newlib/libm/common/log2.c

   1 /* Double-precision log2(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 __log2_data.tab
  37 #define T2 __log2_data.tab2
  38 #define B __log2_data.poly1
  39 #define A __log2_data.poly
  40 #define InvLn2hi __log2_data.invln2hi
  41 #define InvLn2lo __log2_data.invln2lo
  42 #define N (1 << LOG2_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 (log2) (double x)
  54 {
  55   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
  56   double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
  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 LOG2_POLY1_ORDER == 11
  65 # define LO asuint64 (1.0 - 0x1.5b51p-5)
  66 # define HI asuint64 (1.0 + 0x1.6ab2p-5)
  67 #endif
  68   if (unlikely (ix - LO < HI - LO))
  69     {
  70       /* Handle close to 1.0 inputs separately.  */
  71       /* Fix sign of zero with downward rounding when x==1.  */
  72       if (WANT_ROUNDING && unlikely (ix == asuint64 (1.0)))
  73         return 0;
  74       r = x - 1.0;
  75 #if HAVE_FAST_FMA
  76       hi = r * InvLn2hi;
  77       lo = r * InvLn2lo + fma (r, InvLn2hi, -hi);
  78 #else
  79       double_t rhi, rlo;
  80       rhi = asdouble (asuint64 (r) & -1ULL << 32);
  81       rlo = r - rhi;
  82       hi = rhi * InvLn2hi;
  83       lo = rlo * InvLn2hi + r * InvLn2lo;
  84 #endif
  85       r2 = r * r; /* rounding error: 0x1p-62.  */
  86       r4 = r2 * r2;
  87 #if LOG2_POLY1_ORDER == 11
  88       /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma).  */
  89       p = r2 * (B[0] + r * B[1]);
  90       y = hi + p;
  91       lo += hi - y + p;
  92       lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])
  93                   + r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
  94       y += lo;
  95 #endif
  96       return y;
  97     }
  98   if (unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
  99     {
 100       /* x < 0x1p-1022 or inf or nan.  */
 101       if (ix * 2 == 0)
 102         return __math_divzero (1);
 103       if (ix == asuint64 (INFINITY)) /* log(inf) == inf.  */
 104         return x;
 105       if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
 106         return __math_invalid (x);
 107       /* x is subnormal, normalize it.  */
 108       ix = asuint64 (x * 0x1p52);
 109       ix -= 52ULL << 52;
 110     }
 111
 112   /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
 113      The range is split into N subintervals.
 114      The ith subinterval contains z and c is near its center.  */
 115   tmp = ix - OFF;
 116   i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
 117   k = (int64_t) tmp >> 52; /* arithmetic shift */
 118   iz = ix - (tmp & 0xfffULL << 52);
 119   invc = T[i].invc;
 120   logc = T[i].logc;
 121   z = asdouble (iz);
 122   kd = (double_t) k;
 123
 124   /* log2(x) = log2(z/c) + log2(c) + k.  */
 125   /* r ~= z/c - 1, |r| < 1/(2*N).  */
 126 #if HAVE_FAST_FMA
 127   /* rounding error: 0x1p-55/N.  */
 128   r = fma (z, invc, -1.0);
 129   t1 = r * InvLn2hi;
 130   t2 = r * InvLn2lo + fma (r, InvLn2hi, -t1);
 131 #else
 132   double_t rhi, rlo;
 133   /* rounding error: 0x1p-55/N + 0x1p-65.  */
 134   r = (z - T2[i].chi - T2[i].clo) * invc;
 135   rhi = asdouble (asuint64 (r) & -1ULL << 32);
 136   rlo = r - rhi;
 137   t1 = rhi * InvLn2hi;
 138   t2 = rlo * InvLn2hi + r * InvLn2lo;
 139 #endif
 140
 141   /* hi + lo = r/ln2 + log2(c) + k.  */
 142   t3 = kd + logc;
 143   hi = t3 + t1;
 144   lo = t3 - hi + t1 + t2;
 145
 146   /* log2(r+1) = r/ln2 + r^2*poly(r).  */
 147   /* Evaluation is optimized assuming superscalar pipelined execution.  */
 148   r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
 149   r4 = r2 * r2;
 150 #if LOG2_POLY_ORDER == 7
 151   /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
 152      ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma).  */
 153   p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
 154   y = lo + r2 * p + hi;
 155 #endif
 156   return y;
 157 }
 158 #endif