277 lines · cpp
1//===-- Double-precision asin function ------------------------------------===//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#include "src/math/asin.h"10#include "src/__support/math/asin.h"11 12namespace LIBC_NAMESPACE_DECL {13 14LLVM_LIBC_FUNCTION(double, asin, (double x)) {15 using namespace asin_internal;16 using FPBits = fputil::FPBits<double>;17 18 FPBits xbits(x);19 int x_exp = xbits.get_biased_exponent();20 21 // |x| < 0.5.22 if (x_exp < FPBits::EXP_BIAS - 1) {23 // |x| < 2^-26.24 if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 26)) {25 // When |x| < 2^-26, the relative error of the approximation asin(x) ~ x26 // is:27 // |asin(x) - x| / |asin(x)| < |x^3| / (6|x|)28 // = x^2 / 629 // < 2^-5430 // < epsilon(1)/2.31 // So the correctly rounded values of asin(x) are:32 // = x + sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,33 // or (rounding mode = FE_UPWARD and x is34 // negative),35 // = x otherwise.36 // To simplify the rounding decision and make it more efficient, we use37 // fma(x, 2^-54, x) instead.38 // Note: to use the formula x + 2^-54*x to decide the correct rounding, we39 // do need fma(x, 2^-54, x) to prevent underflow caused by 2^-54*x when40 // |x| < 2^-1022. For targets without FMA instructions, when x is close to41 // denormal range, we normalize x,42#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS)43 return x;44#elif defined(LIBC_TARGET_CPU_HAS_FMA_DOUBLE)45 return fputil::multiply_add(x, 0x1.0p-54, x);46#else47 if (xbits.abs().uintval() == 0)48 return x;49 // Get sign(x) * min_normal.50 FPBits eps_bits = FPBits::min_normal();51 eps_bits.set_sign(xbits.sign());52 double eps = eps_bits.get_val();53 double normalize_const = (x_exp == 0) ? eps : 0.0;54 double scaled_normal =55 fputil::multiply_add(x + normalize_const, 0x1.0p54, eps);56 return fputil::multiply_add(scaled_normal, 0x1.0p-54, -normalize_const);57#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS58 }59 60#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS61 return x * asin_eval(x * x);62#else63 unsigned idx;64 DoubleDouble x_sq = fputil::exact_mult(x, x);65 double err = xbits.abs().get_val() * 0x1.0p-51;66 // Polynomial approximation:67 // p ~ asin(x)/x68 69 DoubleDouble p = asin_eval(x_sq, idx, err);70 // asin(x) ~ x * (ASIN_COEFFS[idx][0] + p)71 DoubleDouble r0 = fputil::exact_mult(x, p.hi);72 double r_lo = fputil::multiply_add(x, p.lo, r0.lo);73 74 // Ziv's accuracy test.75 76 double r_upper = r0.hi + (r_lo + err);77 double r_lower = r0.hi + (r_lo - err);78 79 if (LIBC_LIKELY(r_upper == r_lower))80 return r_upper;81 82 // Ziv's accuracy test failed, perform 128-bit calculation.83 84 // Recalculate mod 1/64.85 idx = static_cast<unsigned>(fputil::nearest_integer(x_sq.hi * 0x1.0p6));86 87 // Get x^2 - idx/64 exactly. When FMA is available, double-double88 // multiplication will be correct for all rounding modes. Otherwise we use89 // Float128 directly.90 Float128 x_f128(x);91 92#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE93 // u = x^2 - idx/6494 Float128 u_hi(95 fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, x_sq.hi));96 Float128 u = fputil::quick_add(u_hi, Float128(x_sq.lo));97#else98 Float128 x_sq_f128 = fputil::quick_mul(x_f128, x_f128);99 Float128 u = fputil::quick_add(100 x_sq_f128, Float128(static_cast<double>(idx) * (-0x1.0p-6)));101#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE102 103 Float128 p_f128 = asin_eval(u, idx);104 Float128 r = fputil::quick_mul(x_f128, p_f128);105 106 return static_cast<double>(r);107#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS108 }109 // |x| >= 0.5110 111 double x_abs = xbits.abs().get_val();112 113 // Maintaining the sign:114 constexpr double SIGN[2] = {1.0, -1.0};115 double x_sign = SIGN[xbits.is_neg()];116 117 // |x| >= 1118 if (LIBC_UNLIKELY(x_exp >= FPBits::EXP_BIAS)) {119 // x = +-1, asin(x) = +- pi/2120 if (x_abs == 1.0) {121 // return +- pi/2122 return fputil::multiply_add(x_sign, PI_OVER_TWO.hi,123 x_sign * PI_OVER_TWO.lo);124 }125 // |x| > 1, return NaN.126 if (xbits.is_quiet_nan())127 return x;128 129 // Set domain error for non-NaN input.130 if (!xbits.is_nan())131 fputil::set_errno_if_required(EDOM);132 133 fputil::raise_except_if_required(FE_INVALID);134 return FPBits::quiet_nan().get_val();135 }136 137 // When |x| >= 0.5, we perform range reduction as follow:138 //139 // Assume further that 0.5 <= x < 1, and let:140 // y = asin(x)141 // We will use the double angle formula:142 // cos(2y) = 1 - 2 sin^2(y)143 // and the complement angle identity:144 // x = sin(y) = cos(pi/2 - y)145 // = 1 - 2 sin^2 (pi/4 - y/2)146 // So:147 // sin(pi/4 - y/2) = sqrt( (1 - x)/2 )148 // And hence:149 // pi/4 - y/2 = asin( sqrt( (1 - x)/2 ) )150 // Equivalently:151 // asin(x) = y = pi/2 - 2 * asin( sqrt( (1 - x)/2 ) )152 // Let u = (1 - x)/2, then:153 // asin(x) = pi/2 - 2 * asin( sqrt(u) )154 // Moreover, since 0.5 <= x < 1:155 // 0 < u <= 1/4, and 0 < sqrt(u) <= 0.5,156 // And hence we can reuse the same polynomial approximation of asin(x) when157 // |x| <= 0.5:158 // asin(x) ~ pi/2 - 2 * sqrt(u) * P(u),159 160 // u = (1 - |x|)/2161 double u = fputil::multiply_add(x_abs, -0.5, 0.5);162 // v_hi + v_lo ~ sqrt(u).163 // Let:164 // h = u - v_hi^2 = (sqrt(u) - v_hi) * (sqrt(u) + v_hi)165 // Then:166 // sqrt(u) = v_hi + h / (sqrt(u) + v_hi)167 // ~ v_hi + h / (2 * v_hi)168 // So we can use:169 // v_lo = h / (2 * v_hi).170 // Then,171 // asin(x) ~ pi/2 - 2*(v_hi + v_lo) * P(u)172 double v_hi = fputil::sqrt<double>(u);173 174#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS175 double p = asin_eval(u);176 double r = x_sign * fputil::multiply_add(-2.0 * v_hi, p, PI_OVER_TWO.hi);177 return r;178#else179 180#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE181 double h = fputil::multiply_add(v_hi, -v_hi, u);182#else183 DoubleDouble v_hi_sq = fputil::exact_mult(v_hi, v_hi);184 double h = (u - v_hi_sq.hi) - v_hi_sq.lo;185#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE186 187 // Scale v_lo and v_hi by 2 from the formula:188 // vh = v_hi * 2189 // vl = 2*v_lo = h / v_hi.190 double vh = v_hi * 2.0;191 double vl = h / v_hi;192 193 // Polynomial approximation:194 // p ~ asin(sqrt(u))/sqrt(u)195 unsigned idx;196 double err = vh * 0x1.0p-51;197 198 DoubleDouble p = asin_eval(DoubleDouble{0.0, u}, idx, err);199 200 // Perform computations in double-double arithmetic:201 // asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p)202 DoubleDouble r0 = fputil::quick_mult(DoubleDouble{vl, vh}, p);203 DoubleDouble r = fputil::exact_add(PI_OVER_TWO.hi, -r0.hi);204 205 double r_lo = PI_OVER_TWO.lo - r0.lo + r.lo;206 207 // Ziv's accuracy test.208 209#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE210 double r_upper = fputil::multiply_add(211 r.hi, x_sign, fputil::multiply_add(r_lo, x_sign, err));212 double r_lower = fputil::multiply_add(213 r.hi, x_sign, fputil::multiply_add(r_lo, x_sign, -err));214#else215 r_lo *= x_sign;216 r.hi *= x_sign;217 double r_upper = r.hi + (r_lo + err);218 double r_lower = r.hi + (r_lo - err);219#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE220 221 if (LIBC_LIKELY(r_upper == r_lower))222 return r_upper;223 224 // Ziv's accuracy test failed, we redo the computations in Float128.225 // Recalculate mod 1/64.226 idx = static_cast<unsigned>(fputil::nearest_integer(u * 0x1.0p6));227 228 // After the first step of Newton-Raphson approximating v = sqrt(u), we have229 // that:230 // sqrt(u) = v_hi + h / (sqrt(u) + v_hi)231 // v_lo = h / (2 * v_hi)232 // With error:233 // sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) )234 // = -h^2 / (2*v * (sqrt(u) + v)^2).235 // Since:236 // (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u,237 // we can add another correction term to (v_hi + v_lo) that is:238 // v_ll = -h^2 / (2*v_hi * 4u)239 // = -v_lo * (h / 4u)240 // = -vl * (h / 8u),241 // making the errors:242 // sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3)243 // well beyond 128-bit precision needed.244 245 // Get the rounding error of vl = 2 * v_lo ~ h / vh246 // Get full product of vh * vl247#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE248 double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi;249#else250 DoubleDouble vh_vl = fputil::exact_mult(v_hi, vl);251 double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi;252#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE253 // vll = 2*v_ll = -vl * (h / (4u)).254 double t = h * (-0.25) / u;255 double vll = fputil::multiply_add(vl, t, vl_lo);256 // m_v = -(v_hi + v_lo + v_ll).257 Float128 m_v = fputil::quick_add(258 Float128(vh), fputil::quick_add(Float128(vl), Float128(vll)));259 m_v.sign = Sign::NEG;260 261 // Perform computations in Float128:262 // asin(x) = pi/2 - (v_hi + v_lo + vll) * P(u).263 Float128 y_f128(fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, u));264 265 Float128 p_f128 = asin_eval(y_f128, idx);266 Float128 r0_f128 = fputil::quick_mul(m_v, p_f128);267 Float128 r_f128 = fputil::quick_add(PI_OVER_TWO_F128, r0_f128);268 269 if (xbits.is_neg())270 r_f128.sign = Sign::NEG;271 272 return static_cast<double>(r_f128);273#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS274}275 276} // namespace LIBC_NAMESPACE_DECL277