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e_log.c
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1 /* @(#)e_log.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 #if defined(LIBM_SCCS) && !defined(lint)
14 static const char rcsid[] =
15 "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
16 #endif
17
18 /* __ieee754_log(x)
19 * Return the logrithm of x
20 *
21 * Method :
22 * 1. Argument Reduction: find k and f such that
23 * x = 2^k * (1+f),
24 * where sqrt(2)/2 < 1+f < sqrt(2) .
25 *
26 * 2. Approximation of log(1+f).
27 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
28 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
29 * = 2s + s*R
30 * We use a special Reme algorithm on [0,0.1716] to generate
31 * a polynomial of degree 14 to approximate R The maximum error
32 * of this polynomial approximation is bounded by 2**-58.45. In
33 * other words,
34 * 2 4 6 8 10 12 14
35 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
36 * (the values of Lg1 to Lg7 are listed in the program)
37 * and
38 * | 2 14 | -58.45
39 * | Lg1*s +...+Lg7*s - R(z) | <= 2
40 * | |
41 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
42 * In order to guarantee error in log below 1ulp, we compute log
43 * by
44 * log(1+f) = f - s*(f - R) (if f is not too large)
45 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
46 *
47 * 3. Finally, log(x) = k*ln2 + log(1+f).
48 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
49 * Here ln2 is split into two floating point number:
50 * ln2_hi + ln2_lo,
51 * where n*ln2_hi is always exact for |n| < 2000.
52 *
53 * Special cases:
54 * log(x) is NaN with signal if x < 0 (including -INF) ;
55 * log(+INF) is +INF; log(0) is -INF with signal;
56 * log(NaN) is that NaN with no signal.
57 *
58 * Accuracy:
59 * according to an error analysis, the error is always less than
60 * 1 ulp (unit in the last place).
61 *
62 * Constants:
63 * The hexadecimal values are the intended ones for the following
64 * constants. The decimal values may be used, provided that the
65 * compiler will convert from decimal to binary accurately enough
66 * to produce the hexadecimal values shown.
67 */
68
69 #include "math_libm.h"
70 #include "math_private.h"
71
72 #ifdef __STDC__
73 static const double
74 #else
75 static double
76 #endif
77 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
78 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
79 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
80 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
81 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
82 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
83 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
84 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
85 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
86 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
87
88 #ifdef __STDC__
89 static const double zero = 0.0;
90 #else
91 static double zero = 0.0;
92 #endif
93
94 #ifdef __STDC__
95 double attribute_hidden
96 __ieee754_log(double x)
97 #else
98 double attribute_hidden
99 __ieee754_log(x)
100 double x;
101 #endif
102 {
103 double hfsq, f, s, z, R, w, t1, t2, dk;
104 int32_t k, hx, i, j;
105 u_int32_t lx;
106
107 EXTRACT_WORDS(hx, lx, x);
108
109 k = 0;
110 if (hx < 0x00100000) { /* x < 2**-1022 */
111 if (((hx & 0x7fffffff) | lx) == 0)
112 return -two54 / zero; /* log(+-0)=-inf */
113 if (hx < 0)
114 return (x - x) / zero; /* log(-#) = NaN */
115 k -= 54;
116 x *= two54; /* subnormal number, scale up x */
117 GET_HIGH_WORD(hx, x);
118 }
119 if (hx >= 0x7ff00000)
120 return x + x;
121 k += (hx >> 20) - 1023;
122 hx &= 0x000fffff;
123 i = (hx + 0x95f64) & 0x100000;
124 SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000)); /* normalize x or x/2 */
125 k += (i >> 20);
126 f = x - 1.0;
127 if ((0x000fffff & (2 + hx)) < 3) { /* |f| < 2**-20 */
128 if (f == zero) {
129 if (k == 0)
130 return zero;
131 else {
132 dk = (double) k;
133 return dk * ln2_hi + dk * ln2_lo;
134 }
135 }
136 R = f * f * (0.5 - 0.33333333333333333 * f);
137 if (k == 0)
138 return f - R;
139 else {
140 dk = (double) k;
141 return dk * ln2_hi - ((R - dk * ln2_lo) - f);
142 }
143 }
144 s = f / (2.0 + f);
145 dk = (double) k;
146 z = s * s;
147 i = hx - 0x6147a;
148 w = z * z;
149 j = 0x6b851 - hx;
150 t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
151 t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
152 i |= j;
153 R = t2 + t1;
154 if (i > 0) {
155 hfsq = 0.5 * f * f;
156 if (k == 0)
157 return f - (hfsq - s * (hfsq + R));
158 else
159 return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
160 f);
161 } else {
162 if (k == 0)
163 return f - s * (f - R);
164 else
165 return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
166 }
167 }
GET_HIGH_WORD
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:102
s
GLdouble s
Definition: glew.h:1376
zero
static double zero
Definition: e_log.c:91
t2
t2
Definition: e_log.c:151
f
GLclampf f
Definition: glew.h:3390
k
int32_t k
Definition: e_log.c:102
x
EGLSurface EGLint x
Definition: eglext.h:293
Lg1
static double Lg1
Definition: e_log.c:80
ln2_lo
static double ln2_lo
Definition: e_log.c:78
j
int32_t j
Definition: e_log.c:102
dk
dk
Definition: e_log.c:145
int32_t
long int32_t
Definition: types.h:9
Lg6
static double Lg6
Definition: e_log.c:85
SET_HIGH_WORD
#define SET_HIGH_WORD(d, v)
Definition: math_private.h:130
t1
GLuint GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat t1
Definition: glew.h:11582
u_int32_t
unsigned int u_int32_t
Definition: math_private.h:29
EXTRACT_WORDS
#define EXTRACT_WORDS(ix0, ix1, d)
Definition: math_private.h:92
Lg5
static double Lg5
Definition: e_log.c:84
__ieee754_log
double __ieee754_log(double) attribute_hidden
math_libm.h
Lg4
static double Lg4
Definition: e_log.c:83
hx
int32_t hx
Definition: e_log.c:102
Lg3
static double Lg3
Definition: e_log.c:82
attribute_hidden
double attribute_hidden
lx
u_int32_t lx
Definition: e_log.c:105
z
GLint GLint GLint GLint z
Definition: gl2ext.h:1214
w
GLint GLint GLint GLint GLint w
Definition: gl2ext.h:1215
Lg7
static double Lg7
Definition: e_log.c:86
two54
static double two54
Definition: e_log.c:79
R
R
Definition: e_log.c:153
i
int i
Definition: pngrutil.c:1377
Lg2
static double Lg2
Definition: e_log.c:81
math_private.h
ln2_hi
static double ln2_hi
Definition: e_log.c:77