1 /* 2 * Double-precision log2(x) function. 3 * 4 * Copyright (c) 2018-2024, 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 #include "test_defs.h" 13 #include "test_sig.h" 14 15 #define T __log2_data.tab 16 #define T2 __log2_data.tab2 17 #define B __log2_data.poly1 18 #define A __log2_data.poly 19 #define InvLn2hi __log2_data.invln2hi 20 #define InvLn2lo __log2_data.invln2lo 21 #define N (1 << LOG2_TABLE_BITS) 22 #define OFF 0x3fe6000000000000 23 24 /* Top 16 bits of a double. */ 25 static inline uint32_t 26 top16 (double x) 27 { 28 return asuint64 (x) >> 48; 29 } 30 31 double 32 log2 (double x) 33 { 34 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */ 35 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p; 36 uint64_t ix, iz, tmp; 37 uint32_t top; 38 int k, i; 39 40 ix = asuint64 (x); 41 top = top16 (x); 42 43 #if LOG2_POLY1_ORDER == 11 44 # define LO asuint64 (1.0 - 0x1.5b51p-5) 45 # define HI asuint64 (1.0 + 0x1.6ab2p-5) 46 #endif 47 if (unlikely (ix - LO < HI - LO)) 48 { 49 /* Handle close to 1.0 inputs separately. */ 50 /* Fix sign of zero with downward rounding when x==1. */ 51 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0))) 52 return 0; 53 r = x - 1.0; 54 #if HAVE_FAST_FMA 55 hi = r * InvLn2hi; 56 lo = r * InvLn2lo + fma (r, InvLn2hi, -hi); 57 #else 58 double_t rhi, rlo; 59 rhi = asdouble (asuint64 (r) & -1ULL << 32); 60 rlo = r - rhi; 61 hi = rhi * InvLn2hi; 62 lo = rlo * InvLn2hi + r * InvLn2lo; 63 #endif 64 r2 = r * r; /* rounding error: 0x1p-62. */ 65 r4 = r2 * r2; 66 #if LOG2_POLY1_ORDER == 11 67 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */ 68 p = r2 * (B[0] + r * B[1]); 69 y = hi + p; 70 lo += hi - y + p; 71 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) 72 + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9]))); 73 y += lo; 74 #endif 75 return eval_as_double (y); 76 } 77 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010)) 78 { 79 /* x < 0x1p-1022 or inf or nan. */ 80 if (ix * 2 == 0) 81 return __math_divzero (1); 82 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */ 83 return x; 84 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0) 85 return __math_invalid (x); 86 /* x is subnormal, normalize it. */ 87 ix = asuint64 (x * 0x1p52); 88 ix -= 52ULL << 52; 89 } 90 91 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact. 92 The range is split into N subintervals. 93 The ith subinterval contains z and c is near its center. */ 94 tmp = ix - OFF; 95 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N; 96 k = (int64_t) tmp >> 52; /* arithmetic shift */ 97 iz = ix - (tmp & 0xfffULL << 52); 98 invc = T[i].invc; 99 logc = T[i].logc; 100 z = asdouble (iz); 101 kd = (double_t) k; 102 103 /* log2(x) = log2(z/c) + log2(c) + k. */ 104 /* r ~= z/c - 1, |r| < 1/(2*N). */ 105 #if HAVE_FAST_FMA 106 /* rounding error: 0x1p-55/N. */ 107 r = fma (z, invc, -1.0); 108 t1 = r * InvLn2hi; 109 t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1); 110 #else 111 double_t rhi, rlo; 112 /* rounding error: 0x1p-55/N + 0x1p-65. */ 113 r = (z - T2[i].chi - T2[i].clo) * invc; 114 rhi = asdouble (asuint64 (r) & -1ULL << 32); 115 rlo = r - rhi; 116 t1 = rhi * InvLn2hi; 117 t2 = rlo * InvLn2hi + r * InvLn2lo; 118 #endif 119 120 /* hi + lo = r/ln2 + log2(c) + k. */ 121 t3 = kd + logc; 122 hi = t3 + t1; 123 lo = t3 - hi + t1 + t2; 124 125 /* log2(r+1) = r/ln2 + r^2*poly(r). */ 126 /* Evaluation is optimized assuming superscalar pipelined execution. */ 127 r2 = r * r; /* rounding error: 0x1p-54/N^2. */ 128 r4 = r2 * r2; 129 #if LOG2_POLY_ORDER == 7 130 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma). 131 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */ 132 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]); 133 y = lo + r2 * p + hi; 134 #endif 135 return eval_as_double (y); 136 } 137 #if USE_GLIBC_ABI 138 strong_alias (log2, __log2_finite) 139 hidden_alias (log2, __ieee754_log2) 140 # if LDBL_MANT_DIG == 53 141 long double log2l (long double x) { return log2 (x); } 142 # endif 143 #endif 144 145 TEST_SIG (S, D, 1, log2, 0.01, 11.1) 146 TEST_ULP (log2, 0.05) 147 TEST_ULP_NONNEAREST (log2, 0.5) 148 TEST_INTERVAL (log2, 0, 0xffff000000000000, 10000) 149 TEST_INTERVAL (log2, 0x1p-4, 0x1p4, 40000) 150 TEST_INTERVAL (log2, 0, inf, 40000) 151