1 /* 2 * Double-precision log10(x) function. 3 * 4 * Copyright (c) 2020-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include "math_config.h" 9 #include "pl_sig.h" 10 #include "pl_test.h" 11 12 /* Polynomial coefficients and lookup tables. */ 13 #define T __log10_data.tab 14 #define T2 __log10_data.tab2 15 #define B __log10_data.poly1 16 #define A __log10_data.poly 17 #define Ln2hi __log10_data.ln2hi 18 #define Ln2lo __log10_data.ln2lo 19 #define InvLn10 __log10_data.invln10 20 #define N (1 << LOG10_TABLE_BITS) 21 #define OFF 0x3fe6000000000000 22 #define LO asuint64 (1.0 - 0x1p-4) 23 #define HI asuint64 (1.0 + 0x1.09p-4) 24 25 /* Top 16 bits of a double. */ 26 static inline uint32_t 27 top16 (double x) 28 { 29 return asuint64 (x) >> 48; 30 } 31 32 /* Fast and low accuracy implementation of log10. 33 The implementation is similar to that of math/log, except that: 34 - Polynomials are computed for log10(1+r) with r on same intervals as log. 35 - Lookup parameters are scaled (at runtime) to switch from base e to base 10. 36 Many errors above 1.59 ulp are observed across the whole range of doubles. 37 The greatest observed error is 1.61 ulp, at around 0.965: 38 log10(0x1.dc8710333a29bp-1) got -0x1.fee26884905a6p-6 39 want -0x1.fee26884905a8p-6. */ 40 double 41 log10 (double x) 42 { 43 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 44 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo; 45 uint64_t ix, iz, tmp; 46 uint32_t top; 47 int k, i; 48 49 ix = asuint64 (x); 50 top = top16 (x); 51 52 if (unlikely (ix - LO < HI - LO)) 53 { 54 /* Handle close to 1.0 inputs separately. */ 55 /* Fix sign of zero with downward rounding when x==1. */ 56 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) 57 return 0; 58 r = x - 1.0; 59 r2 = r * r; 60 r3 = r * r2; 61 y = r3 62 * (B[1] + r * B[2] + r2 * B[3] 63 + r3 64 * (B[4] + r * B[5] + r2 * B[6] 65 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10]))); 66 /* Worst-case error is around 0.507 ULP. */ 67 w = r * 0x1p27; 68 double_t rhi = r + w - w; 69 double_t rlo = r - rhi; 70 w = rhi * rhi * B[0]; 71 hi = r + w; 72 lo = r - hi + w; 73 lo += B[0] * rlo * (rhi + r); 74 y += lo; 75 y += hi; 76 /* Scale by 1/ln(10). Polynomial already contains scaling. */ 77 y = y * InvLn10; 78 79 return eval_as_double (y); 80 } 81 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) 82 { 83 /* x < 0x1p-1022 or inf or nan. */ 84 if (ix * 2 == 0) 85 return __math_divzero (1); 86 if (ix == asuint64 (INFINITY)) /* log10(inf) == inf. */ 87 return x; 88 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 89 return __math_invalid (x); 90 /* x is subnormal, normalize it. */ 91 ix = asuint64 (x * 0x1p52); 92 ix -= 52ULL << 52; 93 } 94 95 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 96 The range is split into N subintervals. 97 The ith subinterval contains z and c is near its center. */ 98 tmp = ix - OFF; 99 i = (tmp >> (52 - LOG10_TABLE_BITS)) % N; 100 k = (int64_t) tmp >> 52; /* arithmetic shift. */ 101 iz = ix - (tmp & 0xfffULL << 52); 102 invc = T[i].invc; 103 logc = T[i].logc; 104 z = asdouble (iz); 105 106 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */ 107 /* r ~= z/c - 1, |r| < 1/(2*N). */ 108 #if HAVE_FAST_FMA 109 /* rounding error: 0x1p-55/N. */ 110 r = fma (z, invc, -1.0); 111 #else 112 /* rounding error: 0x1p-55/N + 0x1p-66. */ 113 r = (z - T2[i].chi - T2[i].clo) * invc; 114 #endif 115 kd = (double_t) k; 116 117 /* w = log(c) + k*Ln2hi. */ 118 w = kd * Ln2hi + logc; 119 hi = w + r; 120 lo = w - hi + r + kd * Ln2lo; 121 122 /* log10(x) = (w + r)/log(10) + (log10(1+r) - r/log(10)). */ 123 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 124 125 /* Scale by 1/ln(10). Polynomial already contains scaling. */ 126 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi; 127 y = y * InvLn10; 128 129 return eval_as_double (y); 130 } 131 132 // clang-format off 133 #if USE_GLIBC_ABI 134 strong_alias (log10, __log10_finite) 135 hidden_alias (log10, __ieee754_log10) 136 #if LDBL_MANT_DIG == 53 137 long double 138 log10l (long double x) 139 { 140 return log10 (x); 141 } 142 #endif 143 #endif 144 // clang-format on 145 146 PL_SIG (S, D, 1, log10, 0.01, 11.1) 147 PL_TEST_ULP (log10, 1.11) 148 PL_TEST_INTERVAL (log10, 0, 0xffff000000000000, 10000) 149 PL_TEST_INTERVAL (log10, 0x1p-4, 0x1p4, 40000) 150 PL_TEST_INTERVAL (log10, 0, inf, 40000) 151