src/math/log.c

   1 /*
   2  * Double-precision log(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 "log_data.h"
  12
  13 #define T __log_data.tab
  14 #define T2 __log_data.tab2
  15 #define B __log_data.poly1
  16 #define A __log_data.poly
  17 #define Ln2hi __log_data.ln2hi
  18 #define Ln2lo __log_data.ln2lo
  19 #define N (1 << LOG_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 log(double x)
  29 {
  30         double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
  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 - 0x1p-4)
  38 #define HI asuint64(1.0 + 0x1.09p-4)
  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                 r2 = r * r;
  46                 r3 = r * r2;
  47                 y = r3 *
  48                     (B[1] + r * B[2] + r2 * B[3] +
  49                      r3 * (B[4] + r * B[5] + r2 * B[6] +
  50                            r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
  51                 /* Worst-case error is around 0.507 ULP.  */
  52                 w = r * 0x1p27;
  53                 double_t rhi = r + w - w;
  54                 double_t rlo = r - rhi;
  55                 w = rhi * rhi * B[0]; /* B[0] == -0.5.  */
  56                 hi = r + w;
  57                 lo = r - hi + w;
  58                 lo += B[0] * rlo * (rhi + r);
  59                 y += lo;
  60                 y += hi;
  61                 return eval_as_double(y);
  62         }
  63         if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
  64                 /* x < 0x1p-1022 or inf or nan.  */
  65                 if (ix * 2 == 0)
  66                         return __math_divzero(1);
  67                 if (ix == asuint64(INFINITY)) /* log(inf) == inf.  */
  68                         return x;
  69                 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
  70                         return __math_invalid(x);
  71                 /* x is subnormal, normalize it.  */
  72                 ix = asuint64(x * 0x1p52);
  73                 ix -= 52ULL << 52;
  74         }
  75
  76         /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
  77            The range is split into N subintervals.
  78            The ith subinterval contains z and c is near its center.  */
  79         tmp = ix - OFF;
  80         i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
  81         k = (int64_t)tmp >> 52; /* arithmetic shift */
  82         iz = ix - (tmp & 0xfffULL << 52);
  83         invc = T[i].invc;
  84         logc = T[i].logc;
  85         z = asdouble(iz);
  86
  87         /* log(x) = log1p(z/c-1) + log(c) + k*Ln2.  */
  88         /* r ~= z/c - 1, |r| < 1/(2*N).  */
  89 #if __FP_FAST_FMA
  90         /* rounding error: 0x1p-55/N.  */
  91         r = __builtin_fma(z, invc, -1.0);
  92 #else
  93         /* rounding error: 0x1p-55/N + 0x1p-66.  */
  94         r = (z - T2[i].chi - T2[i].clo) * invc;
  95 #endif
  96         kd = (double_t)k;
  97
  98         /* hi + lo = r + log(c) + k*Ln2.  */
  99         w = kd * Ln2hi + logc;
 100         hi = w + r;
 101         lo = w - hi + r + kd * Ln2lo;
 102
 103         /* log(x) = lo + (log1p(r) - r) + hi.  */
 104         r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
 105         /* Worst case error if |y| > 0x1p-5:
 106            0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
 107            Worst case error if |y| > 0x1p-4:
 108            0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma).  */
 109         y = lo + r2 * A[0] +
 110             r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
 111         return eval_as_double(y);
 112 }