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| 1 | +//===-- Implementation header for atan --------------------------*- 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-exception |
| 6 | +// |
| 7 | +//===----------------------------------------------------------------------===// |
| 8 | + |
| 9 | +#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ATAN_H |
| 10 | +#define LLVM_LIBC_SRC___SUPPORT_MATH_ATAN_H |
| 11 | + |
| 12 | +#include "atan_utils.h" |
| 13 | +#include "src/__support/FPUtil/FEnvImpl.h" |
| 14 | +#include "src/__support/FPUtil/FPBits.h" |
| 15 | +#include "src/__support/FPUtil/double_double.h" |
| 16 | +#include "src/__support/FPUtil/multiply_add.h" |
| 17 | +#include "src/__support/FPUtil/nearest_integer.h" |
| 18 | +#include "src/__support/macros/config.h" |
| 19 | +#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY |
| 20 | + |
| 21 | +namespace LIBC_NAMESPACE_DECL { |
| 22 | + |
| 23 | +namespace math { |
| 24 | + |
| 25 | +// To compute atan(x), we divided it into the following cases: |
| 26 | +// * |x| < 2^-26: |
| 27 | +// Since |x| > atan(|x|) > |x| - |x|^3/3, and |x|^3/3 < ulp(x)/2, we simply |
| 28 | +// return atan(x) = x - sign(x) * epsilon. |
| 29 | +// * 2^-26 <= |x| < 1: |
| 30 | +// We perform range reduction mod 2^-6 = 1/64 as follow: |
| 31 | +// Let k = 2^(-6) * round(|x| * 2^6), then |
| 32 | +// atan(x) = sign(x) * atan(|x|) |
| 33 | +// = sign(x) * (atan(k) + atan((|x| - k) / (1 + |x|*k)). |
| 34 | +// We store atan(k) in a look up table, and perform intermediate steps in |
| 35 | +// double-double. |
| 36 | +// * 1 < |x| < 2^53: |
| 37 | +// First we perform the transformation y = 1/|x|: |
| 38 | +// atan(x) = sign(x) * (pi/2 - atan(1/|x|)) |
| 39 | +// = sign(x) * (pi/2 - atan(y)). |
| 40 | +// Then we compute atan(y) using range reduction mod 2^-6 = 1/64 as the |
| 41 | +// previous case: |
| 42 | +// Let k = 2^(-6) * round(y * 2^6), then |
| 43 | +// atan(y) = atan(k) + atan((y - k) / (1 + y*k)) |
| 44 | +// = atan(k) + atan((1/|x| - k) / (1 + k/|x|) |
| 45 | +// = atan(k) + atan((1 - k*|x|) / (|x| + k)). |
| 46 | +// * |x| >= 2^53: |
| 47 | +// Using the reciprocal transformation: |
| 48 | +// atan(x) = sign(x) * (pi/2 - atan(1/|x|)). |
| 49 | +// We have that: |
| 50 | +// atan(1/|x|) <= 1/|x| <= 2^-53, |
| 51 | +// which is smaller than ulp(pi/2) / 2. |
| 52 | +// So we can return: |
| 53 | +// atan(x) = sign(x) * (pi/2 - epsilon) |
| 54 | + |
| 55 | +LIBC_INLINE static constexpr double atan(double x) { |
| 56 | + |
| 57 | + using namespace atan_internal; |
| 58 | + using FPBits = fputil::FPBits<double>; |
| 59 | + |
| 60 | + constexpr double IS_NEG[2] = {1.0, -1.0}; |
| 61 | + constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54, |
| 62 | + 0x1.921fb54442d18p0}; |
| 63 | + constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54, |
| 64 | + -0x1.921fb54442d18p0}; |
| 65 | + |
| 66 | + FPBits xbits(x); |
| 67 | + bool x_sign = xbits.is_neg(); |
| 68 | + xbits = xbits.abs(); |
| 69 | + uint64_t x_abs = xbits.uintval(); |
| 70 | + int x_exp = |
| 71 | + static_cast<int>(x_abs >> FPBits::FRACTION_LEN) - FPBits::EXP_BIAS; |
| 72 | + |
| 73 | + // |x| < 1. |
| 74 | + if (x_exp < 0) { |
| 75 | + if (LIBC_UNLIKELY(x_exp < -26)) { |
| 76 | +#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 77 | + return x; |
| 78 | +#else |
| 79 | + if (x == 0.0) |
| 80 | + return x; |
| 81 | + // |x| < 2^-26 |
| 82 | + return fputil::multiply_add(-0x1.0p-54, x, x); |
| 83 | +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 84 | + } |
| 85 | + |
| 86 | + double x_d = xbits.get_val(); |
| 87 | + // k = 2^-6 * round(2^6 * |x|) |
| 88 | + double k = fputil::nearest_integer(0x1.0p6 * x_d); |
| 89 | + unsigned idx = static_cast<unsigned>(k); |
| 90 | + k *= 0x1.0p-6; |
| 91 | + |
| 92 | + // numerator = |x| - k |
| 93 | + DoubleDouble num, den; |
| 94 | + num.lo = 0.0; |
| 95 | + num.hi = x_d - k; |
| 96 | + |
| 97 | + // denominator = 1 - k * |x| |
| 98 | + den.hi = fputil::multiply_add(x_d, k, 1.0); |
| 99 | + DoubleDouble prod = fputil::exact_mult(x_d, k); |
| 100 | + // Using Dekker's 2SUM algorithm to compute the lower part. |
| 101 | + den.lo = ((1.0 - den.hi) + prod.hi) + prod.lo; |
| 102 | + |
| 103 | + // x_r = (|x| - k) / (1 + k * |x|) |
| 104 | + DoubleDouble x_r = fputil::div(num, den); |
| 105 | + |
| 106 | + // Approximating atan(x_r) using Taylor polynomial. |
| 107 | + DoubleDouble p = atan_eval(x_r); |
| 108 | + |
| 109 | + // atan(x) = sign(x) * (atan(k) + atan(x_r)) |
| 110 | + // = sign(x) * (atan(k) + atan( (|x| - k) / (1 + k * |x|) )) |
| 111 | +#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 112 | + return IS_NEG[x_sign] * (ATAN_I[idx].hi + (p.hi + (p.lo + ATAN_I[idx].lo))); |
| 113 | +#else |
| 114 | + |
| 115 | + DoubleDouble c0 = fputil::exact_add(ATAN_I[idx].hi, p.hi); |
| 116 | + double c1 = c0.lo + (ATAN_I[idx].lo + p.lo); |
| 117 | + double r = IS_NEG[x_sign] * (c0.hi + c1); |
| 118 | + |
| 119 | + return r; |
| 120 | +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 121 | + } |
| 122 | + |
| 123 | + // |x| >= 2^53 or x is NaN. |
| 124 | + if (LIBC_UNLIKELY(x_exp >= 53)) { |
| 125 | + // x is nan |
| 126 | + if (xbits.is_nan()) { |
| 127 | + if (xbits.is_signaling_nan()) { |
| 128 | + fputil::raise_except_if_required(FE_INVALID); |
| 129 | + return FPBits::quiet_nan().get_val(); |
| 130 | + } |
| 131 | + return x; |
| 132 | + } |
| 133 | + // |x| >= 2^53 |
| 134 | + // atan(x) ~ sign(x) * pi/2. |
| 135 | + if (x_exp >= 53) |
| 136 | +#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 137 | + return IS_NEG[x_sign] * PI_OVER_2.hi; |
| 138 | +#else |
| 139 | + return fputil::multiply_add(IS_NEG[x_sign], PI_OVER_2.hi, |
| 140 | + IS_NEG[x_sign] * PI_OVER_2.lo); |
| 141 | +#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 142 | + } |
| 143 | + |
| 144 | + double x_d = xbits.get_val(); |
| 145 | + double y = 1.0 / x_d; |
| 146 | + |
| 147 | + // k = 2^-6 * round(2^6 / |x|) |
| 148 | + double k = fputil::nearest_integer(0x1.0p6 * y); |
| 149 | + unsigned idx = static_cast<unsigned>(k); |
| 150 | + k *= 0x1.0p-6; |
| 151 | + |
| 152 | + // denominator = |x| + k |
| 153 | + DoubleDouble den = fputil::exact_add(x_d, k); |
| 154 | + // numerator = 1 - k * |x| |
| 155 | + DoubleDouble num; |
| 156 | + num.hi = fputil::multiply_add(-x_d, k, 1.0); |
| 157 | + DoubleDouble prod = fputil::exact_mult(x_d, k); |
| 158 | + // Using Dekker's 2SUM algorithm to compute the lower part. |
| 159 | + num.lo = ((1.0 - num.hi) - prod.hi) - prod.lo; |
| 160 | + |
| 161 | + // x_r = (1/|x| - k) / (1 - k/|x|) |
| 162 | + // = (1 - k * |x|) / (|x| - k) |
| 163 | + DoubleDouble x_r = fputil::div(num, den); |
| 164 | + |
| 165 | + // Approximating atan(x_r) using Taylor polynomial. |
| 166 | + DoubleDouble p = atan_eval(x_r); |
| 167 | + |
| 168 | + // atan(x) = sign(x) * (pi/2 - atan(1/|x|)) |
| 169 | + // = sign(x) * (pi/2 - atan(k) - atan(x_r)) |
| 170 | + // = (-sign(x)) * (-pi/2 + atan(k) + atan((1 - k*|x|)/(|x| - k))) |
| 171 | +#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS |
| 172 | + double lo_part = p.lo + ATAN_I[idx].lo + MPI_OVER_2.lo; |
| 173 | + return IS_NEG[!x_sign] * (MPI_OVER_2.hi + ATAN_I[idx].hi + (p.hi + lo_part)); |
| 174 | +#else |
| 175 | + DoubleDouble c0 = fputil::exact_add(MPI_OVER_2.hi, ATAN_I[idx].hi); |
| 176 | + DoubleDouble c1 = fputil::exact_add(c0.hi, p.hi); |
| 177 | + double c2 = c1.lo + (c0.lo + p.lo) + (ATAN_I[idx].lo + MPI_OVER_2.lo); |
| 178 | + |
| 179 | + double r = IS_NEG[!x_sign] * (c1.hi + c2); |
| 180 | + |
| 181 | + return r; |
| 182 | +#endif |
| 183 | +} |
| 184 | + |
| 185 | +} // namespace math |
| 186 | + |
| 187 | +} // namespace LIBC_NAMESPACE_DECL |
| 188 | + |
| 189 | +#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATAN_H |
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