1 /* 2 * Double-precision log2(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 "math_config.h" 11 12 #define T __log2_data.tab 13 #define T2 __log2_data.tab2 14 #define B __log2_data.poly1 15 #define A __log2_data.poly 16 #define InvLn2hi __log2_data.invln2hi 17 #define InvLn2lo __log2_data.invln2lo 18 #define N (1 << LOG2_TABLE_BITS) 19 #define OFF 0x3fe6000000000000 20 21 /* Top 16 bits of a double. */ 22 static inline uint32_t 23 top16 (double x) 24 { 25 return asuint64 (x) >> 48; 26 } 27 28 double 29 log2 (double x) 30 { 31 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 32 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p; 33 uint64_t ix, iz, tmp; 34 uint32_t top; 35 int k, i; 36 37 ix = asuint64 (x); 38 top = top16 (x); 39 40 #if LOG2_POLY1_ORDER == 11 41 # define LO asuint64 (1.0 - 0x1.5b51p-5) 42 # define HI asuint64 (1.0 + 0x1.6ab2p-5) 43 #endif 44 if (unlikely (ix - LO < HI - LO)) 45 { 46 /* Handle close to 1.0 inputs separately. */ 47 /* Fix sign of zero with downward rounding when x==1. */ 48 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) 49 return 0; 50 r = x - 1.0; 51 #if HAVE_FAST_FMA 52 hi = r * InvLn2hi; 53 lo = r * InvLn2lo + fma (r, InvLn2hi, -hi); 54 #else 55 double_t rhi, rlo; 56 rhi = asdouble (asuint64 (r) & -1ULL << 32); 57 rlo = r - rhi; 58 hi = rhi * InvLn2hi; 59 lo = rlo * InvLn2hi + r * InvLn2lo; 60 #endif 61 r2 = r * r; /* rounding error: 0x1p-62. */ 62 r4 = r2 * r2; 63 #if LOG2_POLY1_ORDER == 11 64 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */ 65 p = r2 * (B[0] + r * B[1]); 66 y = hi + p; 67 lo += hi - y + p; 68 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) 69 + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9]))); 70 y += lo; 71 #endif 72 return eval_as_double (y); 73 } 74 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) 75 { 76 /* x < 0x1p-1022 or inf or nan. */ 77 if (ix * 2 == 0) 78 return __math_divzero (1); 79 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */ 80 return x; 81 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 82 return __math_invalid (x); 83 /* x is subnormal, normalize it. */ 84 ix = asuint64 (x * 0x1p52); 85 ix -= 52ULL << 52; 86 } 87 88 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 89 The range is split into N subintervals. 90 The ith subinterval contains z and c is near its center. */ 91 tmp = ix - OFF; 92 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N; 93 k = (int64_t) tmp >> 52; /* arithmetic shift */ 94 iz = ix - (tmp & 0xfffULL << 52); 95 invc = T[i].invc; 96 logc = T[i].logc; 97 z = asdouble (iz); 98 kd = (double_t) k; 99 100 /* log2(x) = log2(z/c) + log2(c) + k. */ 101 /* r ~= z/c - 1, |r| < 1/(2*N). */ 102 #if HAVE_FAST_FMA 103 /* rounding error: 0x1p-55/N. */ 104 r = fma (z, invc, -1.0); 105 t1 = r * InvLn2hi; 106 t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1); 107 #else 108 double_t rhi, rlo; 109 /* rounding error: 0x1p-55/N + 0x1p-65. */ 110 r = (z - T2[i].chi - T2[i].clo) * invc; 111 rhi = asdouble (asuint64 (r) & -1ULL << 32); 112 rlo = r - rhi; 113 t1 = rhi * InvLn2hi; 114 t2 = rlo * InvLn2hi + r * InvLn2lo; 115 #endif 116 117 /* hi + lo = r/ln2 + log2(c) + k. */ 118 t3 = kd + logc; 119 hi = t3 + t1; 120 lo = t3 - hi + t1 + t2; 121 122 /* log2(r+1) = r/ln2 + r^2*poly(r). */ 123 /* Evaluation is optimized assuming superscalar pipelined execution. */ 124 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 125 r4 = r2 * r2; 126 #if LOG2_POLY_ORDER == 7 127 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma). 128 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */ 129 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]); 130 y = lo + r2 * p + hi; 131 #endif 132 return eval_as_double (y); 133 } 134 #if USE_GLIBC_ABI 135 strong_alias (log2, __log2_finite) 136 hidden_alias (log2, __ieee754_log2) 137 #endif 138
