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 #include <sys/cdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: e_log.c,v 1.12 2002/05/26 22:01:51 wiz 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.h"
70 #include "math_private.h"
71
72 static const double
73 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
74 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
75 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
76 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
77 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
78 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
79 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
80 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
81 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
82 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
83
84 static const double zero = 0.0;
85
86 double
__ieee754_log(double x)87 __ieee754_log(double x)
88 {
89 double hfsq,f,s,z,R,w,t1,t2,dk;
90 int32_t k,hx,i,j;
91 u_int32_t lx;
92
93 EXTRACT_WORDS(hx,lx,x);
94
95 k=0;
96 if (hx < 0x00100000) { /* x < 2**-1022 */
97 if (((hx&0x7fffffff)|lx)==0)
98 return -two54/zero; /* log(+-0)=-inf */
99 if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
100 k -= 54; x *= two54; /* subnormal number, scale up x */
101 GET_HIGH_WORD(hx,x);
102 }
103 if (hx >= 0x7ff00000) return x+x;
104 k += (hx>>20)-1023;
105 hx &= 0x000fffff;
106 i = (hx+0x95f64)&0x100000;
107 SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
108 k += (i>>20);
109 f = x-1.0;
110 if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
111 if(f==zero) { if(k==0) return zero; else {dk=(double)k;
112 return dk*ln2_hi+dk*ln2_lo;}
113 }
114 R = f*f*(0.5-0.33333333333333333*f);
115 if(k==0) return f-R; else {dk=(double)k;
116 return dk*ln2_hi-((R-dk*ln2_lo)-f);}
117 }
118 s = f/(2.0+f);
119 dk = (double)k;
120 z = s*s;
121 i = hx-0x6147a;
122 w = z*z;
123 j = 0x6b851-hx;
124 t1= w*(Lg2+w*(Lg4+w*Lg6));
125 t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
126 i |= j;
127 R = t2+t1;
128 if(i>0) {
129 hfsq=0.5*f*f;
130 if(k==0) return f-(hfsq-s*(hfsq+R)); else
131 return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
132 } else {
133 if(k==0) return f-s*(f-R); else
134 return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
135 }
136 }
137