1 //===-- Single-precision log(x) 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/logf.h" 10 #include "common_constants.h" // Lookup table for (1/f) and log(f) 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/except_value_utils.h" 15 #include "src/__support/FPUtil/multiply_add.h" 16 #include "src/__support/common.h" 17 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 18 #include "src/__support/macros/properties/cpu_features.h" 19 20 // This is an algorithm for log(x) in single precision which is correctly 21 // rounded for all rounding modes, based on the implementation of log(x) from 22 // the RLIBM project at: 23 // https://people.cs.rutgers.edu/~sn349/rlibm 24 25 // Step 1 - Range reduction: 26 // For x = 2^m * 1.mant, log(x) = m * log(2) + log(1.m) 27 // If x is denormal, we normalize it by multiplying x by 2^23 and subtracting 28 // m by 23. 29 30 // Step 2 - Another range reduction: 31 // To compute log(1.mant), let f be the highest 8 bits including the hidden 32 // bit, and d be the difference (1.mant - f), i.e. the remaining 16 bits of the 33 // mantissa. Then we have the following approximation formula: 34 // log(1.mant) = log(f) + log(1.mant / f) 35 // = log(f) + log(1 + d/f) 36 // ~ log(f) + P(d/f) 37 // since d/f is sufficiently small. 38 // log(f) and 1/f are then stored in two 2^7 = 128 entries look-up tables. 39 40 // Step 3 - Polynomial approximation: 41 // To compute P(d/f), we use a single degree-5 polynomial in double precision 42 // which provides correct rounding for all but few exception values. 43 // For more detail about how this polynomial is obtained, please refer to the 44 // paper: 45 // Lim, J. and Nagarakatte, S., "One Polynomial Approximation to Produce 46 // Correctly Rounded Results of an Elementary Function for Multiple 47 // Representations and Rounding Modes", Proceedings of the 49th ACM SIGPLAN 48 // Symposium on Principles of Programming Languages (POPL-2022), Philadelphia, 49 // USA, January 16-22, 2022. 50 // https://people.cs.rutgers.edu/~sn349/papers/rlibmall-popl-2022.pdf 51 52 namespace LIBC_NAMESPACE { 53 54 LLVM_LIBC_FUNCTION(float, logf, (float x)) { 55 constexpr double LOG_2 = 0x1.62e42fefa39efp-1; 56 using FPBits = typename fputil::FPBits<float>; 57 58 FPBits xbits(x); 59 uint32_t x_u = xbits.uintval(); 60 61 int m = -FPBits::EXP_BIAS; 62 63 using fputil::round_result_slightly_down; 64 using fputil::round_result_slightly_up; 65 66 // Small inputs 67 if (x_u < 0x4c5d65a5U) { 68 // Hard-to-round cases. 69 switch (x_u) { 70 case 0x3f7f4d6fU: // x = 0x1.fe9adep-1f 71 return round_result_slightly_up(-0x1.659ec8p-9f); 72 case 0x41178febU: // x = 0x1.2f1fd6p+3f 73 return round_result_slightly_up(0x1.1fcbcep+1f); 74 #ifdef LIBC_TARGET_CPU_HAS_FMA 75 case 0x3f800000U: // x = 1.0f 76 return 0.0f; 77 #else 78 case 0x1e88452dU: // x = 0x1.108a5ap-66f 79 return round_result_slightly_up(-0x1.6d7b18p+5f); 80 #endif // LIBC_TARGET_CPU_HAS_FMA 81 } 82 // Subnormal inputs. 83 if (LIBC_UNLIKELY(x_u < FPBits::min_normal().uintval())) { 84 if (x_u == 0) { 85 // Return -inf and raise FE_DIVBYZERO 86 fputil::set_errno_if_required(ERANGE); 87 fputil::raise_except_if_required(FE_DIVBYZERO); 88 return FPBits::inf(Sign::NEG).get_val(); 89 } 90 // Normalize denormal inputs. 91 xbits = FPBits(xbits.get_val() * 0x1.0p23f); 92 m -= 23; 93 x_u = xbits.uintval(); 94 } 95 } else { 96 // Hard-to-round cases. 97 switch (x_u) { 98 case 0x4c5d65a5U: // x = 0x1.bacb4ap+25f 99 return round_result_slightly_down(0x1.1e0696p+4f); 100 case 0x65d890d3U: // x = 0x1.b121a6p+76f 101 return round_result_slightly_down(0x1.a9a3f2p+5f); 102 case 0x6f31a8ecU: // x = 0x1.6351d8p+95f 103 return round_result_slightly_down(0x1.08b512p+6f); 104 case 0x7a17f30aU: // x = 0x1.2fe614p+117f 105 return round_result_slightly_up(0x1.451436p+6f); 106 #ifndef LIBC_TARGET_CPU_HAS_FMA 107 case 0x500ffb03U: // x = 0x1.1ff606p+33f 108 return round_result_slightly_up(0x1.6fdd34p+4f); 109 case 0x5cd69e88U: // x = 0x1.ad3d1p+58f 110 return round_result_slightly_up(0x1.45c146p+5f); 111 case 0x5ee8984eU: // x = 0x1.d1309cp+62f; 112 return round_result_slightly_up(0x1.5c9442p+5f); 113 #endif // LIBC_TARGET_CPU_HAS_FMA 114 } 115 // Exceptional inputs. 116 if (LIBC_UNLIKELY(x_u > FPBits::max_normal().uintval())) { 117 if (x_u == 0x8000'0000U) { 118 // Return -inf and raise FE_DIVBYZERO 119 fputil::set_errno_if_required(ERANGE); 120 fputil::raise_except_if_required(FE_DIVBYZERO); 121 return FPBits::inf(Sign::NEG).get_val(); 122 } 123 if (xbits.is_neg() && !xbits.is_nan()) { 124 // Return NaN and raise FE_INVALID 125 fputil::set_errno_if_required(EDOM); 126 fputil::raise_except_if_required(FE_INVALID); 127 return FPBits::quiet_nan().get_val(); 128 } 129 // x is +inf or nan 130 return x; 131 } 132 } 133 134 #ifndef LIBC_TARGET_CPU_HAS_FMA 135 // Returning the correct +0 when x = 1.0 for non-FMA targets with FE_DOWNWARD 136 // rounding mode. 137 if (LIBC_UNLIKELY((x_u & 0x007f'ffffU) == 0)) 138 return static_cast<float>( 139 static_cast<double>(m + xbits.get_biased_exponent()) * LOG_2); 140 #endif // LIBC_TARGET_CPU_HAS_FMA 141 142 uint32_t mant = xbits.get_mantissa(); 143 // Extract 7 leading fractional bits of the mantissa 144 int index = mant >> 16; 145 // Add unbiased exponent. Add an extra 1 if the 7 leading fractional bits are 146 // all 1's. 147 m += static_cast<int>((x_u + (1 << 16)) >> 23); 148 149 // Set bits to 1.m 150 xbits.set_biased_exponent(0x7F); 151 152 float u = xbits.get_val(); 153 double v; 154 #ifdef LIBC_TARGET_CPU_HAS_FMA 155 v = static_cast<double>(fputil::multiply_add(u, R[index], -1.0f)); // Exact. 156 #else 157 v = fputil::multiply_add(static_cast<double>(u), RD[index], -1.0); // Exact 158 #endif // LIBC_TARGET_CPU_HAS_FMA 159 160 // Degree-5 polynomial approximation of log generated by Sollya with: 161 // > P = fpminimax(log(1 + x)/x, 4, [|1, D...|], [-2^-8, 2^-7]); 162 constexpr double COEFFS[4] = {-0x1.000000000fe63p-1, 0x1.555556e963c16p-2, 163 -0x1.000028dedf986p-2, 0x1.966681bfda7f7p-3}; 164 double v2 = v * v; // Exact 165 double p2 = fputil::multiply_add(v, COEFFS[3], COEFFS[2]); 166 double p1 = fputil::multiply_add(v, COEFFS[1], COEFFS[0]); 167 double p0 = LOG_R[index] + v; 168 double r = fputil::multiply_add(static_cast<double>(m), LOG_2, 169 fputil::polyeval(v2, p0, p1, p2)); 170 return static_cast<float>(r); 171 } 172 173 } // namespace LIBC_NAMESPACE 174