1 //===-- Half-precision 10^x - 1 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/exp10m1f16.h" 10 #include "expxf16.h" 11 #include "hdr/errno_macros.h" 12 #include "hdr/fenv_macros.h" 13 #include "src/__support/FPUtil/FEnvImpl.h" 14 #include "src/__support/FPUtil/FPBits.h" 15 #include "src/__support/FPUtil/PolyEval.h" 16 #include "src/__support/FPUtil/cast.h" 17 #include "src/__support/FPUtil/except_value_utils.h" 18 #include "src/__support/FPUtil/multiply_add.h" 19 #include "src/__support/FPUtil/rounding_mode.h" 20 #include "src/__support/common.h" 21 #include "src/__support/macros/config.h" 22 #include "src/__support/macros/optimization.h" 23 #include "src/__support/macros/properties/cpu_features.h" 24 25 namespace LIBC_NAMESPACE_DECL { 26 27 static constexpr fputil::ExceptValues<float16, 3> EXP10M1F16_EXCEPTS_LO = {{ 28 // (input, RZ output, RU offset, RD offset, RN offset) 29 // x = 0x1.5c4p-4, exp10m1f16(x) = 0x1.bacp-3 (RZ) 30 {0x2d71U, 0x32ebU, 1U, 0U, 0U}, 31 // x = -0x1.5ep-13, exp10m1f16(x) = -0x1.92cp-12 (RZ) 32 {0x8978U, 0x8e4bU, 0U, 1U, 0U}, 33 // x = -0x1.e2p-10, exp10m1f16(x) = -0x1.14cp-8 (RZ) 34 {0x9788U, 0x9c53U, 0U, 1U, 0U}, 35 }}; 36 37 #ifdef LIBC_TARGET_CPU_HAS_FMA 38 static constexpr size_t N_EXP10M1F16_EXCEPTS_HI = 3; 39 #else 40 static constexpr size_t N_EXP10M1F16_EXCEPTS_HI = 6; 41 #endif 42 43 static constexpr fputil::ExceptValues<float16, N_EXP10M1F16_EXCEPTS_HI> 44 EXP10M1F16_EXCEPTS_HI = {{ 45 // (input, RZ output, RU offset, RD offset, RN offset) 46 // x = 0x1.8f4p-2, exp10m1f16(x) = 0x1.744p+0 (RZ) 47 {0x363dU, 0x3dd1U, 1U, 0U, 0U}, 48 // x = 0x1.95cp-2, exp10m1f16(x) = 0x1.7d8p+0 (RZ) 49 {0x3657U, 0x3df6U, 1U, 0U, 0U}, 50 // x = 0x1.d04p-2, exp10m1f16(x) = 0x1.d7p+0 (RZ) 51 {0x3741U, 0x3f5cU, 1U, 0U, 1U}, 52 #ifndef LIBC_TARGET_CPU_HAS_FMA 53 // x = 0x1.0cp+1, exp10m1f16(x) = 0x1.ec4p+6 (RZ) 54 {0x4030U, 0x57b1U, 1U, 0U, 1U}, 55 // x = 0x1.1b8p+1, exp10m1f16(x) = 0x1.45cp+7 (RZ) 56 {0x406eU, 0x5917U, 1U, 0U, 1U}, 57 // x = 0x1.2f4p+2, exp10m1f16(x) = 0x1.ab8p+15 (RZ) 58 {0x44bdU, 0x7aaeU, 1U, 0U, 1U}, 59 #endif 60 }}; 61 62 LLVM_LIBC_FUNCTION(float16, exp10m1f16, (float16 x)) { 63 using FPBits = fputil::FPBits<float16>; 64 FPBits x_bits(x); 65 66 uint16_t x_u = x_bits.uintval(); 67 uint16_t x_abs = x_u & 0x7fffU; 68 69 // When |x| <= 2^(-3), or |x| >= 11 * log10(2), or x is NaN. 70 if (LIBC_UNLIKELY(x_abs <= 0x3000U || x_abs >= 0x429fU)) { 71 // exp10m1(NaN) = NaN 72 if (x_bits.is_nan()) { 73 if (x_bits.is_signaling_nan()) { 74 fputil::raise_except_if_required(FE_INVALID); 75 return FPBits::quiet_nan().get_val(); 76 } 77 78 return x; 79 } 80 81 // When x >= 16 * log10(2). 82 if (x_u >= 0x44d1U && x_bits.is_pos()) { 83 // exp10m1(+inf) = +inf 84 if (x_bits.is_inf()) 85 return FPBits::inf().get_val(); 86 87 switch (fputil::quick_get_round()) { 88 case FE_TONEAREST: 89 case FE_UPWARD: 90 fputil::set_errno_if_required(ERANGE); 91 fputil::raise_except_if_required(FE_OVERFLOW | FE_INEXACT); 92 return FPBits::inf().get_val(); 93 default: 94 return FPBits::max_normal().get_val(); 95 } 96 } 97 98 // When x < -11 * log10(2). 99 if (x_u > 0xc29fU) { 100 // exp10m1(-inf) = -1 101 if (x_bits.is_inf()) 102 return FPBits::one(Sign::NEG).get_val(); 103 104 // When x >= -0x1.ce4p+1, round(10^x - 1, HP, RN) = -0x1.ffcp-1. 105 if (x_u <= 0xc339U) { 106 return fputil::round_result_slightly_down( 107 fputil::cast<float16>(-0x1.ffcp-1)); 108 } 109 110 // When x < -0x1.ce4p+1, round(10^x - 1, HP, RN) = -1. 111 switch (fputil::quick_get_round()) { 112 case FE_TONEAREST: 113 case FE_DOWNWARD: 114 return FPBits::one(Sign::NEG).get_val(); 115 default: 116 return fputil::cast<float16>(-0x1.ffcp-1); 117 } 118 } 119 120 // When |x| <= 2^(-3). 121 if (x_abs <= 0x3000U) { 122 if (LIBC_UNLIKELY(x_abs == 0)) 123 return x; 124 125 if (auto r = EXP10M1F16_EXCEPTS_LO.lookup(x_u); 126 LIBC_UNLIKELY(r.has_value())) 127 return r.value(); 128 129 float xf = x; 130 // Degree-5 minimax polynomial generated by Sollya with the following 131 // commands: 132 // > display = hexadecimal; 133 // > P = fpminimax((10^x - 1)/x, 4, [|SG...|], [-2^-3, 2^-3]); 134 // > x * P; 135 return fputil::cast<float16>( 136 xf * fputil::polyeval(xf, 0x1.26bb1cp+1f, 0x1.5351c8p+1f, 137 0x1.04704p+1f, 0x1.2ce084p+0f, 0x1.14a6bep-1f)); 138 } 139 } 140 141 // When x is 1, 2, or 3. These are hard-to-round cases with exact results. 142 // 10^4 - 1 = 9'999 is not exactly representable as a float16, but luckily the 143 // polynomial approximation gives the correct result for x = 4 in all 144 // rounding modes. 145 if (LIBC_UNLIKELY((x_u & ~(0x3c00U | 0x4000U | 0x4200U | 0x4400U)) == 0)) { 146 switch (x_u) { 147 case 0x3c00U: // x = 1.0f16 148 return fputil::cast<float16>(9.0); 149 case 0x4000U: // x = 2.0f16 150 return fputil::cast<float16>(99.0); 151 case 0x4200U: // x = 3.0f16 152 return fputil::cast<float16>(999.0); 153 } 154 } 155 156 if (auto r = EXP10M1F16_EXCEPTS_HI.lookup(x_u); LIBC_UNLIKELY(r.has_value())) 157 return r.value(); 158 159 // exp10(x) = exp2((hi + mid) * log2(10)) * exp10(lo) 160 auto [exp2_hi_mid, exp10_lo] = exp10_range_reduction(x); 161 // exp10m1(x) = exp2((hi + mid) * log2(lo)) * exp10(lo) - 1 162 return fputil::cast<float16>( 163 fputil::multiply_add(exp2_hi_mid, exp10_lo, -1.0f)); 164 } 165 166 } // namespace LIBC_NAMESPACE_DECL 167