the
    /// conversion is successful, then the result is exactly the same as the
    /// original value if `S1 >= S0`, and rounded down otherwise.
    ///
    /// This should be implemented as `TryFrom` but that [conflicts with the
    /// standard library
    /// implementation](https://users.rust-lang.org/t/conflicting-implementations-of-trait-from/92994).
    pub fn convert<const P1: usize, const S1: usize>(&self) -> Option<Fixed<P1, S1>> {
        Fixed::try_new_with_exponent(self.0, S1 as i32 - S0 as i32)
    }

    /// Returns this value converted into another type `Fixed<P1, S1>`, or
    /// `None` if this value is outside the range of the target type.  If the
    /// conversion is successful, then the result is exactly the same as the
    /// original value if `S1 >= S0`, and rounded to even otherwise.
    pub fn convert_round_even<const P1: usize, const S1: usize>(&self) -> Option<Fixed<P1, S1>> {
        Fixed::try_new_with_exponent_round_even(self.0, S1 as i32 - S0 as i32)
    }

    /// Returns -1 if this value is less than zero, 0 if this value is zero, and
    /// 1 if this value is greater than zero, in an arbitrary `Fixed` type.
    /// Returns `None` on overflow (if this value is nonzero and `S1 >= P1`).
    pub fn checked_sign_generic<const P1: usize, const S1: usize>(&self) -> Option<Fixed<P1, S1>> {
        let one = Fixed::<P1, S1>::scale();
        match self.0.cmp(&0) {
            Ordering::Less if S1 < P1 => Some(Fixed(-one)),
            Ordering::Equal => Some(Fixed::ZERO),
            Ordering::Greater if S1 < P1 => Some(Fixed(one)),
            _ => None,
        }
    }

    /// Calculates `self + other`, for operands with scale and precision `(S0,P0)` and
    /// `(S1,P1)`, respectively, producing a result with scale and precision
    /// `(S2,P2)`.  The result is calculated exactly if possible and otherwise
    /// rounded toward zero.  Returns `None` if the result is not representable
    /// in the result type.
    pub fn checked_add_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Option<Fixed<P2, S2>> {
        match S0.cmp(&S1) {
            Ordering::Less => {
                let factor = pow10(S1 - S0);
                if let Some(shifted) = self.0.checked_mul(factor)
                    && let Some(sum) = other.0.checked_add(shifted)
                {
                    // This is the common case, where we can shift `self` left
                    // to match `other` and add `other` without intermediate
                    // overflow.
                    Fixed::try_new_with_exponent(sum, S2 as i32 - S1 as i32)
                } else {
                    // If the result type has more digits to the left of the
                    // decimal point than `other`, then we still might be able
                    // to compute a correct answer without ultimate overflow.
                    let result = (I256::from_product(self.0, factor) + I256::from(other.0))
                        .narrowing_div(pow10(S1.saturating_sub(S2)))?;
                    Fixed::try_new_with_exponent(result, (S2.saturating_sub(S1)) as i32)
                }
            }
            Ordering::Equal => {
                if let Some(sum) = self.0.checked_add(other.0) {
                    // This is the common case, where the result fits in `i128`.
                    Fixed::try_new_with_exponent(sum, S2 as i32 - S0 as i32)
                } else if S2 < S0 {
                    // The ultimate result might fit in `i128` but we need
                    // multiple-precision arithmetic.
                    Fixed::try_new(
                        (I256::from(self.0) + I256::from(other.0)).narrowing_div(pow10(S0 - S2))?,
                    )
                } else {
                    // Definitely does not fit.
                    None
                }
            }
            Ordering::Greater => {
                // Mirror of the `Less` case.
                let factor = pow10(S0 - S1);
                if let Some(shifted) = other.0.checked_mul(factor)
                    && let Some(sum) = self.0.checked_add(shifted)
                {
                    Fixed::try_new_with_exponent(sum, S2 as i32 - S0 as i32)
                } else {
                    let result = (I256::from_product(other.0, factor) + I256::from(self.0))
                        .narrowing_div(pow10(S0.saturating_sub(S2)))?;
                    Fixed::try_new_with_exponent(result, S2.saturating_sub(S0) as i32)
                }
            }
        }
    }

    /// Calculates `self - other`, for operands with scale and precision
    /// `(S0,P0)` and `(S1,P1)`, respectively, producing a result with scale and
    /// precision `(S2,P2)`.  The result is calculated exactly if possible and
    /// otherwise rounded toward zero.  Returns `None` if the result is not
    /// representable in the result type.
    pub fn checked_sub_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Option<Fixed<P2, S2>> {
        self.checked_add_generic(-other)
    }

    /// Calculates `self * other`, for operands with scale and precision
    /// `(S0,P0)` and `(S1,P1)`, respectively, producing a result with scale and
    /// precision `(S2,P2)`.  The result is calculated exactly if possible and
    /// otherwise rounded toward zero.  Returns `None` if the result is not
    /// representable in the result type.
    pub fn checked_mul_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Option<Fixed<P2, S2>> {
        Fixed::<P2, S2>::try_new_with_exponent(
            I256::from_product(self.0, other.0)
                .narrowing_div(pow10((S0 + S1).saturating_sub(S2)))?,
            S2.saturating_sub(S0 + S1) as i32,
        )
    }

    /// Calculate `self / other`, for operands with scale and precision
    /// `(S0,P0)` and `(S1,P1)`, respectively, producing a result with scale and
    /// precision `(S2,P2)`.  The result is calculated exactly if possible and
    /// otherwise rounded toward zero.  Returns `None` if the result is not
    /// representable in the result type, or if `other` is zero.
    pub fn checked_div_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Option<Fixed<P2, S2>> {
        if other == 0 {
            None
        } else {
            let shift_left = (S1 + S2).saturating_sub(S0);
            if shift_left > 38 {
                // A shift this big would exceed the range of I256, so we can't
                // calculate it, but the ultimate result would also overflow, so
                // we don't have to.
                None
            } else {
                Fixed::try_new_with_exponent(
                    I256::from_product(self.0, pow10(shift_left)).narrowing_div(other.0)?,
                    -(S0.saturating_sub(S1 + S2) as i32),
                )
            }
        }
    }

    /// Compute `self + other`, rounding toward zero, panicking if overflow
    /// occurs.
    pub fn strict_add_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Fixed<P2, S2> {
        self.checked_add_generic(other).unwrap()
    }

    /// Compute `self - other`, rounding toward zero, panicking if overflow
    /// occurs.
    pub fn strict_sub_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Fixed<P2, S2> {
        self.checked_sub_generic(other).unwrap()
    }

    /// Compute `self * other`, rounding toward zero, panicking if overflow
    /// occurs.
    pub fn strict_mul_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Fixed<P2, S2> {
        self.checked_mul_generic(other).unwrap()
    }

    /// Compute `self / other`, rounding toward zero, panicking if overflow
    /// occurs or if `other` is zero.
    pub fn strict_div_generic<
        const P1: usize,
        const S1: usize,
        const P2: usize,
        const S2: usize,
    >(
        self,
        other: Fixed<P1, S1>,
    ) -> Fixed<P2, S2> {
        self.checked_div_generic(other).unwrap()
    }
}

impl<const P: usize, const S: usize> Zero for Fixed<P, S> {
    fn zero() -> Self {
        Self::ZERO
    }

    fn is_zero(&self) -> bool {
        *self == Self::ZERO
    }
}

impl<const P: usize, const S: usize> One for Fixed<P, S> {
    /// This will panic at compile time if 1 isn't in the range of this type.
    fn one() -> Self {
        Self::ONE
    }
}

impl<const P: usize, const S: usize> TryFrom<f64> for Fixed<P, S> {
    type Error = OutOfRange;

    /// Convert `value` to `Fixed`, rounding toward zero, reporting an error if
    /// `value` is out of range.
    fn try_from(value: f64) -> Result<Self, Self::Error> {
        cast(value * Self::scale() as f64)
            .and_then(Self::try_new)
            .ok_or(OutOfRange)
    }
}

impl<const P: usize, const S: usize> TryFrom<f32> for Fixed<P, S> {
    type Error = OutOfRange;

    /// Convert `value` to `Fixed`, rounding toward zero, reporting an error if
    /// `value` is out of range.
    fn try_from(value: f32) -> Result<Self, Self::Error> {
        cast(value as f64 * Self::scale() as f64)
            .and_then(Self::try_new)
            .ok_or(OutOfRange)
    }
}

impl<const P: usize, const S: usize> From<Fixed<P, S>> for f64 {
    fn from(value: Fixed<P, S>) -> Self {
        value.0 as f64 / Fixed::<P, S>::scale() as f64
    }
}

impl<const P: usize, const S: usize> TryFrom<i128> for Fixed<P, S> {
    type Error = OutOfRange;

    /// Convert `value` to `Fixed`, reporting an error if `value` is out of
    /// range.  This is an exact conversion that cannot lose precision if it
    /// succeeds.
    fn try_from(value: i128) -> Result<Self, Self::Error> {
        if value.unsigned_abs() <= Self::max_u128() {
            Ok(Self(value * Self::scale()))
        } else {
            Err(OutOfRange)
        }
    }
}

macro_rules! try_from_signed_int {
    ($type_name:ty) => {
        impl<const P: usize, const S: usize> TryFrom<$type_name> for Fixed<P, S> {
            type Error = OutOfRange;

            /// Convert `value` to `Fixed`, rounding toward zero, reporting an
            /// error if `value` is out of range.
            fn try_from(value: $type_name) -> Result<Self, Self::Error> {
                (value as i128).try_into()
            }
        }
    };
}

try_from_signed_int!(isize);
try_from_signed_int!(i64);
try_from_signed_int!(i32);
try_from_signed_int!(i16);
try_from_signed_int!(i8);

impl<const P: usize, const S: usize> TryFrom<u128> for Fixed<P, S> {
    type Error = OutOfRange;

    /// Convert `value` to `Fixed`, reporting an error if `value` is out of
    /// range.  This is an exact conversion that cannot lose precision if it
    /// succeeds.
    fn try_from(value: u128) -> Result<Self, Self::Error> {
        if value <= Self::max_i128() as u128 {
            Ok(Self(value as i128 * Self::scale()))
        } else {
            Err(OutOfRange)
        }
    }
}

macro_rules! try_from_unsigned_int {
    ($type_name:ty) => {
        impl<const P: usize, const S: usize> TryFrom<$type_name> for Fixed<P, S> {
            type Error = OutOfRange;

            /// Convert `value` to `Fixed`, reporting an error if `value` is out
            /// of range.  This is an exact conversion that cannot lose
            /// precision if it succeeds.
            fn try_from(value: $type_name) -> Result<Self, Self::Error> {
                (value as u128).try_into()
            }
        }
    };
}

try_from_unsigned_int!(usize);
try_from_unsigned_int!(u64);
try_from_unsigned_int!(u32);
try_from_unsigned_int!(u16);
try_from_unsigned_int!(u8);

macro_rules! min_max_int {
    ($signed_type:ty, $max_signed:ident, $min_signed:ident, $unsigned_type:ty, $max_unsigned:ident) => {
        #[doc = "Returns the maximum `"]
        #[doc = stringify!($signed_type)]
        #[doc = "` that can be converted to this type."]
        pub const fn $max_signed() -> $signed_type {
            if Self::max_i128() > <$signed_type>::MAX as i128 {
                <$signed_type>::MAX
            } else {
                Self::max_i128() as $signed_type
            }
        }

        #[doc = "Returns the minimum `"]
        #[doc = stringify!($signed_type)]
        #[doc = "` that can be converted to this type."]
        pub const fn $min_signed() -> $signed_type {
            -Self::$max_signed()
        }

        #[doc = "Returns the maximum `"]
        #[doc = stringify!($unsigned_type)]
        #[doc = "` that can be converted to this type.\n\nThe minimum is 0."]
        pub const fn $max_unsigned() -> $unsigned_type {
            if Self::max_u128() > <$unsigned_type>::MAX as u128 {
                <$unsigned_type>::MAX
            } else {
                Self::max_u128() as $unsigned_type
            }
        }
    };
}

impl<const P: usize, const S: usize> Fixed<P, S> {
    /// Returns the maximum `i128` that can be converted to this type.
    pub const fn max_i128() -> i128 {
        if P > S { pow10(P - S) - 1 } else { 0 }
    }

    /// Returns the minimum `i128` that can be converted to this type.
    pub const fn min_i128() -> i128 {
        -Self::max_i128()
    }

    /// Returns the maximum `u128` that can be converted to this type.
    ///
    /// The minimum is 0.
    pub const fn max_u128() -> u128 {
        Self::max_i128().cast_unsigned()
    }

    min_max_int!(isize, max_isize, min_isize, usize, max_usize);
    min_max_int!(i64, max_i64, min_i64, u64, max_u64);
    min_max_int!(i32, max_i32, min_i32, u32, max_u32);
    min_max_int!(i16, max_i16, min_i16, u16, max_u16);
    min_max_int!(i8, max_i8, min_i8, u8, max_u8);
}

impl<const P: usize, const S: usize> From<Fixed<P, S>> for i128 {
    /// Convert from `Fixed` to integer, rounding toward zero (the same
    /// semantics as Rust casts from float to integer).
    fn from(value: Fixed<P, S>) -> Self {
        // Integer `/` rounds toward zero in Rust.
        value.0 / <Fixed<P, S>>::scale()
    }
}

macro_rules! try_to_signed_int {
    ($type_name:ty) => {
        impl<const P: usize, const S: usize> TryFrom<Fixed<P, S>> for $type_name {
            type Error = OutOfRange;

            /// Convert from `Fixed` to integer, rounding toward zero (the same
            /// semantics as Rust casts from float to integer).
            fn try_from(value: Fixed<P, S>) -> Result<Self, Self::Error> {
                i128::from(value).try_into().map_err(|_| OutOfRange)
            }
        }
    };
}

try_to_signed_int!(i64);
try_to_signed_int!(i32);
try_to_signed_int!(i16);
try_to_signed_int!(i8);
try_to_signed_int!(isize);

/// This is the same as [try_to_signed_int] except for the documentation
/// comment.
macro_rules! try_to_unsigned_int {
    ($type_name:ty) => {
        impl<const P: usize, const S: usize> TryFrom<Fixed<P, S>> for $type_name {
            type Error = OutOfRange;

            /// Convert from `Fixed` to integer, rounding toward zero (the same
            /// semantics as Rust casts from float to integer).
            ///
            /// Because this rounds toward zero, negative values greater than -1
            /// will convert to 0 instead of an out-of-range error.
            fn try_from(value: Fixed<P, S>) -> Result<Self, Self::Error> {
                i128::from(value).try_into().map_err(|_| OutOfRange)
            }
        }
    };
}

try_to_unsigned_int!(u128);
try_to_unsigned_int!(u64);
try_to_unsigned_int!(u32);
try_to_unsigned_int!(u16);
try_to_unsigned_int!(u8);
try_to_unsigned_int!(usize);

impl<const P: usize, const S: usize> Add for Fixed<P, S> {
    type Output = Self;

    /// Returns the sum, rounding toward zero.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn add(self, other: Self) -> Self::Output {
        self.checked_add(&other).unwrap()
    }
}

impl<const P: usize, const S: usize> Add for &Fixed<P, S> {
    type Output = Fixed<P, S>;

    /// Returns the sum, which is exact if the result is in range.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn add(self, other: Self) -> Self::Output {
        self.checked_add(other).unwrap()
    }
}

impl<const P: usize, const S: usize> CheckedAdd for Fixed<P, S> {
    /// Returns the sum, which is exact, or `None` if the result is out of
    /// range.
    fn checked_add(&self, other: &Self) -> Option<Self> {
        self.checked_add_generic(*other)
    }
}

impl<const P: usize, const S: usize> AddAssign for Fixed<P, S> {
    /// Adds `other` to `self`, which is an exact calculation.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn add_assign(&mut self, other: Self) {
        *self = *self + other;
    }
}

impl<const P: usize, const S: usize> AddAssign<&Fixed<P, S>> for Fixed<P, S> {
    /// Adds `other` to `self`, which is an exact calculation.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn add_assign(&mut self, other: &Fixed<P, S>) {
        *self = *self + *other;
    }
}

impl<const P: usize, const S: usize> Sub for Fixed<P, S> {
    type Output = Self;

    /// Returns the difference, which is exact if the result is in range.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn sub(self, other: Self) -> Self::Output {
        self.checked_sub(&other).unwrap()
    }
}

impl<const P: usize, const S: usize> Sub for &Fixed<P, S> {
    type Output = Fixed<P, S>;

    /// Returns the difference, which is exact if the result is in range.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn sub(self, other: Self) -> Self::Output {
        self.checked_sub(other).unwrap()
    }
}

impl<const P: usize, const S: usize> CheckedSub for Fixed<P, S> {
    /// Returns the difference, which is exact, or `None` if the result is out
    /// of range.
    fn checked_sub(&self, other: &Self) -> Option<Self> {
        self.checked_sub_generic(*other)
    }
}

impl<const P: usize, const S: usize> SubAssign for Fixed<P, S> {
    /// Subtracts `other` from `self`, which is an exact calculation.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn sub_assign(&mut self, other: Self) {
        *self = *self - other;
    }
}

impl<const P: usize, const S: usize> Mul for Fixed<P, S> {
    type Output = Self;

    /// Returns the product, rounding toward zero.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn mul(self, other: Self) -> Self::Output {
        self.checked_mul(&other).unwrap()
    }
}

impl<const P: usize, const S: usize> Mul for &Fixed<P, S> {
    type Output = Fixed<P, S>;

    /// Returns the product, rounding toward zero.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn mul(self, other: Self) -> Self::Output {
        self.checked_mul(other).unwrap()
    }
}

impl<const P: usize, const S: usize> CheckedMul for Fixed<P, S> {
    /// Returns the product, rounding toward zero, or `None` if the result is
    /// out of range.
    fn checked_mul(&self, other: &Self) -> Option<Self> {
        Self::checked_mul_generic(*self, *other)
    }
}

impl<const P: usize, const S: usize> MulAssign for Fixed<P, S> {
    /// Multiplies `self` by `other`, rounding toward zero.
    ///
    /// # Panic
    ///
    /// Panics if the result is out of range.
    fn mul_assign(&mut self, other: Self) {
        *self = *self * other;
    }
}

impl<const P: usize, const S: usize> Div for Fixed<P, S> {
    type Output = Self;

    /// Returns the quotient, rounding toward zero.
    ///
    /// # Panic
    ///
    /// Panics if `other` is zero or the result is out of range.
    fn div(self, other: Self) -> Self::Output {
        self.checked_div(&other).unwrap()
    }
}

impl<const P: usize, const S: usize> Div for &Fixed<P, S> {
    type Output = Fixed<P, S>;

    /// Returns the quotient, rounding toward zero.
    ///
    /// # Panic
    ///
    /// Panics if `other` is zero or the result is out of range.
    fn div(self, other: Self) -> Self::Output {
        self.checked_div(other).unwrap()
    }
}

impl<const P: usize, const S: usize> CheckedDiv for Fixed<P, S> {
    /// Returns the quotient, rounding toward zero, or `None` if `other` is zero
    /// or the result is out of range.
    fn checked_div(&self, other: &Self) -> Option<Self> {
        Self::checked_div_generic(*self, *other)
    }
}

impl<const P: usize, const S: usize> DivAssign for Fixed<P, S> {
    /// Divides `self` by `other`, rounding toward zero.
    ///
    /// # Panic
    ///
    /// Panics if `other` is zero or the result is out of range.
    fn div_assign(&mut self, other: Self) {
        *self = *self / other;
    }
}

impl<const P: usize, const S: usize> Neg for Fixed<P, S> {
    type Output = Self;

    /// Returns `-self`.  This is an exact calculation that cannot overflow.
    fn neg(self) -> Self::Output {
        Self(-self.0)
    }
}

impl<const P: usize, const S: usize> Neg for &Fixed<P, S> {
    type Output = Fixed<P, S>;

    /// Returns `-self`.  This is an exact calculation that cannot overflow.
    fn neg(self) -> Self::Output {
        Fixed(-self.0)
    }
}

impl<const P: usize, const S: usize> FromStr for Fixed<P, S> {
    type Err = ParseDecimalError;

    /// Parses `s` as `Fixed`.
    ///
    /// This accepts the same forms as [f64::from_str], except that it rejects
    /// infinities and NaNs (which `Fixed` does not support), as well as
    /// out-of-range values.  Rounds overprecise values to the nearest
    /// representable value, rounding halfway values to even.
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let (value, exponent) = parse_decimal(s, S as i32)?;
        Self::try_new_with_exponent_round_even(value, exponent).ok_or(ParseDecimalError::OutOfRange)
    }
}

impl<const P: usize, const S: usize> Debug for Fixed<P, S> {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        debug_decimal(self.0, S, f)
    }
}

impl<const P: usize, const S: usize> Display for Fixed<P, S> {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        display_decimal(self.0, S, f)
    }
}

impl<const P: usize, const S: usize> Fixed<P, S> {
    /// Value returned by `Self::MIN.to_unsigned_encoding()`.
    ///
    /// This is always 0.
    pub const UNSIGNED_MIN: u128 = 0;

    /// Value returned by `Self::ZERO.to_unsigned_encoding()`.
    pub const UNSIGNED_ZERO: u128 = pow10(P).cast_unsigned() - 1;

    /// Value returned by `Self::MAX.to_unsigned_encoding()`.
    pub const UNSIGNED_MAX: u128 = pow10(P).cast_unsigned() * 2 - 2;

    /// Returns this value converted to a `u128` in the range
    /// [`Self::UNSIGNED_MIN`] to [`Self::UNSIGNED_MAX`] (inclusive).  The
    /// values returned for any given [`Fixed<P,S>`] are suitable for equality
    /// and order comparison, that is, `a.cmp(&b)` has the same value as
    /// `a.to_unsigned_encoding().cmp(&b.to_unsigned_encoding())`.
    ///
    /// # Usage
    ///
    /// This might only be useful in practice for [DBSP] aggregates, which only
    /// only support unsigned integer values.  We expose it in case it's useful
    /// for some other purpose.
    ///
    /// [DBSP]: https://docs.rs/dbsp/latest/dbsp/
    pub fn to_unsigned_encoding(self) -> u128 {
        Self::UNSIGNED_ZERO.checked_add_signed(self.0).unwrap()
    }

    /// Inverts the transformation of [`to_unsigned_encoding`], returning the
    /// original `Fixed` value (assuming `P` and `S` are the same as before).
    /// Returns `None` if `encoding` is invalid (which cannot happen if
    /// [`to_unsigned_encoding`] returned `encoding`).
    ///
    /// [`to_unsigned_encoding`]: Self::to_unsigned_encoding
    ///
    /// # Usage
    ///
    /// This might only be useful in practice for [DBSP] aggregates, which only
    /// only support unsigned integer values.  We expose it in case it's useful
    /// for some other purpose.
    ///
    /// [DBSP]: https://docs.rs/dbsp/latest/dbsp/
    pub fn from_unsigned_encoding(encoding: u128) -> Option<Self> {
        if encoding < Self::UNSIGNED_ZERO {
            Some(Self(-(Self::UNSIGNED_ZERO - encoding).cast_signed()))
        } else if encoding <= Self::UNSIGNED_MAX {
            Some(Self((encoding - Self::UNSIGNED_ZERO).cast_signed()))
        } else {
            None
        }
    }
}

#[cfg(test)]
mod test {
    use std::str::FromStr;

    use num_traits::{CheckedAdd, CheckedDiv, CheckedMul, CheckedSub};

    use crate::Fixed;

    type F = Fixed<10, 2>;
    fn f(n: f64) -> F {
        Fixed::try_from(n).unwrap()
    }

    fn f38_0(n: f64) -> Fixed<38, 0> {
        Fixed::try_from(n).unwrap()
    }

    fn f38_38(s: &str) -> Fixed<38, 38> {
        Fixed::from_str(s).unwrap()
    }

    #[test]
    fn mul() {
        // A few specific handwritten cases.
        assert_eq!(f(1.23) * f(2.34), f(2.87));
        assert_eq!(f(-1.23) * f(2.34), f(-2.87));
        assert_eq!(f(1.23) * f(-2.34), f(-2.87));
        assert_eq!(f(-1.23) * f(-2.34), f(2.87));

        // General case.
        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<10, 2> = Fixed(b);
                assert_eq!(af * bf, Fixed::<10, 2>(a * b / 100));
            }
        }

        // General case with overflow.
        for a in -999..=999 {
            let af: Fixed<3, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<3, 2> = Fixed(b);
                let c = a * b / 100;
                let expected = (c.unsigned_abs() < 1000).then_some(Fixed(c));
                assert_eq!(af.checked_mul(&bf), expected);
            }
        }
    }

    #[test]
    fn mul_generic() {
        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<10, 3> = Fixed(b);
                let cf: Fixed<10, 5> = af.checked_mul_generic(bf).unwrap();
                assert_eq!(cf, Fixed::<10, 5>(a * b));
                let df: Fixed<10, 6> = af.checked_mul_generic(bf).unwrap();
                assert_eq!(df, Fixed::<10, 6>(a * b * 10));
                let ef: Fixed<10, 0> = af.checked_mul_generic(bf).unwrap();
                assert_eq!(ef, Fixed::<10, 0>(a * b / 100_000));
            }
        }
    }

    #[test]
    fn div() {
        // A few specific handwritten cases.
        assert_eq!(f(1.23) / f(2.34), f(0.52));
        assert_eq!(f(-1.23) / f(2.34), f(-0.52));
        assert_eq!(f(1.23) / f(-2.34), f(-0.52));
        assert_eq!(f(-1.23) / f(-2.34), f(0.52));
        assert_eq!(
            f38_0(1.0)
                .checked_div_generic::<38, 0, 38, 38>(f38_0(7.0))
                .unwrap(),
            f38_38("0.14285714285714285714285714285714285714")
        );

        assert_eq!(
            f38_0(123.0).checked_div_generic::<38, 38, 38, 38>(f38_38("0.456")),
            None
        );

        // General case.
        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<10, 2> = Fixed(b);
                assert_eq!(af.checked_div(&bf), (b != 0).then(|| Fixed(a * 100 / b)));
            }
        }

        // General case with overflow.
        for a in -999..=999 {
            let af: Fixed<3, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<3, 2> = Fixed(b);
                let expected = if b != 0 {
                    let result = a * 100 / b;
                    (result.unsigned_abs() <= 999).then_some(Fixed(result))
                } else {
                    None
                };
                assert_eq!(af.checked_div(&bf), expected);
            }
        }
    }

    #[test]
    fn div_generic() {
        fn test<const P: usize, const S: usize>(a: i128, af: Fixed<P, S>) {
            for b in -999..=999 {
                if b != 0 {
                    let bf: Fixed<10, 3> = Fixed(b);
                    let cf: Fixed<10, 5> = af.checked_div_generic(bf).unwrap();
                    assert_eq!(cf, Fixed::<10, 5>(a * 1_000_000 / b));
                    let df: Fixed<10, 6> = af.checked_div_generic(bf).unwrap();
                    assert_eq!(df, Fixed::<10, 6>(a * 10_000_000 / b));
                    let ef: Fixed<10, 0> = af.checked_div_generic(bf).unwrap();
                    assert_eq!(ef, Fixed::<10, 0>(a * 10 / b));
                }
            }
        }

        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            test(a, af);
            let af2: Fixed<18, 10> = af.convert().unwrap();
            test(a, af2);
        }
    }

    #[test]
    fn add() {
        // A few specific handwritten cases.
        assert_eq!(f(1.23) + f(2.34), f(3.57));
        assert_eq!(f(-1.23) + f(2.34), f(1.11));
        assert_eq!(f(1.23) + f(-2.34), f(-1.11));
        assert_eq!(f(-1.23) + f(-2.34), f(-3.57));

        // Cases that require avoiding intermediate overflow.
        let af: Fixed<38, 35> = Fixed::try_from(999).unwrap();
        let bf: Fixed<37, 34> = Fixed::try_from(999).unwrap();
        let cf: Fixed<36, 30> = af.checked_add_generic(bf).unwrap();
        assert_eq!(cf, 1998);
        let df: Fixed<36, 30> = bf.checked_add_generic(af).unwrap();
        assert_eq!(df, 1998);
        let ef: Fixed<36, 3> = af.checked_add_generic(af).unwrap();
        assert_eq!(ef, 1998);
        let ff: Option<Fixed<38, 35>> = af.checked_add_generic(af);
        assert_eq!(ff, None);

        // General case.
        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<10, 2> = Fixed(b);
                assert_eq!(af + bf, Fixed::<10, 2>(a + b));
            }
        }

        // General case with overflow.
        for a in -999..=999 {
            let af: Fixed<3, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<3, 2> = Fixed(b);
                let c = a + b;
                let expected = (c.unsigned_abs() < 1000).then_some(Fixed(c));
                assert_eq!(af.checked_add(&bf), expected);
            }
        }

        // General case with type conversion.
        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<10, 3> = Fixed(b);
                let cf: Fixed<10, 5> = af.checked_add_generic(bf).unwrap();
                assert_eq!(
                    cf,
                    Fixed::<10, 5>(a * 1000 + b * 100),
                    "{af} + {bf} ?= {cf}"
                );
                let cf: Fixed<10, 5> = bf.checked_add_generic(af).unwrap();
                assert_eq!(
                    cf,
                    Fixed::<10, 5>(a * 1000 + b * 100),
                    "{bf} + {af} ?= {cf}"
                );
                let df: Fixed<10, 6> = af.checked_add_generic(bf).unwrap();
                assert_eq!(
                    df,
                    Fixed::<10, 6>(a * 10_000 + b * 1000),
                    "{af} + {bf} ?= {df}"
                );
                let ef: Fixed<10, 0> = af.checked_add_generic(bf).unwrap();
                assert_eq!(
                    ef,
                    Fixed::<10, 0>((a * 10 + b) / 1000),
                    "{af} + {bf} ?= {ef}"
                );

                let ff: Fixed<10, 2> = Fixed(b);
                let gf: Fixed<10, 1> = af.checked_add_generic(ff).unwrap();
                assert_eq!(gf, Fixed::<10, 1>((a + b) / 10), "{af} + {ff} ?= {gf}");
                let hf: Fixed<10, 3> = af.checked_add_generic(ff).unwrap();
                assert_eq!(hf, Fixed::<10, 3>((a + b) * 10), "{af} + {ff} ?= {hf}");
            }
        }
    }

    #[test]
    fn sub() {
        // A few specific handwritten cases.
        assert_eq!(f(1.23) - f(2.34), f(-1.11));
        assert_eq!(f(-1.23) - f(2.34), f(-3.57));
        assert_eq!(f(1.23) - f(-2.34), f(3.57));
        assert_eq!(f(-1.23) - f(-2.34), f(1.11));

        // General case.
        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<10, 2> = Fixed(b);
                assert_eq!(af - bf, Fixed::<10, 2>(a - b));
            }
        }

        // General case with overflow.
        for a in -999..=999 {
            let af: Fixed<3, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<3, 2> = Fixed(b);
                let c = a - b;
                let expected = (c.unsigned_abs() < 1000).then_some(Fixed(c));
                assert_eq!(af.checked_sub(&bf), expected);
            }
        }

        // General case with type conversion.
        for a in -999..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            for b in -999..=999 {
                let bf: Fixed<10, 3> = Fixed(b);
                let cf: Fixed<10, 5> = af.checked_sub_generic(bf).unwrap();
                assert_eq!(
                    cf,
                    Fixed::<10, 5>(a * 1000 - b * 100),
                    "{af} - {bf} ?= {cf}"
                );
                let cf: Fixed<10, 5> = bf.checked_sub_generic(af).unwrap();
                assert_eq!(
                    cf,
                    Fixed::<10, 5>(b * 100 - a * 1000),
                    "{bf} - {af} ?= {cf}"
                );
                let df: Fixed<10, 6> = af.checked_sub_generic(bf).unwrap();
                assert_eq!(df, Fixed::<10, 6>(a * 10_000 - b * 1000));
            }
        }
    }

    #[test]
    fn powi() {
        assert_eq!(
            Fixed::<10, 8>::from_str("1.12345678")
                .unwrap()
                .powi(8)
                .to_string()
                .as_str(),
            "2.53776238"
        );
        assert_eq!(f(2.0).powi(3), f(8.0));
        assert_eq!(f(-2.0).powi(3), f(-8.0));
        assert_eq!(f(1.7).powi(8), f(69.75));
        assert_eq!(f(1.7).powi(-8), f(0.01));
        assert_eq!(f(0.0).powi(1), f(0.0));
        assert_eq!(f(0.0).checked_powi(0), None);
        assert_eq!(f(0.0).checked_powi(-1), None);
    }

    #[test]
    fn convert() {
        let a = Fixed::<10, 10>::from_str("0.0123456789").unwrap();
        assert_eq!(&a.convert::<10, 0>().unwrap().to_string(), "0");
        assert_eq!(&a.convert::<10, 1>().unwrap().to_string(), "0");
        assert_eq!(&a.convert::<10, 2>().unwrap().to_string(), "0.01");
        assert_eq!(&a.convert::<10, 3>().unwrap().to_string(), "0.012");
        assert_eq!(&a.convert::<10, 4>().unwrap().to_string(), "0.0123");
        assert_eq!(&a.convert::<10, 5>().unwrap().to_string(), "0.01234");
        assert_eq!(&a.convert::<10, 6>().unwrap().to_string(), "0.012345");
        assert_eq!(&a.convert::<10, 7>().unwrap().to_string(), "0.0123456");
        assert_eq!(&a.convert::<10, 8>().unwrap().to_string(), "0.01234567");
        assert_eq!(&a.convert::<10, 9>().unwrap().to_string(), "0.012345678");
        assert_eq!(&a.convert::<10, 10>().unwrap().to_string(), "0.0123456789");
        assert_eq!(&a.convert_round_even::<10, 0>().unwrap().to_string(), "0");
        assert_eq!(&a.convert_round_even::<10, 1>().unwrap().to_string(), "0");
        assert_eq!(
            &a.convert_round_even::<10, 2>().unwrap().to_string(),
            "0.01"
        );
        assert_eq!(
            &a.convert_round_even::<10, 3>().unwrap().to_string(),
            "0.012"
        );
        assert_eq!(
            &a.convert_round_even::<10, 4>().unwrap().to_string(),
            "0.0123"
        );
        assert_eq!(
            &a.convert_round_even::<10, 5>().unwrap().to_string(),
            "0.01235"
        );
        assert_eq!(
            &a.convert_round_even::<10, 6>().unwrap().to_string(),
            "0.012346"
        );
        assert_eq!(
            &a.convert_round_even::<10, 7>().unwrap().to_string(),
            "0.0123457"
        );
        assert_eq!(
            &a.convert_round_even::<10, 8>().unwrap().to_string(),
            "0.01234568"
        );
        assert_eq!(
            &a.convert_round_even::<10, 9>().unwrap().to_string(),
            "0.012345679"
        );
        assert_eq!(
            &a.convert_round_even::<10, 10>().unwrap().to_string(),
            "0.0123456789"
        );

        let b = Fixed::<10, 5>::from_str("12345.67895").unwrap();
        assert_eq!(&b.convert::<10, 0>().unwrap().to_string(), "12345");
        assert_eq!(&b.convert::<10, 1>().unwrap().to_string(), "12345.6");
        assert_eq!(&b.convert::<10, 2>().unwrap().to_string(), "12345.67");
        assert_eq!(&b.convert::<10, 3>().unwrap().to_string(), "12345.678");
        assert_eq!(&b.convert::<10, 4>().unwrap().to_string(), "12345.6789");
        assert_eq!(&b.convert::<10, 5>().unwrap().to_string(), "12345.67895");
        assert_eq!(b.convert::<10, 6>(), None);
        assert_eq!(b.convert::<10, 7>(), None);
        assert_eq!(b.convert::<10, 8>(), None);
        assert_eq!(b.convert::<10, 9>(), None);
        assert_eq!(b.convert::<10, 10>(), None);
        assert_eq!(
            &b.convert_round_even::<10, 0>().unwrap().to_string(),
            "12346"
        );
        assert_eq!(
            &b.convert_round_even::<10, 1>().unwrap().to_string(),
            "12345.7"
        );
        assert_eq!(
            &b.convert_round_even::<10, 2>().unwrap().to_string(),
            "12345.68"
        );
        assert_eq!(
            &b.convert_round_even::<10, 3>().unwrap().to_string(),
            "12345.679"
        );
        assert_eq!(
            &b.convert_round_even::<10, 4>().unwrap().to_string(),
            "12345.679"
        );
        assert_eq!(
            &b.convert_round_even::<10, 5>().unwrap().to_string(),
            "12345.67895"
        );
        assert_eq!(b.convert_round_even::<10, 6>(), None);
        assert_eq!(b.convert_round_even::<10, 7>(), None);
        assert_eq!(b.convert_round_even::<10, 8>(), None);
        assert_eq!(b.convert_round_even::<10, 9>(), None);
        assert_eq!(b.convert_round_even::<10, 10>(), None);
    }

    #[test]
    fn constants() {
        assert_eq!(Fixed::<5, 0>::MAX, Fixed::<5, 0>(99999));
        assert_eq!(Fixed::<5, 0>::MIN, Fixed::<5, 0>(-99999));
        assert_eq!(Fixed::<5, 0>::ZERO, Fixed::<5, 0>(0));
        assert_eq!(Fixed::<5, 0>::ONE, Fixed::<5, 0>(1));

        assert_eq!(Fixed::<5, 1>::MAX, Fixed::<5, 1>(99999));
        assert_eq!(Fixed::<5, 1>::MIN, Fixed::<5, 1>(-99999));
        assert_eq!(Fixed::<5, 1>::ZERO, Fixed::<5, 1>(0));
        assert_eq!(Fixed::<5, 1>::ONE, Fixed::<5, 1>(10));

        assert_eq!(Fixed::<5, 2>::MAX, Fixed::<5, 2>(99999));
        assert_eq!(Fixed::<5, 2>::MIN, Fixed::<5, 2>(-99999));
        assert_eq!(Fixed::<5, 2>::ZERO, Fixed::<5, 2>(0));
        assert_eq!(Fixed::<5, 2>::ONE, Fixed::<5, 2>(100));

        assert_eq!(Fixed::<5, 3>::MAX, Fixed::<5, 3>(99999));
        assert_eq!(Fixed::<5, 3>::MIN, Fixed::<5, 3>(-99999));
        assert_eq!(Fixed::<5, 3>::ZERO, Fixed::<5, 3>(0));
        assert_eq!(Fixed::<5, 3>::ONE, Fixed::<5, 3>(1000));

        assert_eq!(Fixed::<5, 4>::MAX, Fixed::<5, 4>(99999));
        assert_eq!(Fixed::<5, 4>::MIN, Fixed::<5, 4>(-99999));
        assert_eq!(Fixed::<5, 4>::ZERO, Fixed::<5, 4>(0));
        assert_eq!(Fixed::<5, 4>::ONE, Fixed::<5, 4>(10000));

        assert_eq!(Fixed::<5, 5>::MAX, Fixed::<5, 5>(99999));
        assert_eq!(Fixed::<5, 5>::MIN, Fixed::<5, 5>(-99999));
        assert_eq!(Fixed::<5, 5>::ZERO, Fixed::<5, 5>(0));
        // This would panic at compile time.  See [super::_invalid_constant_test].
        //let _ = Fixed::<5, 5>::ONE;
    }

    #[test]
    fn floor() {
        assert_eq!(f(5.0).floor(), f(5.0));
        assert_eq!(f(5.1).floor(), f(5.0));
        assert_eq!(f(5.5).floor(), f(5.0));
        assert_eq!(f(5.9).floor(), f(5.0));
        assert_eq!(f(-5.0).floor(), f(-5.0));
        assert_eq!(f(-5.1).floor(), f(-6.0));
        assert_eq!(f(-5.5).floor(), f(-6.0));
        assert_eq!(f(-5.6).floor(), f(-6.0));
        assert_eq!(f(4.0).floor(), f(4.0));
        assert_eq!(f(4.1).floor(), f(4.0));
        assert_eq!(f(4.5).floor(), f(4.0));
        assert_eq!(f(4.9).floor(), f(4.0));
        assert_eq!(f(-4.0).floor(), f(-4.0));
        assert_eq!(f(-4.1).floor(), f(-5.0));
        assert_eq!(f(-4.5).floor(), f(-5.0));
        assert_eq!(f(-4.6).floor(), f(-5.0));
        assert_eq!(f(-99_999_999.0).floor(), f(-99_999_999.0));
        assert_eq!(f(-99_999_999.1).checked_floor(), None);
        assert_eq!(f(-99_999_999.5).checked_floor(), None);
        assert_eq!(f(-99_999_999.6).checked_floor(), None);
    }

    #[test]
    fn ceil() {
        assert_eq!(f(5.0).ceil(), f(5.0));
        assert_eq!(f(5.1).ceil(), f(6.0));
        assert_eq!(f(5.5).ceil(), f(6.0));
        assert_eq!(f(5.9).ceil(), f(6.0));
        assert_eq!(f(-5.0).ceil(), f(-5.0));
        assert_eq!(f(-5.1).ceil(), f(-5.0));
        assert_eq!(f(-5.5).ceil(), f(-5.0));
        assert_eq!(f(-5.6).ceil(), f(-5.0));
        assert_eq!(f(4.0).ceil(), f(4.0));
        assert_eq!(f(4.1).ceil(), f(5.0));
        assert_eq!(f(4.5).ceil(), f(5.0));
        assert_eq!(f(4.9).ceil(), f(5.0));
        assert_eq!(f(-4.0).ceil(), f(-4.0));
        assert_eq!(f(-4.1).ceil(), f(-4.0));
        assert_eq!(f(-4.5).ceil(), f(-4.0));
        assert_eq!(f(-4.6).ceil(), f(-4.0));
        assert_eq!(f(99_999_999.0).ceil(), f(99_999_999.0));
        assert_eq!(f(99_999_999.1).checked_ceil(), None);
        assert_eq!(f(99_999_999.5).checked_ceil(), None);
        assert_eq!(f(99_999_999.6).checked_ceil(), None);
    }

    #[test]
    fn trunc() {
        fn test(x: f64, expected: f64) {
            assert_eq!(f(x).trunc(), f(expected));
            assert_eq!(f(x).trunc_digits(0), f(expected));
        }

        test(5.0, 5.0);
        test(5.1, 5.0);
        test(5.5, 5.0);
        test(5.9, 5.0);
        test(-5.0, -5.0);
        test(-5.1, -5.0);
        test(-5.5, -5.0);
        test(-5.6, -5.0);
        test(4.0, 4.0);
        test(4.1, 4.0);
        test(4.5, 4.0);
        test(4.9, 4.0);
        test(-4.0, -4.0);
        test(-4.1, -4.0);
        test(-4.5, -4.0);
        test(-4.6, -4.0);
        test(99_999_999.0, 99_999_999.0);
        test(99_999_999.1, 99_999_999.0);
        test(99_999_999.5, 99_999_999.0);
        test(99_999_999.6, 99_999_999.0);
        test(-99_999_999.0, -99_999_999.0);
        test(-99_999_999.1, -99_999_999.0);
        test(-99_999_999.5, -99_999_999.0);
        test(-99_999_999.6, -99_999_999.0);
    }

    #[test]
    fn round() {
        fn test(x: f64, expected: Option<f64>) {
            assert_eq!(f(x).checked_round(0), expected.map(f));
        }
        type F = Fixed<10, 2>;
        fn test1((x, s): (i128, i32), d: i32, expected: Option<(i128, i32)>) {
            assert_eq!(
                F::new(x, s).unwrap().checked_round(d),
                expected.map(|x| F::new(x.0, x.1).unwrap())
            );
        }

        // round(5.1 , 2) = 5.1
        test1((51, 1), 2, Some((51, 1)));
        // round(5.1, 1) = 5.1
        test1((51, 1), 1, Some((51, 1)));
        // round(5.1, 0) = 5.0
        test1((51, 1), 0, Some((50, 1)));
        // round(5.1, -1) = 10
        test1((51, 1), -1, Some((10, 0)));

        // round(2.1, 2) = 2.1
        test1((21, 1), 2, Some((21, 1)));
        // round(2.1, 1) = 2.1
        test1((21, 1), 1, Some((21, 1)));
        // round(2.1, 0) = 2.0
        test1((21, 1), 0, Some((20, 1)));
        // round(2.1, -1) = 0
        test1((21, 1), -1, Some((0, 0)));

        // round(99_999_999.1, 2) = 99_999_999.1
        test1((999_999_991, 1), 2, Some((999_999_991, 1)));
        // round(99_999_999.1, 1) = 99_999_999.1
        test1((999_999_991, 1), 1, Some((999_999_991, 1)));
        // round(99_999_999.1, 0) = 99_999_999.0
        test1((999_999_991, 1), 0, Some((999_999_990, 1)));
        // round(99_999_999.1, -1) = None
        test1((999_999_991, 1), -1, None);

        test(5.0, Some(5.0));
        test(5.1, Some(5.0));
        test(5.5, Some(6.0));
        test(5.9, Some(6.0));
        test(-5.0, Some(-5.0));
        test(-5.1, Some(-5.0));
        test(-5.5, Some(-6.0));
        test(-5.6, Some(-6.0));
        test(4.0, Some(4.0));
        test(4.1, Some(4.0));
        test(4.5, Some(5.0));
        test(4.9, Some(5.0));
        test(-4.0, Some(-4.0));
        test(-4.1, Some(-4.0));
        test(-4.5, Some(-5.0));
        test(-4.6, Some(-5.0));
        test(99_999_999.0, Some(99_999_999.0));
        test(99_999_999.1, Some(99_999_999.0));
        test(99_999_999.5, None);
        test(99_999_999.6, None);
        test(-99_999_999.0, Some(-99_999_999.0));
        test(-99_999_999.1, Some(-99_999_999.0));
        test(-99_999_999.5, None);
        test(-99_999_999.6, None);
    }

    #[test]
    fn trunc_digits() {
        let x = Fixed::<10, 4>(245368746);
        assert_eq!(x.trunc_digits(5).to_string(), "24536.8746");
        assert_eq!(x.trunc_digits(4).to_string(), "24536.8746");
        assert_eq!(x.trunc_digits(3).to_string(), "24536.874");
        assert_eq!(x.trunc_digits(2).to_string(), "24536.87");
        assert_eq!(x.trunc_digits(1).to_string(), "24536.8");
        assert_eq!(x.trunc_digits(0).to_string(), "24536");
        assert_eq!(x.trunc_digits(-1).to_string(), "24530");
        assert_eq!(x.trunc_digits(-2).to_string(), "24500");
        assert_eq!(x.trunc_digits(-3).to_string(), "24000");
        assert_eq!(x.trunc_digits(-4).to_string(), "20000");
        assert_eq!(x.trunc_digits(-5).to_string(), "0");
        assert_eq!(x.trunc_digits(-50).to_string(), "0");

        let x = -x;
        assert_eq!(x.trunc_digits(5).to_string(), "-24536.8746");
        assert_eq!(x.trunc_digits(4).to_string(), "-24536.8746");
        assert_eq!(x.trunc_digits(3).to_string(), "-24536.874");
        assert_eq!(x.trunc_digits(2).to_string(), "-24536.87");
        assert_eq!(x.trunc_digits(1).to_string(), "-24536.8");
        assert_eq!(x.trunc_digits(0).to_string(), "-24536");
        assert_eq!(x.trunc_digits(-1).to_string(), "-24530");
        assert_eq!(x.trunc_digits(-2).to_string(), "-24500");
        assert_eq!(x.trunc_digits(-3).to_string(), "-24000");
        assert_eq!(x.trunc_digits(-4).to_string(), "-20000");
        assert_eq!(x.trunc_digits(-5).to_string(), "0");
        assert_eq!(x.trunc_digits(-50).to_string(), "0");
    }

    #[test]
    fn sign() {
        assert_eq!(f(-0.1).sign(), Fixed::<1, 0>::try_from(-1).unwrap());
        assert_eq!(f(0.0).sign(), Fixed::<1, 0>::try_from(0).unwrap());
        assert_eq!(f(0.5).sign(), Fixed::<1, 0>::try_from(1).unwrap());
    }

    #[test]
    fn sqrt() {
        // A few selected values.
        assert_eq!(f(0.0).sqrt(), f(0.0));
        assert_eq!(f(1.0).sqrt(), f(1.0));
        assert_eq!(f(2.0).sqrt(), f(1.41));
        assert_eq!(f(3.0).sqrt(), f(1.73));
        assert_eq!(f(4.0).sqrt(), f(2.0));
        assert_eq!(f(-1.0).checked_sqrt(), None);

        // General case.
        for a in 0..=999 {
            let af: Fixed<10, 2> = Fixed(a);
            assert_eq!(af.sqrt(), Fixed::<10, 2>((a * 100).isqrt()));
        }
    }

    #[test]
    fn nullable() {
        /// Adds `a` and `b` and returns the sum.  Return `None` if `a` or `b`
        /// is `None` or if their sum is out of range.
        fn nullable_checked_add_generic<
            const PA: usize,
            const SA: usize,
            const PB: usize,
            const SB: usize,
            const PC: usize,
            const SC: usize,
        >(
            a: Option<Fixed<PA, SA>>,
            b: Option<Fixed<PB, SB>>,
        ) -> Option<Fixed<PC, SC>> {
            a.zip(b).and_then(|(a, b)| a.checked_add_generic(b))
        }

        let a: Option<Fixed<10, 2>> = Some("1.23".parse().unwrap());
        let b: Option<Fixed<5, 4>> = Some("4.5678".parse().unwrap());
        let c: Option<Fixed<10, 4>> = nullable_checked_add_generic(a, b);
        assert_eq!(c, Some("5.7978".parse().unwrap()));
    }

    #[test]
    fn to_integer() {
        for x in -9999..=9999 {
            let f = Fixed::<4, 1>(x);
            assert_eq!(i128::from(f), x / 10);
            assert_eq!(i64::try_from(f).unwrap(), (x / 10) as i64);
            assert_eq!(i32::try_from(f).unwrap(), (x / 10) as i32);
            assert_eq!(i16::try_from(f).unwrap(), (x / 10) as i16);
            assert_eq!(
                i8::try_from(f).ok(),
                (-1289..=1279).contains(&x).then_some((x / 10) as i8)
            );
            assert_eq!(
                u128::try_from(f).ok(),
                (x > -10).then_some((x / 10) as u128)
            );
            assert_eq!(u64::try_from(f).ok(), (x > -10).then_some((x / 10) as u64));
            assert_eq!(u32::try_from(f).ok(), (x > -10).then_some((x / 10) as u32));
            assert_eq!(u16::try_from(f).ok(), (x > -10).then_some((x / 10) as u16));
            assert_eq!(
                u8::try_from(f).ok(),
                (-9..=2559).contains(&x).then_some((x / 10) as u8)
            );
        }
    }

    #[test]
    fn compare_against_fixed() {
        fn check_comparisons<const PA: usize, const SA: usize, const PB: usize, const SB: usize>(
            fx: Fixed<PA, SA>,
            fy: Fixed<PB, SB>,
            x: i128,
            y: i128,
        ) {
            assert_eq!(fx == fy, x == y);
            assert_eq!(fx != fy, x != y);
            assert_eq!(fx > fy, x > y);
            assert_eq!(fx >= fy, x >= y);
            assert_eq!(fx < fy, x < y);
            assert_eq!(fx <= fy, x <= y);
        }

        for x in -999..=999 {
            let fx = Fixed::<3, 1>(x);
            for y in -999..=999 {
                check_comparisons(fx, Fixed::<3, 0>(y), x, y * 10);
                check_comparisons(fx, Fixed::<3, 1>(y), x, y);
                check_comparisons(fx, Fixed::<3, 2>(y), x * 10, y);
            }
        }
    }

    #[test]
    fn compare_against_integers() {
        for x in -999..=999 {
            let f = Fixed::<3, 1>(x);
            for y in -100..=100 {
                let expect = x == y * 10;
                assert_eq!(f == y as i8, expect);
                assert_eq!(f == y as i16, expect);
                assert_eq!(f == y as i32, expect);
                assert_eq!(f == y as i64, expect);
                assert_eq!(f == y, expect);
                assert_eq!(f == y as isize, expect);
                if y >= 0 {
                    assert_eq!(f == y as u8, expect);
                    assert_eq!(f == y as u16, expect);
                    assert_eq!(f == y as u32, expect);
                    assert_eq!(f == y as u64, expect);
                    assert_eq!(f == y as u128, expect);
                    assert_eq!(f == y as usize, expect);
                }
            }
        }
    }

    #[test]
    fn unsigned_encoding() {
        type F = Fixed<3, 1>;
        for x in -999..=999 {
            let f = Fixed::<3, 1>(x);
            assert_eq!(F::from_unsigned_encoding(f.to_unsigned_encoding()), Some(f));
        }
        assert_eq!(F::MIN.to_unsigned_encoding(), 0);
        assert_eq!(F::ZERO.to_unsigned_encoding(), 999);
        assert_eq!(F::MAX.to_unsigned_encoding(), 999 * 2);
        assert_eq!(F::from_unsigned_encoding(0), Some(F::MIN));
        assert_eq!(F::from_unsigned_encoding(999), Some(F::ZERO));
        assert_eq!(F::from_unsigned_encoding(999 * 2), Some(F::MAX));
        assert_eq!(F::from_unsigned_encoding(999 * 2 + 1), None);
    }

    #[test]
    fn new() {
        type F1 = Fixed<11, 1>;
        assert_eq!(F1::new(435, -8), None);
        assert_eq!(F1::new(435, -7).unwrap().to_string(), "4350000000");
        assert_eq!(F1::new(435, -6).unwrap().to_string(), "435000000");
        assert_eq!(F1::new(435, -5).unwrap().to_string(), "43500000");
        assert_eq!(F1::new(435, -4).unwrap().to_string(), "4350000");
        assert_eq!(F1::new(435, -3).unwrap().to_string(), "435000");
        assert_eq!(F1::new(435, -2).unwrap().to_string(), "43500");
        assert_eq!(F1::new(435, -1).unwrap().to_string(), "4350");
        assert_eq!(F1::new(435, 0).unwrap().to_string(), "435");
        assert_eq!(F1::new(435, 1).unwrap().to_string(), "43.5");
        assert_eq!(F1::new(435, 2).unwrap().to_string(), "4.3");
        assert_eq!(F1::new(435, 3).unwrap().to_string(), "0.4");
        assert_eq!(F1::new(435, 4).unwrap().to_string(), "0");

        type F2 = Fixed<11, 2>;
        assert_eq!(F2::new(435, -7), None);
        assert_eq!(F2::new(435, -6).unwrap().to_string(), "435000000");
        assert_eq!(F2::new(435, -5).unwrap().to_string(), "43500000");
        assert_eq!(F2::new(435, -4).unwrap().to_string(), "4350000");
        assert_eq!(F2::new(435, -3).unwrap().to_string(), "435000");
        assert_eq!(F2::new(435, -2).unwrap().to_string(), "43500");
        assert_eq!(F2::new(435, -1).unwrap().to_string(), "4350");
        assert_eq!(F2::new(435, 0).unwrap().to_string(), "435");
        assert_eq!(F2::new(435, 1).unwrap().to_string(), "43.5");
        assert_eq!(F2::new(435, 2).unwrap().to_string(), "4.35");
        assert_eq!(F2::new(435, 3).unwrap().to_string(), "0.43");
        assert_eq!(F2::new(435, 4).unwrap().to_string(), "0.04");
        assert_eq!(F2::new(435, 5).unwrap().to_string(), "0");
    }

    #[test]
    fn new_round_even() {
        type F1 = Fixed<11, 1>;
        assert_eq!(F1::new_round_even(435, -8), None);
        assert_eq!(
            F1::new_round_even(435, -7).unwrap().to_string(),
            "4350000000"
        );
        assert_eq!(
            F1::new_round_even(435, -6).unwrap().to_string(),
            "435000000"
        );
        assert_eq!(F1::new_round_even(435, -5).unwrap().to_string(), "43500000");
        assert_eq!(F1::new_round_even(435, -4).unwrap().to_string(), "4350000");
        assert_eq!(F1::new_round_even(435, -3).unwrap().to_string(), "435000");
        assert_eq!(F1::new_round_even(435, -2).unwrap().to_string(), "43500");
        assert_eq!(F1::new_round_even(435, -1).unwrap().to_string(), "4350");
        assert_eq!(F1::new_round_even(435, 0).unwrap().to_string(), "435");
        assert_eq!(F1::new_round_even(435, 1).unwrap().to_string(), "43.5");
        assert_eq!(F1::new_round_even(435, 2).unwrap().to_string(), "4.4");
        assert_eq!(F1::new_round_even(435, 3).unwrap().to_string(), "0.4");
        assert_eq!(F1::new_round_even(435, 4).unwrap().to_string(), "0");

        type F2 = Fixed<11, 2>;
        assert_eq!(F2::new_round_even(435, -7), None);
        assert_eq!(
            F2::new_round_even(435, -6).unwrap().to_string(),
            "435000000"
        );
        assert_eq!(F2::new_round_even(435, -5).unwrap().to_string(), "43500000");
        assert_eq!(F2::new_round_even(435, -4).unwrap().to_string(), "4350000");
        assert_eq!(F2::new_round_even(435, -3).unwrap().to_string(), "435000");
        assert_eq!(F2::new_round_even(435, -2).unwrap().to_string(), "43500");
        assert_eq!(F2::new_round_even(435, -1).unwrap().to_string(), "4350");
        assert_eq!(F2::new_round_even(435, 0).unwrap().to_string(), "435");
        assert_eq!(F2::new_round_even(435, 1).unwrap().to_string(), "43.5");
        assert_eq!(F2::new_round_even(435, 2).unwrap().to_string(), "4.35");
        assert_eq!(F2::new_round_even(435, 3).unwrap().to_string(), "0.44");
        assert_eq!(F2::new_round_even(435, 4).unwrap().to_string(), "0.04");
        assert_eq!(F2::new_round_even(435, 5).unwrap().to_string(), "0");
    }
}