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 "libm.h" 11 #include "log2_data.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 top16(double x) 24 { 25 return asuint64(x) >> 48; 26 } 27 28 double log2(double x) 29 { 30 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p; 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 - 0x1.5b51p-5) 38 #define HI asuint64(1.0 + 0x1.6ab2p-5) 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 #if __FP_FAST_FMA 46 hi = r * InvLn2hi; 47 lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi); 48 #else 49 double_t rhi, rlo; 50 rhi = asdouble(asuint64(r) & -1ULL << 32); 51 rlo = r - rhi; 52 hi = rhi * InvLn2hi; 53 lo = rlo * InvLn2hi + r * InvLn2lo; 54 #endif 55 r2 = r * r; /* rounding error: 0x1p-62. */ 56 r4 = r2 * r2; 57 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */ 58 p = r2 * (B[0] + r * B[1]); 59 y = hi + p; 60 lo += hi - y + p; 61 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) + 62 r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9]))); 63 y += lo; 64 return eval_as_double(y); 65 } 66 if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) { 67 /* x < 0x1p-1022 or inf or nan. */ 68 if (ix * 2 == 0) 69 return __math_divzero(1); 70 if (ix == asuint64(INFINITY)) /* log(inf) == inf. */ 71 return x; 72 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 73 return __math_invalid(x); 74 /* x is subnormal, normalize it. */ 75 ix = asuint64(x * 0x1p52); 76 ix -= 52ULL << 52; 77 } 78 79 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 80 The range is split into N subintervals. 81 The ith subinterval contains z and c is near its center. */ 82 tmp = ix - OFF; 83 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N; 84 k = (int64_t)tmp >> 52; /* arithmetic shift */ 85 iz = ix - (tmp & 0xfffULL << 52); 86 invc = T[i].invc; 87 logc = T[i].logc; 88 z = asdouble(iz); 89 kd = (double_t)k; 90 91 /* log2(x) = log2(z/c) + log2(c) + k. */ 92 /* r ~= z/c - 1, |r| < 1/(2*N). */ 93 #if __FP_FAST_FMA 94 /* rounding error: 0x1p-55/N. */ 95 r = __builtin_fma(z, invc, -1.0); 96 t1 = r * InvLn2hi; 97 t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1); 98 #else 99 double_t rhi, rlo; 100 /* rounding error: 0x1p-55/N + 0x1p-65. */ 101 r = (z - T2[i].chi - T2[i].clo) * invc; 102 rhi = asdouble(asuint64(r) & -1ULL << 32); 103 rlo = r - rhi; 104 t1 = rhi * InvLn2hi; 105 t2 = rlo * InvLn2hi + r * InvLn2lo; 106 #endif 107 108 /* hi + lo = r/ln2 + log2(c) + k. */ 109 t3 = kd + logc; 110 hi = t3 + t1; 111 lo = t3 - hi + t1 + t2; 112 113 /* log2(r+1) = r/ln2 + r^2*poly(r). */ 114 /* Evaluation is optimized assuming superscalar pipelined execution. */ 115 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 116 r4 = r2 * r2; 117 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma). 118 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */ 119 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]); 120 y = lo + r2 * p + hi; 121 return eval_as_double(y); 122 } 123