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1//===-- Implementation header for acospif16 ---------------------*- 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___SUPPORT_MATH_ACOSPIF16_H10#define LLVM_LIBC_SRC___SUPPORT_MATH_ACOSPIF16_H11 12#include "include/llvm-libc-macros/float16-macros.h"13 14#ifdef LIBC_TYPES_HAS_FLOAT1615 16#include "src/__support/FPUtil/FEnvImpl.h"17#include "src/__support/FPUtil/FPBits.h"18#include "src/__support/FPUtil/PolyEval.h"19#include "src/__support/FPUtil/cast.h"20#include "src/__support/FPUtil/multiply_add.h"21#include "src/__support/FPUtil/sqrt.h"22#include "src/__support/macros/optimization.h"23 24namespace LIBC_NAMESPACE_DECL {25 26namespace math {27 28LIBC_INLINE static constexpr float16 acospif16(float16 x) {29 using FPBits = fputil::FPBits<float16>;30 FPBits xbits(x);31 32 uint16_t x_u = xbits.uintval();33 uint16_t x_abs = x_u & 0x7fff;34 uint16_t x_sign = x_u >> 15;35 36 // |x| > 0x1p0, |x| > 1, or x is NaN.37 if (LIBC_UNLIKELY(x_abs > 0x3c00)) {38 // acospif16(NaN) = NaN39 if (xbits.is_nan()) {40 if (xbits.is_signaling_nan()) {41 fputil::raise_except_if_required(FE_INVALID);42 return FPBits::quiet_nan().get_val();43 }44 45 return x;46 }47 48 // 1 < |x| <= +inf49 fputil::raise_except_if_required(FE_INVALID);50 fputil::set_errno_if_required(EDOM);51 52 return FPBits::quiet_nan().get_val();53 }54 55 // |x| == 0x1p0, x is 1 or -156 // if x is (-)1, return 157 // if x is (+)1, return 058 if (LIBC_UNLIKELY(x_abs == 0x3c00))59 return fputil::cast<float16>(x_sign ? 1.0f : 0.0f);60 61 float xf = x;62 float xsq = xf * xf;63 64 // Degree-6 minimax polynomial coefficients of asin(x) generated by Sollya65 // with: > P = fpminimax(asin(x)/(pi * x), [|0, 2, 4, 6, 8|], [|SG...|], [0,66 // 0.5]);67 constexpr float POLY_COEFFS[5] = {0x1.45f308p-2f, 0x1.b2900cp-5f,68 0x1.897e36p-6f, 0x1.9efafcp-7f,69 0x1.06d884p-6f};70 // |x| <= 0x1p-1, |x| <= 0.571 if (x_abs <= 0x3800) {72 // if x is 0, return 0.573 if (LIBC_UNLIKELY(x_abs == 0))74 return fputil::cast<float16>(0.5f);75 76 // Note that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x), then77 // acospi(x) = 0.5 - asin(x)/pi78 float interm =79 fputil::polyeval(xsq, POLY_COEFFS[0], POLY_COEFFS[1], POLY_COEFFS[2],80 POLY_COEFFS[3], POLY_COEFFS[4]);81 82 return fputil::cast<float16>(fputil::multiply_add(-xf, interm, 0.5f));83 }84 85 // When |x| > 0.5, assume that 0.5 < |x| <= 186 //87 // Step-by-step range-reduction proof:88 // 1: Let y = asin(x), such that, x = sin(y)89 // 2: From complimentary angle identity:90 // x = sin(y) = cos(pi/2 - y)91 // 3: Let z = pi/2 - y, such that x = cos(z)92 // 4: From double angle formula; cos(2A) = 1 - 2 * sin^2(A):93 // z = 2A, z/2 = A94 // cos(z) = 1 - 2 * sin^2(z/2)95 // 5: Make sin(z/2) subject of the formula:96 // sin(z/2) = sqrt((1 - cos(z))/2)97 // 6: Recall [3]; x = cos(z). Therefore:98 // sin(z/2) = sqrt((1 - x)/2)99 // 7: Let u = (1 - x)/2100 // 8: Therefore:101 // asin(sqrt(u)) = z/2102 // 2 * asin(sqrt(u)) = z103 // 9: Recall [3]; z = pi/2 - y. Therefore:104 // y = pi/2 - z105 // y = pi/2 - 2 * asin(sqrt(u))106 // 10: Recall [1], y = asin(x). Therefore:107 // asin(x) = pi/2 - 2 * asin(sqrt(u))108 // 11: Recall that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x)109 // Therefore:110 // acos(x) = pi/2 - (pi/2 - 2 * asin(sqrt(u)))111 // acos(x) = 2 * asin(sqrt(u))112 // acospi(x) = 2 * (asin(sqrt(u)) / pi)113 //114 // THE RANGE REDUCTION, HOW?115 // 12: Recall [7], u = (1 - x)/2116 // 13: Since 0.5 < x <= 1, therefore:117 // 0 <= u <= 0.25 and 0 <= sqrt(u) <= 0.5118 //119 // Hence, we can reuse the same [0, 0.5] domain polynomial approximation for120 // Step [11] as `sqrt(u)` is in range.121 // When -1 < x <= -0.5, the identity:122 // acos(x) = pi - acos(-x)123 // acospi(x) = 1 - acos(-x)/pi124 // allows us to compute for the negative x value (lhs)125 // with a positive x value instead (rhs).126 127 float xf_abs = (xf < 0 ? -xf : xf);128 float u = fputil::multiply_add(-0.5f, xf_abs, 0.5f);129 float sqrt_u = fputil::sqrt<float>(u);130 131 float asin_sqrt_u =132 sqrt_u * fputil::polyeval(u, POLY_COEFFS[0], POLY_COEFFS[1],133 POLY_COEFFS[2], POLY_COEFFS[3], POLY_COEFFS[4]);134 135 // Same as acos(x), but devided the expression with pi136 return fputil::cast<float16>(137 x_sign ? fputil::multiply_add(-2.0f, asin_sqrt_u, 1.0f)138 : 2.0f * asin_sqrt_u);139}140 141} // namespace math142 143} // namespace LIBC_NAMESPACE_DECL144 145#endif // LIBC_TYPES_HAS_FLOAT16146 147#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ACOSPIF16_H148