1 /* 2 * Double-precision log(x) function. 3 * 4 * Copyright (c) 2018-2023, Arm Limited. 5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception 6 */ 7 8 #include <float.h> 9 #include <math.h> 10 #include <stdint.h> 11 #include "math_config.h" 12 13 #define T __log_data.tab 14 #define T2 __log_data.tab2 15 #define B __log_data.poly1 16 #define A __log_data.poly 17 #define Ln2hi __log_data.ln2hi 18 #define Ln2lo __log_data.ln2lo 19 #define N (1 << LOG_TABLE_BITS) 20 #define OFF 0x3fe6000000000000 21 22 /* Top 16 bits of a double. */ 23 static inline uint32_t 24 top16 (double x) 25 { 26 return asuint64 (x) >> 48; 27 } 28 29 double 30 optr_aor_log_f64 (double x) 31 { 32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 33 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo; 34 uint64_t ix, iz, tmp; 35 uint32_t top; 36 int k, i; 37 38 ix = asuint64 (x); 39 top = top16 (x); 40 41 #if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11 42 #define LO asuint64 (1.0 - 0x1p-5) 43 #define HI asuint64 (1.0 + 0x1.1p-5) 44 #elif LOG_POLY1_ORDER == 12 45 #define LO asuint64 (1.0 - 0x1p-4) 46 #define HI asuint64 (1.0 + 0x1.09p-4) 47 #endif 48 if (unlikely (ix - LO < HI - LO)) 49 { 50 /* Handle close to 1.0 inputs separately. */ 51 /* Fix sign of zero with downward rounding when x==1. */ 52 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) 53 return 0; 54 r = x - 1.0; 55 r2 = r * r; 56 r3 = r * r2; 57 #if LOG_POLY1_ORDER == 10 58 /* Worst-case error is around 0.516 ULP. */ 59 y = r3 60 * (B[1] + r * B[2] + r2 * B[3] 61 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8]))); 62 w = B[0] * r2; /* B[0] == -0.5. */ 63 hi = r + w; 64 y += r - hi + w; 65 y += hi; 66 #elif LOG_POLY1_ORDER == 11 67 /* Worst-case error is around 0.516 ULP. */ 68 y = r3 69 * (B[1] + r * B[2] 70 + r2 71 * (B[3] + r * B[4] + r2 * B[5] 72 + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9]))); 73 w = B[0] * r2; /* B[0] == -0.5. */ 74 hi = r + w; 75 y += r - hi + w; 76 y += hi; 77 #elif LOG_POLY1_ORDER == 12 78 y = r3 79 * (B[1] + r * B[2] + r2 * B[3] 80 + r3 81 * (B[4] + r * B[5] + r2 * B[6] 82 + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10]))); 83 #if N <= 64 84 /* Worst-case error is around 0.532 ULP. */ 85 w = B[0] * r2; /* B[0] == -0.5. */ 86 hi = r + w; 87 y += r - hi + w; 88 y += hi; 89 #else 90 /* Worst-case error is around 0.507 ULP. */ 91 w = r * 0x1p27; 92 double_t rhi = r + w - w; 93 double_t rlo = r - rhi; 94 w = rhi * rhi * B[0]; /* B[0] == -0.5. */ 95 hi = r + w; 96 lo = r - hi + w; 97 lo += B[0] * rlo * (rhi + r); 98 y += lo; 99 y += hi; 100 #endif 101 #endif 102 return eval_as_double (y); 103 } 104 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) 105 { 106 /* x < 0x1p-1022 or inf or nan. */ 107 if (ix * 2 == 0) 108 return __math_divzero (1); 109 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */ 110 return x; 111 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 112 return __math_invalid (x); 113 /* x is subnormal, normalize it. */ 114 ix = asuint64 (x * 0x1p52); 115 ix -= 52ULL << 52; 116 } 117 118 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 119 The range is split into N subintervals. 120 The ith subinterval contains z and c is near its center. */ 121 tmp = ix - OFF; 122 i = (tmp >> (52 - LOG_TABLE_BITS)) % N; 123 k = (int64_t) tmp >> 52; /* arithmetic shift */ 124 iz = ix - (tmp & 0xfffULL << 52); 125 invc = T[i].invc; 126 logc = T[i].logc; 127 z = asdouble (iz); 128 129 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */ 130 /* r ~= z/c - 1, |r| < 1/(2*N). */ 131 #if HAVE_FAST_FMA 132 /* rounding error: 0x1p-55/N. */ 133 r = fma (z, invc, -1.0); 134 #else 135 /* rounding error: 0x1p-55/N + 0x1p-66. */ 136 r = (z - T2[i].chi - T2[i].clo) * invc; 137 #endif 138 kd = (double_t) k; 139 140 /* hi + lo = r + log(c) + k*Ln2. */ 141 w = kd * Ln2hi + logc; 142 hi = w + r; 143 lo = w - hi + r + kd * Ln2lo; 144 145 /* log(x) = lo + (log1p(r) - r) + hi. */ 146 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 147 /* Worst case error if |y| > 0x1p-5: 148 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma) 149 Worst case error if |y| > 0x1p-4: 150 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */ 151 #if LOG_POLY_ORDER == 6 152 y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi; 153 #elif LOG_POLY_ORDER == 7 154 y = lo 155 + r2 156 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3]) 157 + r2 * r2 * (A[4] + r * A[5])) 158 + hi; 159 #endif 160 return eval_as_double (y); 161 } 162