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[libc][math] Add float-only implementation for sinf / cosf. #161680
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| Original file line number | Diff line number | Diff line change |
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| //===-- Compute sin + cos for small angles ----------------------*- C++ -*-===// | ||
| // | ||
| // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. | ||
| // See https://llvm.org/LICENSE.txt for license information. | ||
| // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception | ||
| // | ||
| //===----------------------------------------------------------------------===// | ||
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| #ifndef LLVM_LIBC_SRC___SUPPORT_MATH_SINCOSF_FLOAT_EVAL_H | ||
| #define LLVM_LIBC_SRC___SUPPORT_MATH_SINCOSF_FLOAT_EVAL_H | ||
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| #include "src/__support/FPUtil/FEnvImpl.h" | ||
| #include "src/__support/FPUtil/FPBits.h" | ||
| #include "src/__support/FPUtil/double_double.h" | ||
| #include "src/__support/FPUtil/multiply_add.h" | ||
| #include "src/__support/FPUtil/nearest_integer.h" | ||
| #include "src/__support/macros/config.h" | ||
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| namespace LIBC_NAMESPACE_DECL { | ||
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| namespace math { | ||
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| namespace sincosf_float_eval { | ||
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| // Since the worst case of `x mod pi` in single precision is > 2^-28, in order | ||
| // to be bounded by 1 ULP, the range reduction accuracy will need to be at | ||
| // lest 2^(-28 - 23) = 2^-51. | ||
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| // For fast small range reduction, we will compute as follow: | ||
| // Let pi ~ c0 + c1 + c2 | ||
| // with |c1| < ulp(c0)/2 and |c2| < ulp(c1)/2 | ||
| // then: | ||
| // k := nearest_int(x * 1/pi); | ||
| // u = (x - k * c0) - k * c1 - k * c2 | ||
| // We requires k * c0, k * c1 to be exactly representable in single precision. | ||
| // Let p_k be the precision of k, then the precision of c0 and c1 are: | ||
| // 24 - p_k, | ||
| // and the ulp of (k * c2) is 2^(-3 * (24 - p_k)). | ||
| // This give us the following bound on the precision of k: | ||
| // 3 * (24 - p_k) >= 51, | ||
| // or equivalently: | ||
| // p_k <= 7. | ||
| // We set the bound for p_k to be 6 so that we can have some more wiggle room | ||
| // for computations. | ||
| LIBC_INLINE static unsigned sincosf_range_reduction_small(float x, float &u) { | ||
| // > display=hexadecimal; | ||
| // > a = round(pi/8, 18, RN); | ||
| // > b = round(pi/8 - a, 18, RN); | ||
| // > c = round(pi/8 - a - b, SG, RN); | ||
| // > round(8/pi, SG, RN); | ||
| constexpr float MPI[3] = {-0x1.921f8p-2f, -0x1.aa22p-21f, -0x1.68c234p-41f}; | ||
| constexpr float ONE_OVER_PI = 0x1.45f306p+1f; | ||
| float prod_hi = x * ONE_OVER_PI; | ||
| float k = fputil::nearest_integer(prod_hi); | ||
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| float y_hi = fputil::multiply_add(k, MPI[0], x); // Exact | ||
| u = fputil::multiply_add(k, MPI[1], y_hi); | ||
| u = fputil::multiply_add(k, MPI[2], u); | ||
| return static_cast<unsigned>(static_cast<int>(k)); | ||
| } | ||
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| // TODO: Add non-FMA version of large range reduction. | ||
| LIBC_INLINE static unsigned sincosf_range_reduction_large(float x, float &u) { | ||
| // > for i from 0 to 13 do { | ||
| // if i < 2 then { pi_inv = 0.25 + 2^(8*(i - 2)) / pi; } | ||
| // else { pi_inv = 2^(8*(i-2)) / pi; }; | ||
| // pn = nearestint(pi_inv); | ||
| // pi_frac = pi_inv - pn; | ||
| // a = round(pi_frac, SG, RN); | ||
| // b = round(pi_frac - a, SG, RN); | ||
| // c = round(pi_frac - a - b, SG, RN); | ||
| // d = round(pi_frac - a - b - c, SG, RN); | ||
| // print("{", 2^3 * a, ",", 2^3 * b, ",", 2^3 * c, ",", 2^3 * d, "},"); | ||
| // }; | ||
| constexpr float EIGHT_OVER_PI[14][4] = { | ||
| {0x1.000146p1f, -0x1.9f246cp-28f, -0x1.bbead6p-54f, -0x1.ec5418p-85f}, | ||
| {0x1.0145f4p1f, -0x1.f246c6p-24f, -0x1.df56bp-49f, -0x1.ec5418p-77f}, | ||
| {0x1.45f306p1f, 0x1.b9391p-24f, 0x1.529fc2p-50f, 0x1.d5f47ep-76f}, | ||
| {0x1.f306dcp1f, 0x1.391054p-24f, 0x1.4fe13ap-49f, 0x1.7d1f54p-74f}, | ||
| {-0x1.f246c6p0f, -0x1.df56bp-25f, -0x1.ec5418p-53f, 0x1.f534dep-78f}, | ||
| {-0x1.236378p1f, 0x1.529fc2p-26f, 0x1.d5f47ep-52f, -0x1.65912p-77f}, | ||
| {0x1.391054p0f, 0x1.4fe13ap-25f, 0x1.7d1f54p-50f, -0x1.6447e4p-75f}, | ||
| {0x1.1054a8p0f, -0x1.ec5418p-29f, 0x1.f534dep-54f, -0x1.f924ecp-81f}, | ||
| {0x1.529fc2p-2f, 0x1.d5f47ep-28f, -0x1.65912p-53f, 0x1.b6c52cp-79f}, | ||
| {-0x1.ac07b2p1f, 0x1.5f47d4p-24f, 0x1.a6ee06p-49f, 0x1.b6295ap-74f}, | ||
| {-0x1.ec5418p-5f, 0x1.f534dep-30f, -0x1.f924ecp-57f, 0x1.5993c4p-82f}, | ||
| {0x1.3abe9p-1f, -0x1.596448p-27f, 0x1.b6c52cp-55f, -0x1.9b0ef2p-80f}, | ||
| {-0x1.505c16p1f, 0x1.a6ee06p-25f, 0x1.b6295ap-50f, -0x1.b0ef1cp-76f}, | ||
| {-0x1.70565ap-1f, 0x1.dc0db6p-26f, 0x1.4acc9ep-53f, 0x1.0e4108p-80f}, | ||
| }; | ||
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| using FPBits = typename fputil::FPBits<float>; | ||
| using fputil::FloatFloat; | ||
| FPBits xbits(x); | ||
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| int x_e_m32 = xbits.get_biased_exponent() - (FPBits::EXP_BIAS + 32); | ||
| unsigned idx = static_cast<unsigned>((x_e_m32 >> 3) + 2); | ||
| // Scale x down by 2^(-(8 * (idx - 2)) | ||
| xbits.set_biased_exponent((x_e_m32 & 7) + FPBits::EXP_BIAS + 32); | ||
| // 2^32 <= |x_reduced| < 2^(32 + 8) = 2^40 | ||
| float x_reduced = xbits.get_val(); | ||
| // x * c_hi = ph.hi + ph.lo exactly. | ||
| FloatFloat ph = fputil::exact_mult<float>(x_reduced, EIGHT_OVER_PI[idx][0]); | ||
| // x * c_mid = pm.hi + pm.lo exactly. | ||
| FloatFloat pm = fputil::exact_mult<float>(x_reduced, EIGHT_OVER_PI[idx][1]); | ||
| // x * c_lo = pl.hi + pl.lo exactly. | ||
| FloatFloat pl = fputil::exact_mult<float>(x_reduced, EIGHT_OVER_PI[idx][2]); | ||
| // Extract integral parts and fractional parts of (ph.lo + pm.hi). | ||
| float sum_hi = ph.lo + pm.hi; | ||
| float k = fputil::nearest_integer(sum_hi); | ||
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| // x * 8/pi mod 1 ~ y_hi + y_mid + y_lo | ||
| float y_hi = (ph.lo - k) + pm.hi; // Exact | ||
| FloatFloat y_mid = fputil::exact_add(pm.lo, pl.hi); | ||
| float y_lo = pl.lo; | ||
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| // y_l = x * c_lo_2 + pl.lo | ||
| float y_l = fputil::multiply_add(x_reduced, EIGHT_OVER_PI[idx][3], y_lo); | ||
| FloatFloat y = fputil::exact_add(y_hi, y_mid.hi); | ||
| y.lo += (y_mid.lo + y_l); | ||
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| // Digits of pi/8, generated by Sollya with: | ||
| // > a = round(pi/8, SG, RN); | ||
| // > b = round(pi/8 - SG, D, RN); | ||
| constexpr FloatFloat PI_OVER_8 = {-0x1.777a5cp-27f, 0x1.921fb6p-2f}; | ||
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| // Error bound: with {a} denote the fractional part of a, i.e.: | ||
| // {a} = a - round(a) | ||
| // Then, | ||
| // | {x * 8/pi} - (y_hi + y_lo) | <= ulp(ulp(y_hi)) <= 2^-47 | ||
| // | {x mod pi/8} - (u.hi + u.lo) | < 2 * 2^-5 * 2^-47 = 2^-51 | ||
| u = fputil::multiply_add(y.hi, PI_OVER_8.hi, y.lo * PI_OVER_8.hi); | ||
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| return static_cast<unsigned>(static_cast<int>(k)); | ||
| } | ||
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| template <bool IS_SIN> LIBC_INLINE static float sincosf_eval(float x) { | ||
| // sin(k * pi/8) for k = 0..15, generated by Sollya with: | ||
| // > for k from 0 to 16 do { | ||
| // print(round(sin(k * pi/8), SG, RN)); | ||
| // }; | ||
| constexpr float SIN_K_PI_OVER_8[16] = { | ||
| 0.0f, 0x1.87de2ap-2f, 0x1.6a09e6p-1f, 0x1.d906bcp-1f, | ||
| 1.0f, 0x1.d906bcp-1f, 0x1.6a09e6p-1f, 0x1.87de2ap-2f, | ||
| 0.0f, -0x1.87de2ap-2f, -0x1.6a09e6p-1f, -0x1.d906bcp-1f, | ||
| -1.0f, -0x1.d906bcp-1f, -0x1.6a09e6p-1f, -0x1.87de2ap-2f, | ||
| }; | ||
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| using FPBits = fputil::FPBits<float>; | ||
| FPBits xbits(x); | ||
| uint32_t x_abs = cpp::bit_cast<uint32_t>(x) & 0x7fff'ffffU; | ||
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| float y; | ||
| unsigned k = 0; | ||
| if (x_abs < 0x4880'0000U) { | ||
| k = sincosf_range_reduction_small(x, y); | ||
| } else { | ||
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| if (LIBC_UNLIKELY(x_abs >= 0x7f80'0000U)) { | ||
| if (xbits.is_signaling_nan()) { | ||
| fputil::raise_except_if_required(FE_INVALID); | ||
| return FPBits::quiet_nan().get_val(); | ||
| } | ||
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| if (x_abs == 0x7f80'0000U) { | ||
| fputil::set_errno_if_required(EDOM); | ||
| fputil::raise_except_if_required(FE_INVALID); | ||
| } | ||
| return x + FPBits::quiet_nan().get_val(); | ||
| } | ||
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| k = sincosf_range_reduction_large(x, y); | ||
| } | ||
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| float sin_k = SIN_K_PI_OVER_8[k & 15]; | ||
| // cos(k * pi/8) = sin(k * pi/8 + pi/2) = sin((k + 4) * pi/8). | ||
| // cos_k = cos(k * pi/8) | ||
| float cos_k = SIN_K_PI_OVER_8[(k + 4) & 15]; | ||
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| float y_sq = y * y; | ||
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| // Polynomial approximation of sin(y) and cos(y) for |y| <= pi/16: | ||
| // | ||
| // Using Taylor polynomial for sin(y): | ||
| // sin(y) ~ y - y^3 / 6 + y^5 / 120 | ||
| // Using minimax polynomial generated by Sollya for cos(y) with: | ||
| // > Q = fpminimax(cos(x), [|0, 2, 4|], [|1, SG...|], [0, pi/16]); | ||
| // | ||
| // Error bounds: | ||
| // * For sin(y) | ||
| // > P = x - SG(1/6)*x^3 + SG(1/120) * x^5; | ||
| // > dirtyinfnorm((sin(x) - P)/sin(x), [-pi/16, pi/16]); | ||
| // 0x1.825...p-27 | ||
| // * For cos(y) | ||
| // > Q = fpminimax(cos(x), [|0, 2, 4|], [|1, SG...|], [0, pi/16]); | ||
| // > dirtyinfnorm((sin(x) - P)/sin(x), [-pi/16, pi/16]); | ||
| // 0x1.aa8...p-29 | ||
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| // p1 = y^2 * 1/120 - 1/6 | ||
| float p1 = fputil::multiply_add(y_sq, 0x1.111112p-7f, -0x1.555556p-3f); | ||
| // q1 = y^2 * coeff(Q, 4) + coeff(Q, 2) | ||
| float q1 = fputil::multiply_add(y_sq, 0x1.54b8bep-5f, -0x1.ffffc4p-2f); | ||
| float y3 = y_sq * y; | ||
| // c1 ~ cos(y) | ||
| float c1 = fputil::multiply_add(y_sq, q1, 1.0f); | ||
| // s1 ~ sin(y) | ||
| float s1 = fputil::multiply_add(y3, p1, y); | ||
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| if constexpr (IS_SIN) { | ||
| // sin(x) = cos(k * pi/8) * sin(y) + sin(k * pi/8) * cos(y). | ||
| return fputil::multiply_add(cos_k, s1, sin_k * c1); | ||
| } else { | ||
| // cos(x) = cos(k * pi/8) * cos(y) - sin(k * pi/8) * sin(y). | ||
| return fputil::multiply_add(cos_k, c1, -sin_k * s1); | ||
| } | ||
| } | ||
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| } // namespace sincosf_float_eval | ||
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| } // namespace math | ||
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| } // namespace LIBC_NAMESPACE_DECL | ||
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| #endif // LLVM_LIBC_SRC___SUPPORT_MATH_SINCOSF_FLOAT_EVAL_H | ||
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