about summary refs log tree commit diff
path: root/sysdeps/ieee754/dbl-64/e_log2.c
blob: e4a6aff9a3b5aa68c03603bf4cb8bb78507ea9c6 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
/* Adapted for log2 by Ulrich Drepper <drepper@cygnus.com>.  */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

/* __ieee754_log2(x)
 * Return the logarithm to base 2 of x
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *			x = 2^k * (1+f),
 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *		 = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *	a polynomial of degree 14 to approximate R The maximum error
 *	of this polynomial approximation is bounded by 2**-58.45. In
 *	other words,
 *			2      4      6      8      10      12      14
 *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *	(the values of Lg1 to Lg7 are listed in the program)
 *	and
 *	    |      2          14          |     -58.45
 *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 *	    |                             |
 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *	In order to guarantee error in log below 1ulp, we compute log
 *	by
 *		log(1+f) = f - s*(f - R)	(if f is not too large)
 *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
 *
 *	3. Finally,  log(x) = k + log(1+f).
 *			    = k+(f-(hfsq-(s*(hfsq+R))))
 *
 * Special cases:
 *	log2(x) is NaN with signal if x < 0 (including -INF) ;
 *	log2(+INF) is +INF; log(0) is -INF with signal;
 *	log2(NaN) is that NaN with no signal.
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include <math.h>
#include <math_private.h>
#include <fix-int-fp-convert-zero.h>

static const double ln2 = 0.69314718055994530942;
static const double two54 = 1.80143985094819840000e+16; /* 43500000 00000000 */
static const double Lg1 = 6.666666666666735130e-01;     /* 3FE55555 55555593 */
static const double Lg2 = 3.999999999940941908e-01;     /* 3FD99999 9997FA04 */
static const double Lg3 = 2.857142874366239149e-01;     /* 3FD24924 94229359 */
static const double Lg4 = 2.222219843214978396e-01;     /* 3FCC71C5 1D8E78AF */
static const double Lg5 = 1.818357216161805012e-01;     /* 3FC74664 96CB03DE */
static const double Lg6 = 1.531383769920937332e-01;     /* 3FC39A09 D078C69F */
static const double Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */

static const double zero = 0.0;

double
__ieee754_log2 (double x)
{
  double hfsq, f, s, z, R, w, t1, t2, dk;
  int32_t k, hx, i, j;
  uint32_t lx;

  EXTRACT_WORDS (hx, lx, x);

  k = 0;
  if (hx < 0x00100000)
    {                           /* x < 2**-1022  */
      if (__glibc_unlikely (((hx & 0x7fffffff) | lx) == 0))
	return -two54 / fabs (x);        /* log(+-0)=-inf */
      if (__glibc_unlikely (hx < 0))
	return (x - x) / (x - x);       /* log(-#) = NaN */
      k -= 54;
      x *= two54;               /* subnormal number, scale up x */
      GET_HIGH_WORD (hx, x);
    }
  if (__glibc_unlikely (hx >= 0x7ff00000))
    return x + x;
  k += (hx >> 20) - 1023;
  hx &= 0x000fffff;
  i = (hx + 0x95f64) & 0x100000;
  SET_HIGH_WORD (x, hx | (i ^ 0x3ff00000));     /* normalize x or x/2 */
  k += (i >> 20);
  dk = (double) k;
  f = x - 1.0;
  if ((0x000fffff & (2 + hx)) < 3)
    {                           /* |f| < 2**-20 */
      if (f == zero)
	{
	  if (FIX_INT_FP_CONVERT_ZERO && dk == 0.0)
	    dk = 0.0;
	  return dk;
	}
      R = f * f * (0.5 - 0.33333333333333333 * f);
      return dk - (R - f) / ln2;
    }
  s = f / (2.0 + f);
  z = s * s;
  i = hx - 0x6147a;
  w = z * z;
  j = 0x6b851 - hx;
  t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
  t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
  i |= j;
  R = t2 + t1;
  if (i > 0)
    {
      hfsq = 0.5 * f * f;
      return dk - ((hfsq - (s * (hfsq + R))) - f) / ln2;
    }
  else
    {
      return dk - ((s * (f - R)) - f) / ln2;
    }
}

strong_alias (__ieee754_log2, __log2_finite)