zenilib  0.5.3.0
e_log.c
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1 /* @(#)e_log.c 5.1 93/09/24 */
2 /*
3  * ====================================================
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
100  double x;
101 #endif
102 {
103  double hfsq, f, s, z, R, w, t1, t2, dk;
104  int32_t k, hx, i, j;
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 }
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:102
GLdouble s
Definition: glew.h:1376
static double zero
Definition: e_log.c:91
t2
Definition: e_log.c:151
GLclampf f
Definition: glew.h:3390
int32_t k
Definition: e_log.c:102
EGLSurface EGLint x
Definition: eglext.h:293
static double Lg1
Definition: e_log.c:80
static double ln2_lo
Definition: e_log.c:78
int32_t j
Definition: e_log.c:102
dk
Definition: e_log.c:145
long int32_t
Definition: types.h:9
static double Lg6
Definition: e_log.c:85
#define SET_HIGH_WORD(d, v)
Definition: math_private.h:130
GLuint GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat GLfloat t1
Definition: glew.h:11582
unsigned int u_int32_t
Definition: math_private.h:29
#define EXTRACT_WORDS(ix0, ix1, d)
Definition: math_private.h:92
static double Lg5
Definition: e_log.c:84
double __ieee754_log(double) attribute_hidden
static double Lg4
Definition: e_log.c:83
int32_t hx
Definition: e_log.c:102
static double Lg3
Definition: e_log.c:82
double attribute_hidden
u_int32_t lx
Definition: e_log.c:105
GLint GLint GLint GLint z
Definition: gl2ext.h:1214
GLint GLint GLint GLint GLint w
Definition: gl2ext.h:1215
static double Lg7
Definition: e_log.c:86
static double two54
Definition: e_log.c:79
R
Definition: e_log.c:153
int i
Definition: pngrutil.c:1377
static double Lg2
Definition: e_log.c:81
static double ln2_hi
Definition: e_log.c:77