src/math/log2.c

   1 /*
   2  * Double-precision log2(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 "log2_data.h"
  12
  13 #define T __log2_data.tab
  14 #define T2 __log2_data.tab2
  15 #define B __log2_data.poly1
  16 #define A __log2_data.poly
  17 #define InvLn2hi __log2_data.invln2hi
  18 #define InvLn2lo __log2_data.invln2lo
  19 #define N (1 << LOG2_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 log2(double x)
  29 {
  30         double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
  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 - 0x1.5b51p-5)
  38 #define HI asuint64(1.0 + 0x1.6ab2p-5)
  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 #if __FP_FAST_FMA
  46                 hi = r * InvLn2hi;
  47                 lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi);
  48 #else
  49                 double_t rhi, rlo;
  50                 rhi = asdouble(asuint64(r) & -1ULL << 32);
  51                 rlo = r - rhi;
  52                 hi = rhi * InvLn2hi;
  53                 lo = rlo * InvLn2hi + r * InvLn2lo;
  54 #endif
  55                 r2 = r * r; /* rounding error: 0x1p-62.  */
  56                 r4 = r2 * r2;
  57                 /* Worst-case error is less than 0.54 ULP (0.55 ULP without fma).  */
  58                 p = r2 * (B[0] + r * B[1]);
  59                 y = hi + p;
  60                 lo += hi - y + p;
  61                 lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) +
  62                             r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
  63                 y += lo;
  64                 return eval_as_double(y);
  65         }
  66         if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
  67                 /* x < 0x1p-1022 or inf or nan.  */
  68                 if (ix * 2 == 0)
  69                         return __math_divzero(1);
  70                 if (ix == asuint64(INFINITY)) /* log(inf) == inf.  */
  71                         return x;
  72                 if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
  73                         return __math_invalid(x);
  74                 /* x is subnormal, normalize it.  */
  75                 ix = asuint64(x * 0x1p52);
  76                 ix -= 52ULL << 52;
  77         }
  78
  79         /* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
  80            The range is split into N subintervals.
  81            The ith subinterval contains z and c is near its center.  */
  82         tmp = ix - OFF;
  83         i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
  84         k = (int64_t)tmp >> 52; /* arithmetic shift */
  85         iz = ix - (tmp & 0xfffULL << 52);
  86         invc = T[i].invc;
  87         logc = T[i].logc;
  88         z = asdouble(iz);
  89         kd = (double_t)k;
  90
  91         /* log2(x) = log2(z/c) + log2(c) + k.  */
  92         /* r ~= z/c - 1, |r| < 1/(2*N).  */
  93 #if __FP_FAST_FMA
  94         /* rounding error: 0x1p-55/N.  */
  95         r = __builtin_fma(z, invc, -1.0);
  96         t1 = r * InvLn2hi;
  97         t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1);
  98 #else
  99         double_t rhi, rlo;
 100         /* rounding error: 0x1p-55/N + 0x1p-65.  */
 101         r = (z - T2[i].chi - T2[i].clo) * invc;
 102         rhi = asdouble(asuint64(r) & -1ULL << 32);
 103         rlo = r - rhi;
 104         t1 = rhi * InvLn2hi;
 105         t2 = rlo * InvLn2hi + r * InvLn2lo;
 106 #endif
 107
 108         /* hi + lo = r/ln2 + log2(c) + k.  */
 109         t3 = kd + logc;
 110         hi = t3 + t1;
 111         lo = t3 - hi + t1 + t2;
 112
 113         /* log2(r+1) = r/ln2 + r^2*poly(r).  */
 114         /* Evaluation is optimized assuming superscalar pipelined execution.  */
 115         r2 = r * r; /* rounding error: 0x1p-54/N^2.  */
 116         r4 = r2 * r2;
 117         /* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
 118            ~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma).  */
 119         p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
 120         y = lo + r2 * p + hi;
 121         return eval_as_double(y);
 122 }