use crate::{ f16::F16, ieee754::FloatEncoding, traits::{Compare, Context, ConvertFrom, ConvertTo, Float, Select}, }; mod consts { #![allow(clippy::excessive_precision)] /// coefficients of taylor series for `sin(pi * x)` centered at `0` /// generated using: /// ```maxima,text /// fpprec:50$ /// sinpi: bfloat(taylor(sin(%pi*x),x,0,19))$ /// for i: 1 step 2 thru 19 do /// printf(true, "pub(crate) const SINPI_KERNEL_TAYLOR_~d: f64 = ~a;~%", i, ssubst("e", "b", string(coeff(sinpi, x, i))))$ /// ``` pub(crate) const SINPI_KERNEL_TAYLOR_1: f64 = 3.1415926535897932384626433832795028841971693993751e0; pub(crate) const SINPI_KERNEL_TAYLOR_3: f64 = -5.1677127800499700292460525111835658670375480943142e0; pub(crate) const SINPI_KERNEL_TAYLOR_5: f64 = 2.550164039877345443856177583695296720669172555234e0; pub(crate) const SINPI_KERNEL_TAYLOR_7: f64 = -5.9926452932079207688773938354604004601536358636814e-1; pub(crate) const SINPI_KERNEL_TAYLOR_9: f64 = 8.2145886611128228798802365523698344807837460797753e-2; pub(crate) const SINPI_KERNEL_TAYLOR_11: f64 = -7.370430945714350777259089957290781501211638236021e-3; pub(crate) const SINPI_KERNEL_TAYLOR_13: f64 = 4.6630280576761256442062891447027174382819981361599e-4; pub(crate) const SINPI_KERNEL_TAYLOR_15: f64 = -2.1915353447830215827384652057094188859248708765956e-5; pub(crate) const SINPI_KERNEL_TAYLOR_17: f64 = 7.9520540014755127847832068624575890327682459384282e-7; pub(crate) const SINPI_KERNEL_TAYLOR_19: f64 = -2.2948428997269873110203872385571587856074785581088e-8; /// coefficients of taylor series for `cos(pi * x)` centered at `0` /// generated using: /// ```maxima,text /// fpprec:50$ /// cospi: bfloat(taylor(cos(%pi*x),x,0,18))$ /// for i: 0 step 2 thru 18 do /// printf(true, "pub(crate) const COSPI_KERNEL_TAYLOR_~d: f64 = ~a;~%", i, ssubst("e", "b", string(coeff(cospi, x, i))))$ /// ``` pub(crate) const COSPI_KERNEL_TAYLOR_0: f64 = 1.0e0; pub(crate) const COSPI_KERNEL_TAYLOR_2: f64 = -4.9348022005446793094172454999380755676568497036204e0; pub(crate) const COSPI_KERNEL_TAYLOR_4: f64 = 4.0587121264167682181850138620293796354053160696952e0; pub(crate) const COSPI_KERNEL_TAYLOR_6: f64 = -1.3352627688545894958753047828505831928711354556681e0; pub(crate) const COSPI_KERNEL_TAYLOR_8: f64 = 2.3533063035889320454187935277546542154506893530856e-1; pub(crate) const COSPI_KERNEL_TAYLOR_10: f64 = -2.5806891390014060012598294252898849657186441048147e-2; pub(crate) const COSPI_KERNEL_TAYLOR_12: f64 = 1.9295743094039230479033455636859576401684718150003e-3; pub(crate) const COSPI_KERNEL_TAYLOR_14: f64 = -1.0463810492484570711801672835223932761029733149091e-4; pub(crate) const COSPI_KERNEL_TAYLOR_16: f64 = 4.3030695870329470072978237149669233008960901556009e-6; pub(crate) const COSPI_KERNEL_TAYLOR_18: f64 = -1.387895246221377211446808750399309343777037849978e-7; } /// computes `sin(pi * x)` for `-0.25 <= x <= 0.25` /// not guaranteed to give correct sign for zero result /// has an error of up to 2ULP pub fn sin_pi_kernel_f16(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 { let x_sq = x * x; let mut v: Ctx::VecF16 = ctx.make(consts::SINPI_KERNEL_TAYLOR_5.to()); v = v.mul_add_fast(x_sq, ctx.make(consts::SINPI_KERNEL_TAYLOR_3.to())); v = v.mul_add_fast(x_sq, ctx.make(consts::SINPI_KERNEL_TAYLOR_1.to())); v * x } /// computes `cos(pi * x)` for `-0.25 <= x <= 0.25` /// has an error of up to 2ULP pub fn cos_pi_kernel_f16(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 { let x_sq = x * x; let mut v: Ctx::VecF16 = ctx.make(consts::COSPI_KERNEL_TAYLOR_4.to()); v = v.mul_add_fast(x_sq, ctx.make(consts::COSPI_KERNEL_TAYLOR_2.to())); v.mul_add_fast(x_sq, ctx.make(consts::COSPI_KERNEL_TAYLOR_0.to())) } /// computes `(sin(pi * x), cos(pi * x))` /// not guaranteed to give correct sign for zero results /// has an error of up to 2ULP pub fn sin_cos_pi_f16(ctx: Ctx, x: Ctx::VecF16) -> (Ctx::VecF16, Ctx::VecF16) { let two_f16: Ctx::VecF16 = ctx.make(2.0.to()); let one_half: Ctx::VecF16 = ctx.make(0.5.to()); let max_contiguous_integer: Ctx::VecF16 = ctx.make((1u16 << (F16::MANTISSA_FIELD_WIDTH + 1)).to()); // if `x` is finite and bigger than `max_contiguous_integer`, then x is an even integer let in_range = x.abs().lt(max_contiguous_integer); // use `lt` so nans are counted as out-of-range let is_finite = x.is_finite(); let nan: Ctx::VecF16 = ctx.make(f32::NAN.to()); let zero_f16: Ctx::VecF16 = ctx.make(0.to()); let one_f16: Ctx::VecF16 = ctx.make(1.to()); let zero_i16: Ctx::VecI16 = ctx.make(0.to()); let one_i16: Ctx::VecI16 = ctx.make(1.to()); let two_i16: Ctx::VecI16 = ctx.make(2.to()); let out_of_range_sin = is_finite.select(zero_f16, nan); let out_of_range_cos = is_finite.select(one_f16, nan); let xi = (x * two_f16).round(); let xk = x - xi * one_half; let sk = sin_pi_kernel_f16(ctx, xk); let ck = cos_pi_kernel_f16(ctx, xk); let xi = Ctx::VecI16::cvt_from(xi); let bit_0_clear = (xi & one_i16).eq(zero_i16); let st = bit_0_clear.select(sk, ck); let ct = bit_0_clear.select(ck, sk); let s = (xi & two_i16).eq(zero_i16).select(st, -st); let c = ((xi + one_i16) & two_i16).eq(zero_i16).select(ct, -ct); ( in_range.select(s, out_of_range_sin), in_range.select(c, out_of_range_cos), ) } /// computes `sin(pi * x)` /// not guaranteed to give correct sign for zero results /// has an error of up to 2ULP pub fn sin_pi_f16(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 { sin_cos_pi_f16(ctx, x).0 } /// computes `cos(pi * x)` /// not guaranteed to give correct sign for zero results /// has an error of up to 2ULP pub fn cos_pi_f16(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 { sin_cos_pi_f16(ctx, x).1 } #[cfg(test)] mod tests { use super::*; use crate::{ f16::F16, scalar::{Scalar, Value}, }; use std::f64; struct CheckUlpCallbackArg { distance_in_ulp: I, x: F, expected: F, result: F, } #[track_caller] fn check_ulp_f16( x: F16, is_ok: impl Fn(CheckUlpCallbackArg) -> bool, fn_f16: impl Fn(F16) -> F16, fn_f64: impl Fn(f64) -> f64, ) { let x_f64: f64 = x.to(); let expected_f64 = fn_f64(x_f64); let expected: F16 = expected_f64.to(); let result = fn_f16(x); if result == expected { return; } if result.is_nan() && expected.is_nan() { return; } let distance_in_ulp = (expected.to_bits() as i32 - result.to_bits() as i32).unsigned_abs(); if !result.is_nan() && !expected.is_nan() && is_ok(CheckUlpCallbackArg { distance_in_ulp, x, expected, result, }) { return; } panic!( "error is too big: \ x = {x:?} {x_bits:#X}, \ result = {result:?} {result_bits:#X}, \ expected = {expected:?} {expected_bits:#X}, \ distance_in_ulp = {distance_in_ulp}", x = x, x_bits = x.to_bits(), result = result, result_bits = result.to_bits(), expected = expected, expected_bits = expected.to_bits(), distance_in_ulp = distance_in_ulp, ); } #[test] #[cfg_attr( not(feature = "f16"), should_panic(expected = "f16 feature is not enabled") )] fn test_sin_pi_kernel_f16() { let check = |x| { check_ulp_f16( x, |arg| arg.distance_in_ulp <= if arg.expected == 0.to() { 0 } else { 2 }, |x| sin_pi_kernel_f16(Scalar, Value(x)).0, |x| (f64::consts::PI * x).sin(), ) }; let quarter = F16::to_bits(0.25f32.to()); for bits in (0..=quarter).rev() { check(F16::from_bits(bits)); check(-F16::from_bits(bits)); } } #[test] #[cfg_attr( not(feature = "f16"), should_panic(expected = "f16 feature is not enabled") )] fn test_cos_pi_kernel_f16() { let check = |x| { check_ulp_f16( x, |arg| arg.distance_in_ulp <= 2 && arg.result <= 1.to(), |x| cos_pi_kernel_f16(Scalar, Value(x)).0, |x| (f64::consts::PI * x).cos(), ) }; let quarter = F16::to_bits(0.25f32.to()); for bits in (0..=quarter).rev() { check(F16::from_bits(bits)); check(-F16::from_bits(bits)); } } fn sin_cos_pi_check_ulp_callback_f16(arg: CheckUlpCallbackArg) -> bool { if f32::cvt_from(arg.x) % 0.5 == 0.0 { arg.distance_in_ulp == 0 } else { arg.distance_in_ulp <= 2 && arg.result.abs() <= 1.to() } } #[test] #[cfg_attr( not(feature = "f16"), should_panic(expected = "f16 feature is not enabled") )] fn test_sin_pi_f16() { for bits in 0..=u16::MAX { check_ulp_f16( F16::from_bits(bits), sin_cos_pi_check_ulp_callback_f16, |x| sin_pi_f16(Scalar, Value(x)).0, |x| (f64::consts::PI * x).sin(), ); } } #[test] #[cfg_attr( not(feature = "f16"), should_panic(expected = "f16 feature is not enabled") )] fn test_cos_pi_f16() { for bits in 0..=u16::MAX { check_ulp_f16( F16::from_bits(bits), sin_cos_pi_check_ulp_callback_f16, |x| cos_pi_f16(Scalar, Value(x)).0, |x| (f64::consts::PI * x).cos(), ); } } }