1 /* 2 * Double-precision log(x) function. 3 * 4 * Copyright (c) 2018, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8 #include <math.h> 9 #include <stdint.h> 10 #include "libm.h" 11 #include "log_data.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 top16(double x) 24 { 25 return asuint64(x) >> 48; 26 } 27 28 double log(double x) 29 { 30 double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo; 31 uint64_t ix, iz, tmp; 32 uint32_t top; 33 int k, i; 34 35 ix = asuint64(x); 36 top = top16(x); 37 #define LO asuint64(1.0 - 0x1p-4) 38 #define HI asuint64(1.0 + 0x1.09p-4) 39 if (predict_false(ix - LO < HI - LO)) { 40 /* Handle close to 1.0 inputs separately. */ 41 /* Fix sign of zero with downward rounding when x==1. */ 42 if (WANT_ROUNDING && predict_false(ix == asuint64(1.0))) 43 return 0; 44 r = x - 1.0; 45 r2 = r * r; 46 r3 = r * r2; 47 y = r3 * 48 (B[1] + r * B[2] + r2 * B[3] + 49 r3 * (B[4] + r * B[5] + r2 * B[6] + 50 r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10]))); 51 /* Worst-case error is around 0.507 ULP. */ 52 w = r * 0x1p27; 53 double_t rhi = r + w - w; 54 double_t rlo = r - rhi; 55 w = rhi * rhi * B[0]; /* B[0] == -0.5. */ 56 hi = r + w; 57 lo = r - hi + w; 58 lo += B[0] * rlo * (rhi + r); 59 y += lo; 60 y += hi; 61 return eval_as_double(y); 62 } 63 if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) { 64 /* x < 0x1p-1022 or inf or nan. */ 65 if (ix * 2 == 0) 66 return __math_divzero(1); 67 if (ix == asuint64(INFINITY)) /* log(inf) == inf. */ 68 return x; 69 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 70 return __math_invalid(x); 71 /* x is subnormal, normalize it. */ 72 ix = asuint64(x * 0x1p52); 73 ix -= 52ULL << 52; 74 } 75 76 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 77 The range is split into N subintervals. 78 The ith subinterval contains z and c is near its center. */ 79 tmp = ix - OFF; 80 i = (tmp >> (52 - LOG_TABLE_BITS)) % N; 81 k = (int64_t)tmp >> 52; /* arithmetic shift */ 82 iz = ix - (tmp & 0xfffULL << 52); 83 invc = T[i].invc; 84 logc = T[i].logc; 85 z = asdouble(iz); 86 87 /* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */ 88 /* r ~= z/c - 1, |r| < 1/(2*N). */ 89 #if __FP_FAST_FMA 90 /* rounding error: 0x1p-55/N. */ 91 r = __builtin_fma(z, invc, -1.0); 92 #else 93 /* rounding error: 0x1p-55/N + 0x1p-66. */ 94 r = (z - T2[i].chi - T2[i].clo) * invc; 95 #endif 96 kd = (double_t)k; 97 98 /* hi + lo = r + log(c) + k*Ln2. */ 99 w = kd * Ln2hi + logc; 100 hi = w + r; 101 lo = w - hi + r + kd * Ln2lo; 102 103 /* log(x) = lo + (log1p(r) - r) + hi. */ 104 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 105 /* Worst case error if |y| > 0x1p-5: 106 0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma) 107 Worst case error if |y| > 0x1p-4: 108 0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */ 109 y = lo + r2 * A[0] + 110 r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi; 111 return eval_as_double(y); 112 } 113