1 /* Adapted for log2 by Ulrich Drepper <drepper@cygnus.com>. */
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 /* __ieee754_log2(x)
14 * Return the logarithm to base 2 of x
15 *
16 * Method :
17 * 1. Argument Reduction: find k and f such that
18 * x = 2^k * (1+f),
19 * where sqrt(2)/2 < 1+f < sqrt(2) .
20 *
21 * 2. Approximation of log(1+f).
22 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
23 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
24 * = 2s + s*R
25 * We use a special Reme algorithm on [0,0.1716] to generate
26 * a polynomial of degree 14 to approximate R The maximum error
27 * of this polynomial approximation is bounded by 2**-58.45. In
28 * other words,
29 * 2 4 6 8 10 12 14
30 * R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
31 * (the values of Lg1 to Lg7 are listed in the program)
32 * and
33 * | 2 14 | -58.45
34 * | Lg1*s +...+Lg7*s - R(z) | <= 2
35 * | |
36 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
37 * In order to guarantee error in log below 1ulp, we compute log
38 * by
39 * log(1+f) = f - s*(f - R) (if f is not too large)
40 * log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
41 *
42 * 3. Finally, log(x) = k + log(1+f).
43 * = k+(f-(hfsq-(s*(hfsq+R))))
44 *
45 * Special cases:
46 * log2(x) is NaN with signal if x < 0 (including -INF) ;
47 * log2(+INF) is +INF; log(0) is -INF with signal;
48 * log2(NaN) is that NaN with no signal.
49 *
50 * Constants:
51 * The hexadecimal values are the intended ones for the following
52 * constants. The decimal values may be used, provided that the
53 * compiler will convert from decimal to binary accurately enough
54 * to produce the hexadecimal values shown.
55 */
56
57 #include "math.h"
58 #include "math_private.h"
59
60 static const double
61 ln2 = 0.69314718055994530942,
62 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
63 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
64 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
65 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
66 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
67 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
68 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
69 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
70
71 static const double zero = 0.0;
72
__ieee754_log2(double x)73 double __ieee754_log2(double x)
74 {
75 double hfsq,f,s,z,R,w,t1,t2,dk;
76 int32_t k,hx,i,j;
77 u_int32_t lx;
78
79 EXTRACT_WORDS(hx,lx,x);
80
81 k=0;
82 if (hx < 0x00100000) { /* x < 2**-1022 */
83 if (((hx&0x7fffffff)|lx)==0)
84 return -two54/(x-x); /* log(+-0)=-inf */
85 if (hx<0) return (x-x)/(x-x); /* log(-#) = NaN */
86 k -= 54; x *= two54; /* subnormal number, scale up x */
87 GET_HIGH_WORD(hx,x);
88 }
89 if (hx >= 0x7ff00000) return x+x;
90 k += (hx>>20)-1023;
91 hx &= 0x000fffff;
92 i = (hx+0x95f64)&0x100000;
93 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
94 k += (i>>20);
95 dk = (double) k;
96 f = x-1.0;
97 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
98 if(f==zero) return dk;
99 R = f*f*(0.5-0.33333333333333333*f);
100 return dk-(R-f)/ln2;
101 }
102 s = f/(2.0+f);
103 z = s*s;
104 i = hx-0x6147a;
105 w = z*z;
106 j = 0x6b851-hx;
107 t1= w*(Lg2+w*(Lg4+w*Lg6));
108 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
109 i |= j;
110 R = t2+t1;
111 if(i>0) {
112 hfsq=0.5*f*f;
113 return dk-((hfsq-(s*(hfsq+R)))-f)/ln2;
114 } else {
115 return dk-((s*(f-R))-f)/ln2;
116 }
117 }
118 strong_alias(__ieee754_log2,log2)
119 libm_hidden_def(log2)
120