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1/*2 * Double-precision log2(x) function.3 *4 * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.5 * See https://llvm.org/LICENSE.txt for license information.6 * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception7 */8 9#include <float.h>10#include <math.h>11#include <stdint.h>12#include "math_config.h"13 14#define T __log2_data.tab15#define T2 __log2_data.tab216#define B __log2_data.poly117#define A __log2_data.poly18#define InvLn2hi __log2_data.invln2hi19#define InvLn2lo __log2_data.invln2lo20#define N (1 << LOG2_TABLE_BITS)21#define OFF 0x3fe600000000000022 23/* Top 16 bits of a double. */24static inline uint32_t25top16 (double x)26{27 return asuint64 (x) >> 48;28}29 30double31log2 (double x)32{33 /* double_t for better performance on targets with FLT_EVAL_METHOD==2. */34 double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;35 uint64_t ix, iz, tmp;36 uint32_t top;37 int k, i;38 39 ix = asuint64 (x);40 top = top16 (x);41 42#if LOG2_POLY1_ORDER == 1143# define LO asuint64 (1.0 - 0x1.5b51p-5)44# define HI asuint64 (1.0 + 0x1.6ab2p-5)45#endif46 if (unlikely (ix - LO < HI - LO))47 {48 /* Handle close to 1.0 inputs separately. */49 /* Fix sign of zero with downward rounding when x==1. */50 if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))51 return 0;52 r = x - 1.0;53#if HAVE_FAST_FMA54 hi = r * InvLn2hi;55 lo = r * InvLn2lo + fma (r, InvLn2hi, -hi);56#else57 double_t rhi, rlo;58 rhi = asdouble (asuint64 (r) & -1ULL << 32);59 rlo = r - rhi;60 hi = rhi * InvLn2hi;61 lo = rlo * InvLn2hi + r * InvLn2lo;62#endif63 r2 = r * r; /* rounding error: 0x1p-62. */64 r4 = r2 * r2;65#if LOG2_POLY1_ORDER == 1166 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */67 p = r2 * (B[0] + r * B[1]);68 y = hi + p;69 lo += hi - y + p;70 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])71 + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));72 y += lo;73#endif74 return eval_as_double (y);75 }76 if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))77 {78 /* x < 0x1p-1022 or inf or nan. */79 if (ix * 2 == 0)80 return __math_divzero (1);81 if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */82 return x;83 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)84 return __math_invalid (x);85 /* x is subnormal, normalize it. */86 ix = asuint64 (x * 0x1p52);87 ix -= 52ULL << 52;88 }89 90 /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.91 The range is split into N subintervals.92 The ith subinterval contains z and c is near its center. */93 tmp = ix - OFF;94 i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;95 k = (int64_t) tmp >> 52; /* arithmetic shift */96 iz = ix - (tmp & 0xfffULL << 52);97 invc = T[i].invc;98 logc = T[i].logc;99 z = asdouble (iz);100 kd = (double_t) k;101 102 /* log2(x) = log2(z/c) + log2(c) + k. */103 /* r ~= z/c - 1, |r| < 1/(2*N). */104#if HAVE_FAST_FMA105 /* rounding error: 0x1p-55/N. */106 r = fma (z, invc, -1.0);107 t1 = r * InvLn2hi;108 t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1);109#else110 double_t rhi, rlo;111 /* rounding error: 0x1p-55/N + 0x1p-65. */112 r = (z - T2[i].chi - T2[i].clo) * invc;113 rhi = asdouble (asuint64 (r) & -1ULL << 32);114 rlo = r - rhi;115 t1 = rhi * InvLn2hi;116 t2 = rlo * InvLn2hi + r * InvLn2lo;117#endif118 119 /* hi + lo = r/ln2 + log2(c) + k. */120 t3 = kd + logc;121 hi = t3 + t1;122 lo = t3 - hi + t1 + t2;123 124 /* log2(r+1) = r/ln2 + r^2*poly(r). */125 /* Evaluation is optimized assuming superscalar pipelined execution. */126 r2 = r * r; /* rounding error: 0x1p-54/N^2. */127 r4 = r2 * r2;128#if LOG2_POLY_ORDER == 7129 /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).130 ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */131 p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);132 y = lo + r2 * p + hi;133#endif134 return eval_as_double (y);135}136#if USE_GLIBC_ABI137strong_alias (log2, __log2_finite)138hidden_alias (log2, __ieee754_log2)139# if LDBL_MANT_DIG == 53140long double log2l (long double x) { return log2 (x); }141# endif142#endif143