newlib/libm/common/log.c

   1 /* Double-precision log(x) function.
   2    Copyright (c) 2018 Arm Ltd.  All rights reserved.
   3
   4    SPDX-License-Identifier: BSD-3-Clause
   5
   6    Redistribution and use in source and binary forms, with or without
   7    modification, are permitted provided that the following conditions
   8    are met:
   9    1. Redistributions of source code must retain the above copyright
  10       notice, this list of conditions and the following disclaimer.
  11    2. Redistributions in binary form must reproduce the above copyright
  12       notice, this list of conditions and the following disclaimer in the
  13       documentation and/or other materials provided with the distribution.
  14    3. The name of the company may not be used to endorse or promote
  15       products derived from this software without specific prior written
  16       permission.
  17
  18    THIS SOFTWARE IS PROVIDED BY ARM LTD ``AS IS'' AND ANY EXPRESS OR IMPLIED
  19    WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
  20    MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
  21    IN NO EVENT SHALL ARM LTD BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  22    SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
  23    TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
  24    PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
  25    LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
  26    NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
  27    SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
  28
  29 #include "fdlibm.h"
  30 #if !__OBSOLETE_MATH
  31
  32 #include <math.h>
  33 #include <stdint.h>
  34 #include "math_config.h"
  35
  36 #define T __log_data.tab
  37 #define T2 __log_data.tab2
  38 #define B __log_data.poly1
  39 #define A __log_data.poly
  40 #define Ln2hi __log_data.ln2hi
  41 #define Ln2lo __log_data.ln2lo
  42 #define N (1 << LOG_TABLE_BITS)
  43 #define OFF 0x3fe6000000000000
  44
  45 /* Top 16 bits of a double.  */
  46 static inline uint32_t
  47 top16 (double x)
  48 {
  49   return asuint64 (x) >> 48;
  50 }
  51
  52 double
  53 log (double x)
  54 {
  55   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  56   double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
  57   uint64_t ix, iz, tmp;
  58   uint32_t top;
  59   int k, i;
  60
  61   ix = asuint64 (x);
  62   top = top16 (x);
  63
  64 #if LOG_POLY1_ORDER == 10 || LOG_POLY1_ORDER == 11
  65 # define LO asuint64 (1.0 - 0x1p-5)
  66 # define HI asuint64 (1.0 + 0x1.1p-5)
  67 #elif LOG_POLY1_ORDER == 12
  68 # define LO asuint64 (1.0 - 0x1p-4)
  69 # define HI asuint64 (1.0 + 0x1.09p-4)
  70 #endif
  71   if (unlikely (ix - LO < HI - LO))
  72     {
  73       /* Handle close to 1.0 inputs separately.  */
  74       /* Fix sign of zero with downward rounding when x==1.  */
  75       if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
  76         return 0;
  77       r = x - 1.0;
  78       r2 = r * r;
  79       r3 = r * r2;
  80 #if LOG_POLY1_ORDER == 10
  81       /* Worst-case error is around 0.516 ULP.  */
  82       y = r3 * (B[1] + r * B[2] + r2 * B[3]
  83                 + r3 * (B[4] + r * B[5] + r2 * B[6] + r3 * (B[7] + r * B[8])));
  84       w = B[0] * r2; /* B[0] == -0.5.  */
  85       hi = r + w;
  86       y += r - hi + w;
  87       y += hi;
  88 #elif LOG_POLY1_ORDER == 11
  89       /* Worst-case error is around 0.516 ULP.  */
  90       y = r3 * (B[1] + r * B[2]
  91                 + r2 * (B[3] + r * B[4] + r2 * B[5]
  92                         + r3 * (B[6] + r * B[7] + r2 * B[8] + r3 * B[9])));
  93       w = B[0] * r2; /* B[0] == -0.5.  */
  94       hi = r + w;
  95       y += r - hi + w;
  96       y += hi;
  97 #elif LOG_POLY1_ORDER == 12
  98       y = r3 * (B[1] + r * B[2] + r2 * B[3]
  99                 + r3 * (B[4] + r * B[5] + r2 * B[6]
 100                         + r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
 101 # if N <= 64
 102       /* Worst-case error is around 0.532 ULP.  */
 103       w = B[0] * r2; /* B[0] == -0.5.  */
 104       hi = r + w;
 105       y += r - hi + w;
 106       y += hi;
 107 # else
 108       /* Worst-case error is around 0.507 ULP.  */
 109       w = r * 0x1p27;
 110       double_t rhi = r + w - w;
 111       double_t rlo = r - rhi;
 112       w = rhi * rhi * B[0]; /* B[0] == -0.5.  */
 113       hi = r + w;
 114       lo = r - hi + w;
 115       lo += B[0] * rlo * (rhi + r);
 116       y += lo;
 117       y += hi;
 118 # endif
 119 #endif
 120       return y;
 121     }
 122   if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
 123     {
 124       /* x < 0x1p-1022 or inf or nan.  */
 125       if (ix * 2 == 0)
 126         return __math_divzero (1);
 127       if (ix == asuint64 (INFINITY)) /* log(inf) == inf.  */
 128         return x;
 129       if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
 130         return __math_invalid (x);
 131       /* x is subnormal, normalize it.  */
 132       ix = asuint64 (x * 0x1p52);
 133       ix -= 52ULL << 52;
 134     }
 135
 136   /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
 137      The range is split into N subintervals.
 138      The ith subinterval contains z and c is near its center.  */
 139   tmp = ix - OFF;
 140   i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
 141   k = (int64_t) tmp >> 52; /* arithmetic shift */
 142   iz = ix - (tmp & 0xfffULL << 52);
 143   invc = T[i].invc;
 144   logc = T[i].logc;
 145   z = asdouble (iz);
 146
 147   /* log(x) = log1p(z/c-1) + log(c) + k*Ln2.  */
 148   /* r ~= z/c - 1, |r| < 1/(2*N).  */
 149 #if HAVE_FAST_FMA
 150   /* rounding error: 0x1p-55/N.  */
 151   r = fma (z, invc, -1.0);
 152 #else
 153   /* rounding error: 0x1p-55/N + 0x1p-66.  */
 154   r = (z - T2[i].chi - T2[i].clo) * invc;
 155 #endif
 156   kd = (double_t) k;
 157
 158   /* hi + lo = r + log(c) + k*Ln2.  */
 159   w = kd * Ln2hi + logc;
 160   hi = w + r;
 161   lo = w - hi + r + kd * Ln2lo;
 162
 163   /* log(x) = lo + (log1p(r) - r) + hi.  */
 164   r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
 165   /* Worst case error if |y| > 0x1p-5:
 166      0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
 167      Worst case error if |y| > 0x1p-4:
 168      0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma).  */
 169 #if LOG_POLY_ORDER == 6
 170   y = lo + r2 * A[0] + r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
 171 #elif LOG_POLY_ORDER == 7
 172   y = lo
 173       + r2 * (A[0] + r * A[1] + r2 * (A[2] + r * A[3])
 174               + r2 * r2 * (A[4] + r * A[5]))
 175       + hi;
 176 #endif
 177   return y;
 178 }
 179 #endif