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/config.h" 18 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY 19 #include "src/__support/macros/properties/cpu_features.h" 20 21 // This is an algorithm for log(x) in single precision which is correctly 22 // rounded for all rounding modes, based on the implementation of log(x) from 23 // the RLIBM project at: 24 // https://people.cs.rutgers.edu/~sn349/rlibm 25 26 // Step 1 - Range reduction: 27 // For x = 2^m * 1.mant, log(x) = m * log(2) + log(1.m) 28 // If x is denormal, we normalize it by multiplying x by 2^23 and subtracting 29 // m by 23. 30 31 // Step 2 - Another range reduction: 32 // To compute log(1.mant), let f be the highest 8 bits including the hidden 33 // bit, and d be the difference (1.mant - f), i.e. the remaining 16 bits of the 34 // mantissa. Then we have the following approximation formula: 35 // log(1.mant) = log(f) + log(1.mant / f) 36 // = log(f) + log(1 + d/f) 37 // ~ log(f) + P(d/f) 38 // since d/f is sufficiently small. 39 // log(f) and 1/f are then stored in two 2^7 = 128 entries look-up tables. 40 41 // Step 3 - Polynomial approximation: 42 // To compute P(d/f), we use a single degree-5 polynomial in double precision 43 // which provides correct rounding for all but few exception values. 44 // For more detail about how this polynomial is obtained, please refer to the 45 // paper: 46 // Lim, J. and Nagarakatte, S., "One Polynomial Approximation to Produce 47 // Correctly Rounded Results of an Elementary Function for Multiple 48 // Representations and Rounding Modes", Proceedings of the 49th ACM SIGPLAN 49 // Symposium on Principles of Programming Languages (POPL-2022), Philadelphia, 50 // USA, January 16-22, 2022. 51 // https://people.cs.rutgers.edu/~sn349/papers/rlibmall-popl-2022.pdf 52 53 namespace LIBC_NAMESPACE_DECL { 54 55 LLVM_LIBC_FUNCTION(float, logf, (float x)) { 56 constexpr double LOG_2 = 0x1.62e42fefa39efp-1; 57 using FPBits = typename fputil::FPBits<float>; 58 59 FPBits xbits(x); 60 uint32_t x_u = xbits.uintval(); 61 62 int m = -FPBits::EXP_BIAS; 63 64 using fputil::round_result_slightly_down; 65 using fputil::round_result_slightly_up; 66 67 // Small inputs 68 if (x_u < 0x4c5d65a5U) { 69 // Hard-to-round cases. 70 switch (x_u) { 71 case 0x3f7f4d6fU: // x = 0x1.fe9adep-1f 72 return round_result_slightly_up(-0x1.659ec8p-9f); 73 case 0x41178febU: // x = 0x1.2f1fd6p+3f 74 return round_result_slightly_up(0x1.1fcbcep+1f); 75 #ifdef LIBC_TARGET_CPU_HAS_FMA 76 case 0x3f800000U: // x = 1.0f 77 return 0.0f; 78 #else 79 case 0x1e88452dU: // x = 0x1.108a5ap-66f 80 return round_result_slightly_up(-0x1.6d7b18p+5f); 81 #endif // LIBC_TARGET_CPU_HAS_FMA 82 } 83 // Subnormal inputs. 84 if (LIBC_UNLIKELY(x_u < FPBits::min_normal().uintval())) { 85 if (x == 0.0f) { 86 // Return -inf and raise FE_DIVBYZERO 87 fputil::set_errno_if_required(ERANGE); 88 fputil::raise_except_if_required(FE_DIVBYZERO); 89 return FPBits::inf(Sign::NEG).get_val(); 90 } 91 // Normalize denormal inputs. 92 xbits = FPBits(xbits.get_val() * 0x1.0p23f); 93 m -= 23; 94 x_u = xbits.uintval(); 95 } 96 } else { 97 // Hard-to-round cases. 98 switch (x_u) { 99 case 0x4c5d65a5U: // x = 0x1.bacb4ap+25f 100 return round_result_slightly_down(0x1.1e0696p+4f); 101 case 0x65d890d3U: // x = 0x1.b121a6p+76f 102 return round_result_slightly_down(0x1.a9a3f2p+5f); 103 case 0x6f31a8ecU: // x = 0x1.6351d8p+95f 104 return round_result_slightly_down(0x1.08b512p+6f); 105 case 0x7a17f30aU: // x = 0x1.2fe614p+117f 106 return round_result_slightly_up(0x1.451436p+6f); 107 #ifndef LIBC_TARGET_CPU_HAS_FMA 108 case 0x500ffb03U: // x = 0x1.1ff606p+33f 109 return round_result_slightly_up(0x1.6fdd34p+4f); 110 case 0x5cd69e88U: // x = 0x1.ad3d1p+58f 111 return round_result_slightly_up(0x1.45c146p+5f); 112 case 0x5ee8984eU: // x = 0x1.d1309cp+62f; 113 return round_result_slightly_up(0x1.5c9442p+5f); 114 #endif // LIBC_TARGET_CPU_HAS_FMA 115 } 116 // Exceptional inputs. 117 if (LIBC_UNLIKELY(x_u > FPBits::max_normal().uintval())) { 118 if (x_u == 0x8000'0000U) { 119 // Return -inf and raise FE_DIVBYZERO 120 fputil::set_errno_if_required(ERANGE); 121 fputil::raise_except_if_required(FE_DIVBYZERO); 122 return FPBits::inf(Sign::NEG).get_val(); 123 } 124 if (xbits.is_neg() && !xbits.is_nan()) { 125 // Return NaN and raise FE_INVALID 126 fputil::set_errno_if_required(EDOM); 127 fputil::raise_except_if_required(FE_INVALID); 128 return FPBits::quiet_nan().get_val(); 129 } 130 // x is +inf or nan 131 return x; 132 } 133 } 134 135 #ifndef LIBC_TARGET_CPU_HAS_FMA 136 // Returning the correct +0 when x = 1.0 for non-FMA targets with FE_DOWNWARD 137 // rounding mode. 138 if (LIBC_UNLIKELY((x_u & 0x007f'ffffU) == 0)) 139 return static_cast<float>( 140 static_cast<double>(m + xbits.get_biased_exponent()) * LOG_2); 141 #endif // LIBC_TARGET_CPU_HAS_FMA 142 143 uint32_t mant = xbits.get_mantissa(); 144 // Extract 7 leading fractional bits of the mantissa 145 int index = mant >> 16; 146 // Add unbiased exponent. Add an extra 1 if the 7 leading fractional bits are 147 // all 1's. 148 m += static_cast<int>((x_u + (1 << 16)) >> 23); 149 150 // Set bits to 1.m 151 xbits.set_biased_exponent(0x7F); 152 153 float u = xbits.get_val(); 154 double v; 155 #ifdef LIBC_TARGET_CPU_HAS_FMA 156 v = static_cast<double>(fputil::multiply_add(u, R[index], -1.0f)); // Exact. 157 #else 158 v = fputil::multiply_add(static_cast<double>(u), RD[index], -1.0); // Exact 159 #endif // LIBC_TARGET_CPU_HAS_FMA 160 161 // Degree-5 polynomial approximation of log generated by Sollya with: 162 // > P = fpminimax(log(1 + x)/x, 4, [|1, D...|], [-2^-8, 2^-7]); 163 constexpr double COEFFS[4] = {-0x1.000000000fe63p-1, 0x1.555556e963c16p-2, 164 -0x1.000028dedf986p-2, 0x1.966681bfda7f7p-3}; 165 double v2 = v * v; // Exact 166 double p2 = fputil::multiply_add(v, COEFFS[3], COEFFS[2]); 167 double p1 = fputil::multiply_add(v, COEFFS[1], COEFFS[0]); 168 double p0 = LOG_R[index] + v; 169 double r = fputil::multiply_add(static_cast<double>(m), LOG_2, 170 fputil::polyeval(v2, p0, p1, p2)); 171 return static_cast<float>(r); 172 } 173 174 } // namespace LIBC_NAMESPACE_DECL 175