1 //===-- Implementation of exp2m1f 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/exp2m1f.h" 10 #include "src/__support/FPUtil/FEnvImpl.h" 11 #include "src/__support/FPUtil/FPBits.h" 12 #include "src/__support/FPUtil/PolyEval.h" 13 #include "src/__support/FPUtil/except_value_utils.h" 14 #include "src/__support/FPUtil/multiply_add.h" 15 #include "src/__support/FPUtil/rounding_mode.h" 16 #include "src/__support/common.h" 17 #include "src/__support/macros/config.h" 18 #include "src/__support/macros/optimization.h" 19 #include "src/__support/macros/properties/cpu_features.h" 20 #include "src/errno/libc_errno.h" 21 22 #include "explogxf.h" 23 24 namespace LIBC_NAMESPACE_DECL { 25 26 static constexpr size_t N_EXCEPTS_LO = 8; 27 28 static constexpr fputil::ExceptValues<float, N_EXCEPTS_LO> EXP2M1F_EXCEPTS_LO = 29 {{ 30 // (input, RZ output, RU offset, RD offset, RN offset) 31 // x = 0x1.36dc8ep-36, exp2m1f(x) = 0x1.aef212p-37 (RZ) 32 {0x2d9b'6e47U, 0x2d57'7909U, 1U, 0U, 0U}, 33 // x = 0x1.224936p-19, exp2m1f(x) = 0x1.926c0ep-20 (RZ) 34 {0x3611'249bU, 0x35c9'3607U, 1U, 0U, 1U}, 35 // x = 0x1.d16d2p-20, exp2m1f(x) = 0x1.429becp-20 (RZ) 36 {0x35e8'b690U, 0x35a1'4df6U, 1U, 0U, 1U}, 37 // x = 0x1.17949ep-14, exp2m1f(x) = 0x1.8397p-15 (RZ) 38 {0x388b'ca4fU, 0x3841'cb80U, 1U, 0U, 1U}, 39 // x = -0x1.9c3e1ep-38, exp2m1f(x) = -0x1.1dbeacp-38 (RZ) 40 {0xacce'1f0fU, 0xac8e'df56U, 0U, 1U, 0U}, 41 // x = -0x1.4d89b4p-32, exp2m1f(x) = -0x1.ce61b6p-33 (RZ) 42 {0xafa6'c4daU, 0xaf67'30dbU, 0U, 1U, 1U}, 43 // x = -0x1.a6eac4p-10, exp2m1f(x) = -0x1.24fadap-10 (RZ) 44 {0xbad3'7562U, 0xba92'7d6dU, 0U, 1U, 1U}, 45 // x = -0x1.e7526ep-6, exp2m1f(x) = -0x1.4e53dep-6 (RZ) 46 {0xbcf3'a937U, 0xbca7'29efU, 0U, 1U, 1U}, 47 }}; 48 49 static constexpr size_t N_EXCEPTS_HI = 3; 50 51 static constexpr fputil::ExceptValues<float, N_EXCEPTS_HI> EXP2M1F_EXCEPTS_HI = 52 {{ 53 // (input, RZ output, RU offset, RD offset, RN offset) 54 // x = 0x1.16a972p-1, exp2m1f(x) = 0x1.d545b2p-2 (RZ) 55 {0x3f0b'54b9U, 0x3eea'a2d9U, 1U, 0U, 0U}, 56 // x = -0x1.9f12acp-5, exp2m1f(x) = -0x1.1ab68cp-5 (RZ) 57 {0xbd4f'8956U, 0xbd0d'5b46U, 0U, 1U, 0U}, 58 // x = -0x1.de7b9cp-5, exp2m1f(x) = -0x1.4508f4p-5 (RZ) 59 {0xbd6f'3dceU, 0xbd22'847aU, 0U, 1U, 1U}, 60 }}; 61 62 LLVM_LIBC_FUNCTION(float, exp2m1f, (float x)) { 63 using FPBits = fputil::FPBits<float>; 64 FPBits xbits(x); 65 66 uint32_t x_u = xbits.uintval(); 67 uint32_t x_abs = x_u & 0x7fff'ffffU; 68 69 // When |x| >= 128, or x is nan, or |x| <= 2^-5 70 if (LIBC_UNLIKELY(x_abs >= 0x4300'0000U || x_abs <= 0x3d00'0000U)) { 71 // |x| <= 2^-5 72 if (x_abs <= 0x3d00'0000U) { 73 if (auto r = EXP2M1F_EXCEPTS_LO.lookup(x_u); LIBC_UNLIKELY(r.has_value())) 74 return r.value(); 75 76 // Minimax polynomial generated by Sollya with: 77 // > display = hexadecimal; 78 // > fpminimax((2^x - 1)/x, 5, [|D...|], [-2^-5, 2^-5]); 79 constexpr double COEFFS[] = { 80 0x1.62e42fefa39f3p-1, 0x1.ebfbdff82c57bp-3, 0x1.c6b08d6f2d7aap-5, 81 0x1.3b2ab6fc92f5dp-7, 0x1.5d897cfe27125p-10, 0x1.43090e61e6af1p-13}; 82 double xd = x; 83 double xsq = xd * xd; 84 double c0 = fputil::multiply_add(xd, COEFFS[1], COEFFS[0]); 85 double c1 = fputil::multiply_add(xd, COEFFS[3], COEFFS[2]); 86 double c2 = fputil::multiply_add(xd, COEFFS[5], COEFFS[4]); 87 double p = fputil::polyeval(xsq, c0, c1, c2); 88 return static_cast<float>(p * xd); 89 } 90 91 // x >= 128, or x is nan 92 if (xbits.is_pos()) { 93 if (xbits.is_finite()) { 94 int rounding = fputil::quick_get_round(); 95 if (rounding == FE_DOWNWARD || rounding == FE_TOWARDZERO) 96 return FPBits::max_normal().get_val(); 97 98 fputil::set_errno_if_required(ERANGE); 99 fputil::raise_except_if_required(FE_OVERFLOW); 100 } 101 102 // x >= 128 and 2^x - 1 rounds to +inf, or x is +inf or nan 103 return x + FPBits::inf().get_val(); 104 } 105 } 106 107 if (LIBC_UNLIKELY(x <= -25.0f)) { 108 // 2^(-inf) - 1 = -1 109 if (xbits.is_inf()) 110 return -1.0f; 111 // 2^nan - 1 = nan 112 if (xbits.is_nan()) 113 return x; 114 115 int rounding = fputil::quick_get_round(); 116 if (rounding == FE_UPWARD || rounding == FE_TOWARDZERO) 117 return -0x1.ffff'fep-1f; // -1.0f + 0x1.0p-24f 118 119 fputil::set_errno_if_required(ERANGE); 120 fputil::raise_except_if_required(FE_UNDERFLOW); 121 return -1.0f; 122 } 123 124 if (auto r = EXP2M1F_EXCEPTS_HI.lookup(x_u); LIBC_UNLIKELY(r.has_value())) 125 return r.value(); 126 127 // For -25 < x < 128, to compute 2^x, we perform the following range 128 // reduction: find hi, mid, lo such that: 129 // x = hi + mid + lo, in which: 130 // hi is an integer, 131 // 0 <= mid * 2^5 < 32 is an integer, 132 // -2^(-6) <= lo <= 2^(-6). 133 // In particular, 134 // hi + mid = round(x * 2^5) * 2^(-5). 135 // Then, 136 // 2^x = 2^(hi + mid + lo) = 2^hi * 2^mid * 2^lo. 137 // 2^mid is stored in the lookup table of 32 elements. 138 // 2^lo is computed using a degree-4 minimax polynomial generated by Sollya. 139 // We perform 2^hi * 2^mid by simply add hi to the exponent field of 2^mid. 140 141 // kf = (hi + mid) * 2^5 = round(x * 2^5) 142 float kf; 143 int k; 144 #ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT 145 kf = fputil::nearest_integer(x * 32.0f); 146 k = static_cast<int>(kf); 147 #else 148 constexpr float HALF[2] = {0.5f, -0.5f}; 149 k = static_cast<int>(fputil::multiply_add(x, 32.0f, HALF[x < 0.0f])); 150 kf = static_cast<float>(k); 151 #endif // LIBC_TARGET_CPU_HAS_NEAREST_INT 152 153 // lo = x - (hi + mid) = x - kf * 2^(-5) 154 double lo = fputil::multiply_add(-0x1.0p-5f, kf, x); 155 156 // hi = floor(kf * 2^(-4)) 157 // exp2_hi = shift hi to the exponent field of double precision. 158 int64_t exp2_hi = 159 static_cast<int64_t>(static_cast<uint64_t>(k >> ExpBase::MID_BITS) 160 << fputil::FPBits<double>::FRACTION_LEN); 161 // mh = 2^hi * 2^mid 162 // mh_bits = bit field of mh 163 int64_t mh_bits = ExpBase::EXP_2_MID[k & ExpBase::MID_MASK] + exp2_hi; 164 double mh = fputil::FPBits<double>(static_cast<uint64_t>(mh_bits)).get_val(); 165 166 // Degree-4 polynomial approximating (2^x - 1)/x generated by Sollya with: 167 // > display = hexadecimal; 168 // > fpminimax((2^x - 1)/x, 4, [|D...|], [-2^-6, 2^-6]); 169 constexpr double COEFFS[5] = {0x1.62e42fefa39efp-1, 0x1.ebfbdff8131c4p-3, 170 0x1.c6b08d7061695p-5, 0x1.3b2b1bee74b2ap-7, 171 0x1.5d88091198529p-10}; 172 double lo_sq = lo * lo; 173 double c1 = fputil::multiply_add(lo, COEFFS[0], 1.0); 174 double c2 = fputil::multiply_add(lo, COEFFS[2], COEFFS[1]); 175 double c3 = fputil::multiply_add(lo, COEFFS[4], COEFFS[3]); 176 double exp2_lo = fputil::polyeval(lo_sq, c1, c2, c3); 177 // 2^x - 1 = 2^(hi + mid + lo) - 1 178 // = 2^(hi + mid) * 2^lo - 1 179 // ~ mh * (1 + lo * P(lo)) - 1 180 // = mh * exp2_lo - 1 181 return static_cast<float>(fputil::multiply_add(exp2_lo, mh, -1.0)); 182 } 183 184 } // namespace LIBC_NAMESPACE_DECL 185