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use crate::double::{DenormDouble, NormDouble, SemiDouble};
use crate::generic::{reduce_pi_2, ReducePi2};
use crate::traits::{Float, Int as _, Like};
pub(crate) trait SinCos<L = Like<Self>>: Float {
fn frac_1_6_ex() -> SemiDouble<Self>;
/// Calculates `(sin(x) - x - x^3 * K3, K3)`
///
/// Where:
/// * `x2 = x^2`
/// * `x5 = x^5`
/// * `K3 ~= -1/6`
fn sin_poly(x2: Self, x5: Self) -> (Self, Self);
/// Calculates `sin(x) - x + x^3 * 1/6`
///
/// Where:
/// * `x2 = x^2`
/// * `x5 = x^5`
fn sin_poly_ex(x2: Self, x5: Self) -> Self;
/// Calculates `cos(x) + 0.5 * x^2 - 1`
///
/// Where:
/// * `x2 = x^2`
/// * `x4 = x^4`
fn cos_poly(x2: Self, x4: Self) -> Self;
}
pub(crate) fn sin<F: SinCos + ReducePi2>(x: F) -> F {
let e = x.raw_exp();
if e == F::MAX_RAW_EXP {
// sin(inf or nan) = nan
F::NAN
} else if e == F::RawExp::ZERO {
// subnormal or zero, sin(x) ~= x
// also handles sin(-0) = -0
x
} else {
let (n, y_hi, y_lo) = reduce_pi_2(x);
match n {
0 => sin_inner(y_hi, y_lo),
1 => cos_inner(y_hi, y_lo),
2 => -sin_inner(y_hi, y_lo),
3 => -cos_inner(y_hi, y_lo),
_ => unreachable!(),
}
}
}
pub(crate) fn cos<F: SinCos + ReducePi2>(x: F) -> F {
let e = x.raw_exp();
if e == F::MAX_RAW_EXP {
// cos(inf or nan) = nan
F::NAN
} else if e == F::RawExp::ZERO {
// subnormal or zero, cos(x) ~= 1
F::one()
} else {
let (n, y_hi, y_lo) = reduce_pi_2(x);
match n {
0 => cos_inner(y_hi, y_lo),
1 => -sin_inner(y_hi, y_lo),
2 => -cos_inner(y_hi, y_lo),
3 => sin_inner(y_hi, y_lo),
_ => unreachable!(),
}
}
}
pub(crate) fn sin_cos<F: SinCos + ReducePi2>(x: F) -> (F, F) {
let e = x.raw_exp();
if e == F::MAX_RAW_EXP {
// sin(inf or nan) = nan
// cos(inf or nan) = nan
(F::NAN, F::NAN)
} else if e == F::RawExp::ZERO {
// subnormal or zero
// sin(x) ~= x
// cos(x) ~= 1
// also handles sin(-0) = -0
(x, F::one())
} else {
let (n, y_hi, y_lo) = reduce_pi_2(x);
let sin = sin_inner(y_hi, y_lo);
let cos = cos_inner(y_hi, y_lo);
match n {
0 => (sin, cos),
1 => (cos, -sin),
2 => (-sin, -cos),
3 => (-cos, sin),
_ => unreachable!(),
}
}
}
/// Calculates `sin(x_hi + x_lo)`, where
/// `x_lo` is very small and `|x_hi| <= π/4`
pub(super) fn sin_inner<F: SinCos>(x_hi: F, x_lo: F) -> F {
// sin(x_hi + x_lo) = sin(x_hi) * cos(x_lo) + cos(x_hi) * sin(x_lo)
// x_lo is small, so sin(x_lo) ~= x_lo and cos(x_lo) ~= 1,
// then sin(x_hi + x_lo) ~= sin(x_hi) + cos(x_hi) * x_lo
//
// sin(x) is calculated with a polynomial.
// cos(x) * x_lo ~= (1 - 0.5*x^2) * x_lo
let x2 = x_hi * x_hi;
let x3 = x2 * x_hi;
let x5 = x3 * x2;
// t1 = sin(x_hi) - x_hi - x_hi^3 * k3
let (t1, k3) = F::sin_poly(x2, x5);
// sin(x_hi + x_lo) ~= sin(x) + (1 - 0.5 * x^2) * x_lo
// = x + t1 + x^3 * k3 + (1 - 0.5 * x^2) * x_lo
x_hi + (x3 * k3 + (t1 + (x_lo - F::half() * x2 * x_lo)))
}
pub(super) fn hi_lo_sin_inner<F: SinCos>(x: NormDouble<F>) -> DenormDouble<F> {
// sin(x) is calculated with a polynomial.
let x_semi = x.to_semi();
let x2 = x_semi.square();
let x2_single = x2.to_single();
let x3 = x2.to_semi() * x_semi;
let x5 = x3.to_single() * x2_single;
// t1 = sin(x) - x + x^3 / 6
let t1 = F::sin_poly_ex(x2_single, x5);
// sin(x) = t1 + x - x^3 / 6
let x3k3 = x3.to_semi() * (-F::frac_1_6_ex());
x.to_denorm().qadd2(x3k3 + t1)
}
/// Calculates `cos(x_hi + x_lo)`, where
/// `x_lo` is very small and `|x_hi| <= π/4`
pub(super) fn cos_inner<F: SinCos>(x_hi: F, x_lo: F) -> F {
// cos(x_hi + x_lo) = cos(x_hi) * cos(x_lo) - sin(x_hi) * sin(x_lo)
// x_lo is small, so sin(x_lo) ~= x_lo and cos(x_lo) ~= 1
// then
// cos(x_hi + x_lo) ~= cos(x_hi) - sin(x_hi) * x_lo
//
// cos(x_hi) is calculated with a polynomial.
// sin(x_hi) * x_lo ~= x_hi * x_lo
// t1 = cos(x_hi) + 0.5 * x_hi^2 - 1
let x2 = x_hi * x_hi;
let x4 = x2 * x2;
let t1 = F::cos_poly(x2, x4);
// cos(x_hi + x_lo) = t1 + 1 - 0.5 * x^2 - x_hi * x_lo
let t2 = DenormDouble::new_qsub11(F::one(), F::half() * x2);
t2.lsub(x_hi * x_lo).ladd(t1).to_single()
}
pub(super) fn hi_lo_cos_inner<F: SinCos>(x: NormDouble<F>) -> DenormDouble<F> {
// cos(x) is calculated with a polynomial.
// t1 = cos(x) + 0.5 * x^2 - 1
let x2 = x.to_semi().square();
let x2_single = x2.to_single();
let x4 = x2_single * x2_single;
let t1 = F::cos_poly(x2_single, x4);
// t2 = 1 - 0.5 * x^2
let t2 = DenormDouble::new_qsub12(F::one(), x2.pmul1(F::half()));
// cos(x) = t1 + t2
t2.qadd1(t1)
}
#[cfg(test)]
mod tests {
use crate::traits::Float;
use crate::FloatMath;
fn test<F: Float + FloatMath>() {
use crate::{cos, sin, sin_cos};
let test_nan = |arg: F| {
let sin1 = sin(arg);
let cos1 = cos(arg);
let (sin2, cos2) = sin_cos(arg);
assert_is_nan!(sin1);
assert_is_nan!(cos1);
assert_is_nan!(sin2);
assert_is_nan!(cos2);
};
let test_value = |arg: F, expected_sin: F, expected_cos: F| {
let sin1 = sin(arg);
let cos1 = cos(arg);
let (sin2, cos2) = sin_cos(arg);
assert_total_eq!(sin1, expected_sin);
assert_total_eq!(cos1, expected_cos);
assert_total_eq!(sin2, expected_sin);
assert_total_eq!(cos2, expected_cos);
};
test_nan(F::NAN);
test_nan(F::INFINITY);
test_nan(F::neg_infinity());
test_value(F::ZERO, F::ZERO, F::one());
test_value(-F::ZERO, -F::ZERO, F::one());
}
#[test]
fn test_f32() {
test::<f32>();
}
#[cfg(feature = "soft-float")]
#[test]
fn test_soft_f32() {
test::<crate::SoftF32>();
}
#[test]
fn test_f64() {
test::<f64>();
}
#[cfg(feature = "soft-float")]
#[test]
fn test_soft_f64() {
test::<crate::SoftF64>();
}
}