1 /* 2 * Double-precision log2(x) function. 3 * 4 * Copyright (c) 2018, Arm Limited. 5 * SPDX-License-Identifier: MIT 6 */ 7 8 #include <float.h> 9 #include <math.h> 10 #include <stdint.h> 11 #include "math_config.h" 12 13 #define T __log2_data.tab 14 #define T2 __log2_data.tab2 15 #define B __log2_data.poly1 16 #define A __log2_data.poly 17 #define InvLn2hi __log2_data.invln2hi 18 #define InvLn2lo __log2_data.invln2lo 19 #define N (1 << LOG2_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 log2 (double x) 31 { 32 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 33 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p; 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 LOG2_POLY1_ORDER == 11 42 # define LO asuint64 (1.0 - 0x1.5b51p-5) 43 # define HI asuint64 (1.0 + 0x1.6ab2p-5) 44 #endif 45 if (unlikely (ix - LO < HI - LO)) 46 { 47 /* Handle close to 1.0 inputs separately. */ 48 /* Fix sign of zero with downward rounding when x==1. */ 49 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) 50 return 0; 51 r = x - 1.0; 52 #if HAVE_FAST_FMA 53 hi = r * InvLn2hi; 54 lo = r * InvLn2lo + fma (r, InvLn2hi, -hi); 55 #else 56 double_t rhi, rlo; 57 rhi = asdouble (asuint64 (r) & -1ULL << 32); 58 rlo = r - rhi; 59 hi = rhi * InvLn2hi; 60 lo = rlo * InvLn2hi + r * InvLn2lo; 61 #endif 62 r2 = r * r; /* rounding error: 0x1p-62. */ 63 r4 = r2 * r2; 64 #if LOG2_POLY1_ORDER == 11 65 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */ 66 p = r2 * (B[0] + r * B[1]); 67 y = hi + p; 68 lo += hi - y + p; 69 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) 70 + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9]))); 71 y += lo; 72 #endif 73 return eval_as_double (y); 74 } 75 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) 76 { 77 /* x < 0x1p-1022 or inf or nan. */ 78 if (ix * 2 == 0) 79 return __math_divzero (1); 80 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */ 81 return x; 82 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 83 return __math_invalid (x); 84 /* x is subnormal, normalize it. */ 85 ix = asuint64 (x * 0x1p52); 86 ix -= 52ULL << 52; 87 } 88 89 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 90 The range is split into N subintervals. 91 The ith subinterval contains z and c is near its center. */ 92 tmp = ix - OFF; 93 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N; 94 k = (int64_t) tmp >> 52; /* arithmetic shift */ 95 iz = ix - (tmp & 0xfffULL << 52); 96 invc = T[i].invc; 97 logc = T[i].logc; 98 z = asdouble (iz); 99 kd = (double_t) k; 100 101 /* log2(x) = log2(z/c) + log2(c) + k. */ 102 /* r ~= z/c - 1, |r| < 1/(2*N). */ 103 #if HAVE_FAST_FMA 104 /* rounding error: 0x1p-55/N. */ 105 r = fma (z, invc, -1.0); 106 t1 = r * InvLn2hi; 107 t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1); 108 #else 109 double_t rhi, rlo; 110 /* rounding error: 0x1p-55/N + 0x1p-65. */ 111 r = (z - T2[i].chi - T2[i].clo) * invc; 112 rhi = asdouble (asuint64 (r) & -1ULL << 32); 113 rlo = r - rhi; 114 t1 = rhi * InvLn2hi; 115 t2 = rlo * InvLn2hi + r * InvLn2lo; 116 #endif 117 118 /* hi + lo = r/ln2 + log2(c) + k. */ 119 t3 = kd + logc; 120 hi = t3 + t1; 121 lo = t3 - hi + t1 + t2; 122 123 /* log2(r+1) = r/ln2 + r^2*poly(r). */ 124 /* Evaluation is optimized assuming superscalar pipelined execution. */ 125 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 126 r4 = r2 * r2; 127 #if LOG2_POLY_ORDER == 7 128 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma). 129 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */ 130 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]); 131 y = lo + r2 * p + hi; 132 #endif 133 return eval_as_double (y); 134 } 135 #if USE_GLIBC_ABI 136 strong_alias (log2, __log2_finite) 137 hidden_alias (log2, __ieee754_log2) 138 # if LDBL_MANT_DIG == 53 139 long double log2l (long double x) { return log2 (x); } 140 # endif 141 #endif 142