src/algorithms/trig_pi.rs

   1 use crate::{
   2     f16::F16,
   3     ieee754::FloatEncoding,
   4     traits::{Compare, Context, ConvertFrom, ConvertTo, Float, Select},
   5 };
   6
   7 mod consts {
   8     #![allow(clippy::excessive_precision)]
   9
  10     /// coefficients of taylor series for `sin(pi * x)` centered at `0`
  11     /// generated using:
  12     /// ```maxima,text
  13     /// fpprec:50$
  14     /// sinpi: bfloat(taylor(sin(%pi*x),x,0,19))$
  15     /// for i: 1 step 2 thru 19 do
  16     ///     printf(true, "pub(crate) const SINPI_KERNEL_TAYLOR_~d: f64 = ~a;~%", i, ssubst("e", "b", string(coeff(sinpi, x, i))))$
  17     /// ```
  18     pub(crate) const SINPI_KERNEL_TAYLOR_1: f64 =
  19         3.1415926535897932384626433832795028841971693993751e0;
  20     pub(crate) const SINPI_KERNEL_TAYLOR_3: f64 =
  21         -5.1677127800499700292460525111835658670375480943142e0;
  22     pub(crate) const SINPI_KERNEL_TAYLOR_5: f64 =
  23         2.550164039877345443856177583695296720669172555234e0;
  24     pub(crate) const SINPI_KERNEL_TAYLOR_7: f64 =
  25         -5.9926452932079207688773938354604004601536358636814e-1;
  26     pub(crate) const SINPI_KERNEL_TAYLOR_9: f64 =
  27         8.2145886611128228798802365523698344807837460797753e-2;
  28     pub(crate) const SINPI_KERNEL_TAYLOR_11: f64 =
  29         -7.370430945714350777259089957290781501211638236021e-3;
  30     pub(crate) const SINPI_KERNEL_TAYLOR_13: f64 =
  31         4.6630280576761256442062891447027174382819981361599e-4;
  32     pub(crate) const SINPI_KERNEL_TAYLOR_15: f64 =
  33         -2.1915353447830215827384652057094188859248708765956e-5;
  34     pub(crate) const SINPI_KERNEL_TAYLOR_17: f64 =
  35         7.9520540014755127847832068624575890327682459384282e-7;
  36     pub(crate) const SINPI_KERNEL_TAYLOR_19: f64 =
  37         -2.2948428997269873110203872385571587856074785581088e-8;
  38
  39     /// coefficients of taylor series for `cos(pi * x)` centered at `0`
  40     /// generated using:
  41     /// ```maxima,text
  42     /// fpprec:50$
  43     /// cospi: bfloat(taylor(cos(%pi*x),x,0,18))$
  44     /// for i: 0 step 2 thru 18 do
  45     ///     printf(true, "pub(crate) const COSPI_KERNEL_TAYLOR_~d: f64 = ~a;~%", i, ssubst("e", "b", string(coeff(cospi, x, i))))$
  46     /// ```
  47     pub(crate) const COSPI_KERNEL_TAYLOR_0: f64 = 1.0e0;
  48     pub(crate) const COSPI_KERNEL_TAYLOR_2: f64 =
  49         -4.9348022005446793094172454999380755676568497036204e0;
  50     pub(crate) const COSPI_KERNEL_TAYLOR_4: f64 =
  51         4.0587121264167682181850138620293796354053160696952e0;
  52     pub(crate) const COSPI_KERNEL_TAYLOR_6: f64 =
  53         -1.3352627688545894958753047828505831928711354556681e0;
  54     pub(crate) const COSPI_KERNEL_TAYLOR_8: f64 =
  55         2.3533063035889320454187935277546542154506893530856e-1;
  56     pub(crate) const COSPI_KERNEL_TAYLOR_10: f64 =
  57         -2.5806891390014060012598294252898849657186441048147e-2;
  58     pub(crate) const COSPI_KERNEL_TAYLOR_12: f64 =
  59         1.9295743094039230479033455636859576401684718150003e-3;
  60     pub(crate) const COSPI_KERNEL_TAYLOR_14: f64 =
  61         -1.0463810492484570711801672835223932761029733149091e-4;
  62     pub(crate) const COSPI_KERNEL_TAYLOR_16: f64 =
  63         4.3030695870329470072978237149669233008960901556009e-6;
  64     pub(crate) const COSPI_KERNEL_TAYLOR_18: f64 =
  65         -1.387895246221377211446808750399309343777037849978e-7;
  66 }
  67
  68 /// computes `sin(pi * x)` for `-0.25 <= x <= 0.25`
  69 /// not guaranteed to give correct sign for zero result
  70 /// has an error of up to 2ULP
  71 pub fn sin_pi_kernel_f16<Ctx: Context>(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 {
  72     let x_sq = x * x;
  73     let mut v: Ctx::VecF16 = ctx.make(consts::SINPI_KERNEL_TAYLOR_5.to());
  74     v = v.mul_add_fast(x_sq, ctx.make(consts::SINPI_KERNEL_TAYLOR_3.to()));
  75     v = v.mul_add_fast(x_sq, ctx.make(consts::SINPI_KERNEL_TAYLOR_1.to()));
  76     v * x
  77 }
  78
  79 /// computes `cos(pi * x)` for `-0.25 <= x <= 0.25`
  80 /// has an error of up to 2ULP
  81 pub fn cos_pi_kernel_f16<Ctx: Context>(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 {
  82     let x_sq = x * x;
  83     let mut v: Ctx::VecF16 = ctx.make(consts::COSPI_KERNEL_TAYLOR_4.to());
  84     v = v.mul_add_fast(x_sq, ctx.make(consts::COSPI_KERNEL_TAYLOR_2.to()));
  85     v.mul_add_fast(x_sq, ctx.make(consts::COSPI_KERNEL_TAYLOR_0.to()))
  86 }
  87
  88 /// computes `(sin(pi * x), cos(pi * x))`
  89 /// not guaranteed to give correct sign for zero results
  90 /// has an error of up to 2ULP
  91 pub fn sin_cos_pi_f16<Ctx: Context>(ctx: Ctx, x: Ctx::VecF16) -> (Ctx::VecF16, Ctx::VecF16) {
  92     let two_f16: Ctx::VecF16 = ctx.make(2.0.to());
  93     let one_half: Ctx::VecF16 = ctx.make(0.5.to());
  94     let max_contiguous_integer: Ctx::VecF16 =
  95         ctx.make((1u16 << (F16::MANTISSA_FIELD_WIDTH + 1)).to());
  96     // if `x` is finite and bigger than `max_contiguous_integer`, then x is an even integer
  97     let in_range = x.abs().lt(max_contiguous_integer); // use `lt` so nans are counted as out-of-range
  98     let is_finite = x.is_finite();
  99     let nan: Ctx::VecF16 = ctx.make(f32::NAN.to());
 100     let zero_f16: Ctx::VecF16 = ctx.make(0.to());
 101     let one_f16: Ctx::VecF16 = ctx.make(1.to());
 102     let zero_i16: Ctx::VecI16 = ctx.make(0.to());
 103     let one_i16: Ctx::VecI16 = ctx.make(1.to());
 104     let two_i16: Ctx::VecI16 = ctx.make(2.to());
 105     let out_of_range_sin = is_finite.select(zero_f16, nan);
 106     let out_of_range_cos = is_finite.select(one_f16, nan);
 107     let xi = (x * two_f16).round();
 108     let xk = x - xi * one_half;
 109     let sk = sin_pi_kernel_f16(ctx, xk);
 110     let ck = cos_pi_kernel_f16(ctx, xk);
 111     let xi = Ctx::VecI16::cvt_from(xi);
 112     let bit_0_clear = (xi & one_i16).eq(zero_i16);
 113     let st = bit_0_clear.select(sk, ck);
 114     let ct = bit_0_clear.select(ck, sk);
 115     let s = (xi & two_i16).eq(zero_i16).select(st, -st);
 116     let c = ((xi + one_i16) & two_i16).eq(zero_i16).select(ct, -ct);
 117     (
 118         in_range.select(s, out_of_range_sin),
 119         in_range.select(c, out_of_range_cos),
 120     )
 121 }
 122
 123 /// computes `sin(pi * x)`
 124 /// not guaranteed to give correct sign for zero results
 125 /// has an error of up to 2ULP
 126 pub fn sin_pi_f16<Ctx: Context>(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 {
 127     sin_cos_pi_f16(ctx, x).0
 128 }
 129
 130 /// computes `cos(pi * x)`
 131 /// not guaranteed to give correct sign for zero results
 132 /// has an error of up to 2ULP
 133 pub fn cos_pi_f16<Ctx: Context>(ctx: Ctx, x: Ctx::VecF16) -> Ctx::VecF16 {
 134     sin_cos_pi_f16(ctx, x).1
 135 }
 136
 137 #[cfg(test)]
 138 mod tests {
 139     use super::*;
 140     use crate::{
 141         f16::F16,
 142         scalar::{Scalar, Value},
 143     };
 144     use std::f64;
 145
 146     struct CheckUlpCallbackArg<F, I> {
 147         distance_in_ulp: I,
 148         x: F,
 149         expected: F,
 150         result: F,
 151     }
 152
 153     #[track_caller]
 154     fn check_ulp_f16(
 155         x: F16,
 156         is_ok: impl Fn(CheckUlpCallbackArg<F16, u32>) -> bool,
 157         fn_f16: impl Fn(F16) -> F16,
 158         fn_f64: impl Fn(f64) -> f64,
 159     ) {
 160         let x_f64: f64 = x.to();
 161         let expected_f64 = fn_f64(x_f64);
 162         let expected: F16 = expected_f64.to();
 163         let result = fn_f16(x);
 164         if result == expected {
 165             return;
 166         }
 167         if result.is_nan() && expected.is_nan() {
 168             return;
 169         }
 170         let distance_in_ulp = (expected.to_bits() as i32 - result.to_bits() as i32).unsigned_abs();
 171         if !result.is_nan()
 172             && !expected.is_nan()
 173             && is_ok(CheckUlpCallbackArg {
 174                 distance_in_ulp,
 175                 x,
 176                 expected,
 177                 result,
 178             })
 179         {
 180             return;
 181         }
 182         panic!(
 183             "error is too big: \
 184                 x = {x:?} {x_bits:#X}, \
 185                 result = {result:?} {result_bits:#X}, \
 186                 expected = {expected:?} {expected_bits:#X}, \
 187                 distance_in_ulp = {distance_in_ulp}",
 188             x = x,
 189             x_bits = x.to_bits(),
 190             result = result,
 191             result_bits = result.to_bits(),
 192             expected = expected,
 193             expected_bits = expected.to_bits(),
 194             distance_in_ulp = distance_in_ulp,
 195         );
 196     }
 197
 198     #[test]
 199     #[cfg_attr(
 200         not(feature = "f16"),
 201         should_panic(expected = "f16 feature is not enabled")
 202     )]
 203     fn test_sin_pi_kernel_f16() {
 204         let check = |x| {
 205             check_ulp_f16(
 206                 x,
 207                 |arg| arg.distance_in_ulp <= if arg.expected == 0.to() { 0 } else { 2 },
 208                 |x| sin_pi_kernel_f16(Scalar, Value(x)).0,
 209                 |x| (f64::consts::PI * x).sin(),
 210             )
 211         };
 212         let quarter = F16::to_bits(0.25f32.to());
 213         for bits in (0..=quarter).rev() {
 214             check(F16::from_bits(bits));
 215             check(-F16::from_bits(bits));
 216         }
 217     }
 218
 219     #[test]
 220     #[cfg_attr(
 221         not(feature = "f16"),
 222         should_panic(expected = "f16 feature is not enabled")
 223     )]
 224     fn test_cos_pi_kernel_f16() {
 225         let check = |x| {
 226             check_ulp_f16(
 227                 x,
 228                 |arg| arg.distance_in_ulp <= 2 && arg.result <= 1.to(),
 229                 |x| cos_pi_kernel_f16(Scalar, Value(x)).0,
 230                 |x| (f64::consts::PI * x).cos(),
 231             )
 232         };
 233         let quarter = F16::to_bits(0.25f32.to());
 234         for bits in (0..=quarter).rev() {
 235             check(F16::from_bits(bits));
 236             check(-F16::from_bits(bits));
 237         }
 238     }
 239
 240     fn sin_cos_pi_check_ulp_callback_f16(arg: CheckUlpCallbackArg<F16, u32>) -> bool {
 241         if f32::cvt_from(arg.x) % 0.5 == 0.0 {
 242             arg.distance_in_ulp == 0
 243         } else {
 244             arg.distance_in_ulp <= 2 && arg.result.abs() <= 1.to()
 245         }
 246     }
 247
 248     #[test]
 249     #[cfg_attr(
 250         not(feature = "f16"),
 251         should_panic(expected = "f16 feature is not enabled")
 252     )]
 253     fn test_sin_pi_f16() {
 254         for bits in 0..=u16::MAX {
 255             check_ulp_f16(
 256                 F16::from_bits(bits),
 257                 sin_cos_pi_check_ulp_callback_f16,
 258                 |x| sin_pi_f16(Scalar, Value(x)).0,
 259                 |x| (f64::consts::PI * x).sin(),
 260             );
 261         }
 262     }
 263
 264     #[test]
 265     #[cfg_attr(
 266         not(feature = "f16"),
 267         should_panic(expected = "f16 feature is not enabled")
 268     )]
 269     fn test_cos_pi_f16() {
 270         for bits in 0..=u16::MAX {
 271             check_ulp_f16(
 272                 F16::from_bits(bits),
 273                 sin_cos_pi_check_ulp_callback_f16,
 274                 |x| cos_pi_f16(Scalar, Value(x)).0,
 275                 |x| (f64::consts::PI * x).cos(),
 276             );
 277         }
 278     }
 279 }