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| 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-exception |
| 6 | +// |
| 7 | +//===----------------------------------------------------------------------===// |
| 8 | + |
| 9 | +#include "src/math/asin.h" |
| 10 | +#include "asin_utils.h" |
| 11 | +#include "src/__support/FPUtil/FEnvImpl.h" |
| 12 | +#include "src/__support/FPUtil/FPBits.h" |
| 13 | +#include "src/__support/FPUtil/PolyEval.h" |
| 14 | +#include "src/__support/FPUtil/double_double.h" |
| 15 | +#include "src/__support/FPUtil/dyadic_float.h" |
| 16 | +#include "src/__support/FPUtil/multiply_add.h" |
| 17 | +#include "src/__support/FPUtil/sqrt.h" |
| 18 | +#include "src/__support/macros/config.h" |
| 19 | +#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY |
| 20 | +#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA |
| 21 | + |
| 22 | +namespace LIBC_NAMESPACE_DECL { |
| 23 | + |
| 24 | +using DoubleDouble = fputil::DoubleDouble; |
| 25 | +using Float128 = fputil::DyadicFloat<128>; |
| 26 | + |
| 27 | +LLVM_LIBC_FUNCTION(double, asin, (double x)) { |
| 28 | + using FPBits = typename fputil::FPBits<double>; |
| 29 | + |
| 30 | + FPBits xbits(x); |
| 31 | + int x_exp = xbits.get_biased_exponent(); |
| 32 | + |
| 33 | + // |x| < 0.5. |
| 34 | + if (x_exp < FPBits::EXP_BIAS - 1) { |
| 35 | + // |x| < 2^-26. |
| 36 | + if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 26)) { |
| 37 | + // When |x| < 2^-26, the relative error of the approximation asin(x) ~ x |
| 38 | + // is: |
| 39 | + // |asin(x) - x| / |asin(x)| < |x^3| / (6|x|) |
| 40 | + // = x^2 / 6 |
| 41 | + // < 2^-54 |
| 42 | + // < epsilon(1)/2. |
| 43 | + // So the correctly rounded values of asin(x) are: |
| 44 | + // = x + sign(x)*eps(x) if rounding mode = FE_TOWARDZERO, |
| 45 | + // or (rounding mode = FE_UPWARD and x is |
| 46 | + // negative), |
| 47 | + // = x otherwise. |
| 48 | + // To simplify the rounding decision and make it more efficient, we use |
| 49 | + // fma(x, 2^-54, x) instead. |
| 50 | + // Note: to use the formula x + 2^-54*x to decide the correct rounding, we |
| 51 | + // do need fma(x, 2^-54, x) to prevent underflow caused by 2^-54*x when |
| 52 | + // |x| < 2^-1022. For targets without FMA instructions, when x is close to |
| 53 | + // denormal range, we normalize x, |
| 54 | +#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) |
| 55 | + return x; |
| 56 | +#elif defined(LIBC_TARGET_CPU_HAS_FMA_DOUBLE) |
| 57 | + return fputil::multiply_add(x, 0x1.0p-54, x); |
| 58 | +#else |
| 59 | + if (x == 0.0) |
| 60 | + return x; |
| 61 | + // Get sign(x) * min_normal. |
| 62 | + FPBits eps_bits = FPBits::min_normal(); |
| 63 | + eps_bits.set_sign(xbits.sign()); |
| 64 | + double eps = eps_bits.get_val(); |
| 65 | + double normalize_const = (x_exp == 0) ? eps : 0.0; |
| 66 | + double scaled_normal = |
| 67 | + fputil::multiply_add(x + normalize_const, 0x1.0p54, eps); |
| 68 | + return fputil::multiply_add(scaled_normal, 0x1.0p-54, -normalize_const); |
| 69 | +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 70 | + } |
| 71 | + |
| 72 | + unsigned idx; |
| 73 | + DoubleDouble x_sq = fputil::exact_mult(x, x); |
| 74 | + double err = x * 0x1.0p-52; |
| 75 | + // Polynomial approximation: |
| 76 | + // p ~ asin(x)/x - ASIN_COEFFS[idx][0] |
| 77 | + double p = asin_eval(x_sq, idx, err); |
| 78 | + // asin(x) ~ x * (ASIN_COEFFS[idx][0] + p) |
| 79 | + DoubleDouble r0 = fputil::exact_mult(x, ASIN_COEFFS[idx][0]); |
| 80 | + double r_lo = fputil::multiply_add(x, p, r0.lo); |
| 81 | + |
| 82 | +#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 83 | + return r0.hi + r_lo; |
| 84 | +#else |
| 85 | + // Ziv's accuracy test. |
| 86 | + |
| 87 | + double r_upper = r0.hi + (r_lo + err); |
| 88 | + double r_lower = r0.hi + (r_lo - err); |
| 89 | + |
| 90 | + if (LIBC_LIKELY(r_upper == r_lower)) |
| 91 | + return r_upper; |
| 92 | + |
| 93 | + // Ziv's accuracy test failed, perform 128-bit calculation. |
| 94 | + |
| 95 | + // Get x^2 - idx/64 exactly. When FMA is available, double-double |
| 96 | + // multiplication will be correct for all rounding modes. Otherwise we use |
| 97 | + // Float128 directly. |
| 98 | + Float128 x_f128(x); |
| 99 | + |
| 100 | +#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE |
| 101 | + // u = x^2 - idx/64 |
| 102 | + Float128 u_hi( |
| 103 | + fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, x_sq.hi)); |
| 104 | + Float128 u = fputil::quick_add(u_hi, Float128(x_sq.lo)); |
| 105 | +#else |
| 106 | + Float128 x_sq_f128 = fputil::quick_mul(x_f128, x_f128); |
| 107 | + Float128 u = fputil::quick_add( |
| 108 | + x_sq_f128, Float128(static_cast<double>(idx) * (-0x1.0p-6))); |
| 109 | +#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE |
| 110 | + |
| 111 | + Float128 p_f128 = asin_eval(u, idx); |
| 112 | + Float128 r = fputil::quick_mul(x_f128, p_f128); |
| 113 | + |
| 114 | + return static_cast<double>(r); |
| 115 | +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 116 | + } |
| 117 | + // |x| >= 0.5 |
| 118 | + |
| 119 | + double x_abs = xbits.abs().get_val(); |
| 120 | + |
| 121 | + // |x| >= 1 |
| 122 | + if (LIBC_UNLIKELY(x_exp >= FPBits::EXP_BIAS)) { |
| 123 | + // x = +-1, asin(x) = +- pi/2 |
| 124 | + if (x_abs == 1.0) { |
| 125 | + // return +- pi/2 |
| 126 | + } |
| 127 | + // |x| > 1, return NaN. |
| 128 | + if (xbits.is_finite()) { |
| 129 | + fputil::set_errno_if_required(EDOM); |
| 130 | + fputil::raise_except_if_required(FE_INVALID); |
| 131 | + } |
| 132 | + return FPBits::quiet_nan().get_val(); |
| 133 | + } |
| 134 | + |
| 135 | + // When |x| >= 0.5, we perform range reduction as follow: |
| 136 | + // |
| 137 | + // Assume further that 0.5 < x <= 1, and let: |
| 138 | + // y = asin(x) |
| 139 | + // We will use the double angle formula: |
| 140 | + // cos(2y) = 1 - 2 sin^2(y) |
| 141 | + // and the complement angle identity: |
| 142 | + // x = sin(y) = cos(pi/2 - y) |
| 143 | + // = 1 - 2 sin^2 (pi/4 - y/2) |
| 144 | + // So: |
| 145 | + // sin(pi/4 - y/2) = sqrt( (1 - x)/2 ) |
| 146 | + // And hence: |
| 147 | + // pi/4 - y/2 = asin( sqrt( (1 - x)/2 ) ) |
| 148 | + // Equivalently: |
| 149 | + // asin(x) = y = pi/2 - 2 * asin( sqrt( (1 - x)/2 ) ) |
| 150 | + // Let u = (1 - x)/2, then: |
| 151 | + // asin(x) = pi/2 - 2 * asin( sqrt(u) ) |
| 152 | + // Moreover, since 0.5 < x <= 1: |
| 153 | + // 0 <= u < 1/4, and 0 <= sqrt(u) < 0.5, |
| 154 | + // And hence we can reuse the same polynomial approximation of asin(x) when |
| 155 | + // |x| <= 0.5: |
| 156 | + // asin(x) ~ pi/2 - 2 * sqrt(u) * P(u), |
| 157 | + |
| 158 | + // u = (1 - |x|)/2 |
| 159 | + double u = fputil::multiply_add(x_abs, -0.5, 0.5); |
| 160 | + // v_hi + v_lo ~ sqrt(u). |
| 161 | + // Let: |
| 162 | + // h = u - v_hi^2 = (sqrt(u) - v_hi) * (sqrt(u) + v_hi) |
| 163 | + // Then: |
| 164 | + // sqrt(u) = v_hi + h / (sqrt(u) + v_hi) |
| 165 | + // ~ v_hi + h / (2 * v_hi) |
| 166 | + // So we can use: |
| 167 | + // v_lo = h / (2 * v_hi). |
| 168 | + // Then, |
| 169 | + // asin(x) ~ pi/2 - 2*(v_hi + v_lo) * P(u) |
| 170 | + double v_hi = fputil::sqrt<double>(u); |
| 171 | +#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE |
| 172 | + double h = fputil::multiply_add(v_hi, -v_hi, u); |
| 173 | +#else |
| 174 | + DoubleDouble v_hi_sq = fputil::exact_mult(v_hi, v_hi); |
| 175 | + double h = (u - v_hi_sq.hi) - v_hi_sq.lo; |
| 176 | +#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE |
| 177 | + |
| 178 | + // Scale v_lo and v_hi by 2 from the formula: |
| 179 | + // vh = v_hi * 2 |
| 180 | + // vl = 2*v_lo = h / v_hi. |
| 181 | + double vh = v_hi * 2.0; |
| 182 | + double vl = h / v_hi; |
| 183 | + |
| 184 | + // Polynomial approximation: |
| 185 | + // p ~ asin(sqrt(u))/sqrt(u) - ASIN_COEFFS[idx][0] |
| 186 | + unsigned idx; |
| 187 | + [[maybe_unused]] double err = vh * 0x1.0p-52; |
| 188 | + double p = asin_eval(DoubleDouble{0.0, u}, idx, err); |
| 189 | + |
| 190 | + // Perform computations in double-double arithmetic: |
| 191 | + // asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p) |
| 192 | + DoubleDouble r0 = fputil::exact_mult(vh, ASIN_COEFFS[idx][0]); |
| 193 | + DoubleDouble r = fputil::exact_add(PI_OVER_TWO.hi, -r0.hi); |
| 194 | + |
| 195 | + // Combining all the lower terms. |
| 196 | + double lo = r.lo - fputil::multiply_add(vl, ASIN_COEFFS[idx][0], |
| 197 | + r0.lo - PI_OVER_TWO.lo); |
| 198 | + double r_lo = fputil::multiply_add(vh, -p, lo); |
| 199 | + |
| 200 | +#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 201 | +#else |
| 202 | + // Ziv's accuracy test. |
| 203 | + |
| 204 | + double r_upper = r.hi + (r_lo + err); |
| 205 | + double r_lower = r.hi + (r_lo - err); |
| 206 | + |
| 207 | + if (LIBC_LIKELY(r_upper == r_lower)) |
| 208 | + return r_upper; |
| 209 | + |
| 210 | + // Ziv's accuracy test failed, we redo the computations in Float128. |
| 211 | + |
| 212 | + // After the first step of Newton-Raphson approximating v = sqrt(u), we have |
| 213 | + // that: |
| 214 | + // sqrt(u) = v_hi + h / (sqrt(u) + v_hi) |
| 215 | + // v_lo = h / (2 * v_hi) |
| 216 | + // With error: |
| 217 | + // sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) ) |
| 218 | + // = -h^2 / (2*v * (sqrt(u) + v)^2). |
| 219 | + // Since: |
| 220 | + // (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u, |
| 221 | + // we can add another correction term to (v_hi + v_lo) that is: |
| 222 | + // v_ll = -h^2 / (2*v_hi * 4u) |
| 223 | + // = -v_lo * (h / 4u) |
| 224 | + // = -vl * (h / 8u), |
| 225 | + // making the errors: |
| 226 | + // sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3) |
| 227 | + // well beyond 128-bit precision needed. |
| 228 | + |
| 229 | + // Get the rounding error of vl = 2 * v_lo ~ h / vh |
| 230 | + // Get full product of vh * vl |
| 231 | +#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE |
| 232 | + double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi; |
| 233 | +#else |
| 234 | + DoubleDouble vh_vl = fputil::exact_mult(v_hi, vl); |
| 235 | + double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi; |
| 236 | +#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE |
| 237 | + // vll = 2*v_ll = -vl * (h / (4u)). |
| 238 | + double t = h * (-0.25) / u; |
| 239 | + double vll = fputil::multiply_add(vl, t, vl_lo); |
| 240 | + // m_v = -(v_hi + v_lo + v_ll). |
| 241 | + Float128 m_v = fputil::quick_add( |
| 242 | + Float128(vh), fputil::quick_add(Float128(vl), Float128(vll))); |
| 243 | + m_v.sign = Sign::NEG; |
| 244 | + |
| 245 | + // Perform computations in Float128: |
| 246 | + // asin(x) = pi/2 - (v_hi + v_lo + vll) * P(u). |
| 247 | + Float128 y_f128(fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, u)); |
| 248 | + |
| 249 | + Float128 p_f128 = asin_eval(y_f128, idx); |
| 250 | + Float128 r0_f128 = fputil::quick_mul(m_v, p_f128); |
| 251 | + Float128 r_f128 = fputil::quick_add(PI_OVER_TWO_F128, r0_f128); |
| 252 | + |
| 253 | + return static_cast<double>(r_f128); |
| 254 | +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 255 | +} |
| 256 | + |
| 257 | +} // namespace LIBC_NAMESPACE_DECL |
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