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1//===-- Compute sin + cos for small angles ----------------------*- C++ -*-===//2//3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.4// See https://llvm.org/LICENSE.txt for license information.5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception6//7//===----------------------------------------------------------------------===//8 9#ifndef LLVM_LIBC_SRC_MATH_GENERIC_SINCOS_EVAL_H10#define LLVM_LIBC_SRC_MATH_GENERIC_SINCOS_EVAL_H11 12#include "src/__support/FPUtil/PolyEval.h"13#include "src/__support/FPUtil/double_double.h"14#include "src/__support/FPUtil/dyadic_float.h"15#include "src/__support/FPUtil/multiply_add.h"16#include "src/__support/integer_literals.h"17#include "src/__support/macros/config.h"18 19namespace LIBC_NAMESPACE_DECL {20 21namespace math {22 23namespace sincos_eval_internal {24 25using fputil::DoubleDouble;26using Float128 = fputil::DyadicFloat<128>;27 28LIBC_INLINE double sincos_eval(const DoubleDouble &u, DoubleDouble &sin_u,29 DoubleDouble &cos_u) {30 // Evaluate sin(y) = sin(x - k * (pi/128))31 // We use the degree-7 Taylor approximation:32 // sin(y) ~ y - y^3/3! + y^5/5! - y^7/7!33 // Then the error is bounded by:34 // |sin(y) - (y - y^3/3! + y^5/5! - y^7/7!)| < |y|^9/9! < 2^-54/9! < 2^-72.35 // For y ~ u_hi + u_lo, fully expanding the polynomial and drop any terms36 // < ulp(u_hi^3) gives us:37 // y - y^3/3! + y^5/5! - y^7/7! = ...38 // ~ u_hi + u_hi^3 * (-1/6 + u_hi^2 * (1/120 - u_hi^2 * 1/5040)) +39 // + u_lo (1 + u_hi^2 * (-1/2 + u_hi^2 / 24))40 double u_hi_sq = u.hi * u.hi; // Error < ulp(u_hi^2) < 2^(-6 - 52) = 2^-58.41 // p1 ~ 1/120 + u_hi^2 / 5040.42 double p1 = fputil::multiply_add(u_hi_sq, -0x1.a01a01a01a01ap-13,43 0x1.1111111111111p-7);44 // q1 ~ -1/2 + u_hi^2 / 24.45 double q1 = fputil::multiply_add(u_hi_sq, 0x1.5555555555555p-5, -0x1.0p-1);46 double u_hi_3 = u_hi_sq * u.hi;47 // p2 ~ -1/6 + u_hi^2 (1/120 - u_hi^2 * 1/5040)48 double p2 = fputil::multiply_add(u_hi_sq, p1, -0x1.5555555555555p-3);49 // q2 ~ 1 + u_hi^2 (-1/2 + u_hi^2 / 24)50 double q2 = fputil::multiply_add(u_hi_sq, q1, 1.0);51 double sin_lo = fputil::multiply_add(u_hi_3, p2, u.lo * q2);52 // Overall, |sin(y) - (u_hi + sin_lo)| < 2*ulp(u_hi^3) < 2^-69.53 54 // Evaluate cos(y) = cos(x - k * (pi/128))55 // We use the degree-8 Taylor approximation:56 // cos(y) ~ 1 - y^2/2 + y^4/4! - y^6/6! + y^8/8!57 // Then the error is bounded by:58 // |cos(y) - (...)| < |y|^10/10! < 2^-8159 // For y ~ u_hi + u_lo, fully expanding the polynomial and drop any terms60 // < ulp(u_hi^3) gives us:61 // 1 - y^2/2 + y^4/4! - y^6/6! + y^8/8! = ...62 // ~ 1 - u_hi^2/2 + u_hi^4(1/24 + u_hi^2 (-1/720 + u_hi^2/40320)) +63 // + u_hi u_lo (-1 + u_hi^2/6)64 // We compute 1 - u_hi^2 accurately:65 // v_hi + v_lo ~ 1 - u_hi^2/266 // with error <= 2^-105.67 double u_hi_neg_half = (-0.5) * u.hi;68 DoubleDouble v;69 70#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE71 v.hi = fputil::multiply_add(u.hi, u_hi_neg_half, 1.0);72 v.lo = 1.0 - v.hi; // Exact73 v.lo = fputil::multiply_add(u.hi, u_hi_neg_half, v.lo);74#else75 DoubleDouble u_hi_sq_neg_half = fputil::exact_mult(u.hi, u_hi_neg_half);76 v = fputil::exact_add(1.0, u_hi_sq_neg_half.hi);77 v.lo += u_hi_sq_neg_half.lo;78#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE79 80 // r1 ~ -1/720 + u_hi^2 / 4032081 double r1 = fputil::multiply_add(u_hi_sq, 0x1.a01a01a01a01ap-16,82 -0x1.6c16c16c16c17p-10);83 // s1 ~ -1 + u_hi^2 / 684 double s1 = fputil::multiply_add(u_hi_sq, 0x1.5555555555555p-3, -1.0);85 double u_hi_4 = u_hi_sq * u_hi_sq;86 double u_hi_u_lo = u.hi * u.lo;87 // r2 ~ 1/24 + u_hi^2 (-1/720 + u_hi^2 / 40320)88 double r2 = fputil::multiply_add(u_hi_sq, r1, 0x1.5555555555555p-5);89 // s2 ~ v_lo + u_hi * u_lo * (-1 + u_hi^2 / 6)90 double s2 = fputil::multiply_add(u_hi_u_lo, s1, v.lo);91 double cos_lo = fputil::multiply_add(u_hi_4, r2, s2);92 // Overall, |cos(y) - (v_hi + cos_lo)| < 2*ulp(u_hi^4) < 2^-75.93 94 sin_u = fputil::exact_add(u.hi, sin_lo);95 cos_u = fputil::exact_add(v.hi, cos_lo);96 97 return fputil::multiply_add(fputil::FPBits<double>(u_hi_3).abs().get_val(),98 0x1.0p-51, 0x1.0p-105);99}100 101LIBC_INLINE void sincos_eval(const Float128 &u, Float128 &sin_u,102 Float128 &cos_u) {103 Float128 u_sq = fputil::quick_mul(u, u);104 105 // sin(u) ~ x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - x^11/11! + x^13/13!106 constexpr Float128 SIN_COEFFS[] = {107 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1108 {Sign::NEG, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // -1/3!109 {Sign::POS, -134, 0x88888888'88888888'88888888'88888889_u128}, // 1/5!110 {Sign::NEG, -140, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // -1/7!111 {Sign::POS, -146, 0xb8ef1d2a'b6399c7d'560e4472'800b8ef2_u128}, // 1/9!112 {Sign::NEG, -153, 0xd7322b3f'aa271c7f'3a3f25c1'bee38f10_u128}, // -1/11!113 {Sign::POS, -160, 0xb092309d'43684be5'1c198e91'd7b4269e_u128}, // 1/13!114 };115 116 // cos(u) ~ 1 - x^2/2 + x^4/4! - x^6/6! + x^8/8! - x^10/10! + x^12/12!117 constexpr Float128 COS_COEFFS[] = {118 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128}, // 1.0119 {Sign::NEG, -128, 0x80000000'00000000'00000000'00000000_u128}, // 1/2120 {Sign::POS, -132, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128}, // 1/4!121 {Sign::NEG, -137, 0xb60b60b6'0b60b60b'60b60b60'b60b60b6_u128}, // 1/6!122 {Sign::POS, -143, 0xd00d00d0'0d00d00d'00d00d00'd00d00d0_u128}, // 1/8!123 {Sign::NEG, -149, 0x93f27dbb'c4fae397'780b69f5'333c725b_u128}, // 1/10!124 {Sign::POS, -156, 0x8f76c77f'c6c4bdaa'26d4c3d6'7f425f60_u128}, // 1/12!125 };126 127 sin_u = fputil::quick_mul(u, fputil::polyeval(u_sq, SIN_COEFFS[0],128 SIN_COEFFS[1], SIN_COEFFS[2],129 SIN_COEFFS[3], SIN_COEFFS[4],130 SIN_COEFFS[5], SIN_COEFFS[6]));131 cos_u = fputil::polyeval(u_sq, COS_COEFFS[0], COS_COEFFS[1], COS_COEFFS[2],132 COS_COEFFS[3], COS_COEFFS[4], COS_COEFFS[5],133 COS_COEFFS[6]);134}135 136} // namespace sincos_eval_internal137 138} // namespace math139 140} // namespace LIBC_NAMESPACE_DECL141 142#endif // LLVM_LIBC_SRC_MATH_GENERIC_SINCOSF_EVAL_H143