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s_log1p.c (5264B)
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1 /* @(#)s_log1p.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 #ifndef lint
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_log1p.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
15 #endif
16
17 /* double log1p(double x)
18 *
19 * Method :
20 * 1. Argument Reduction: find k and f such that
21 * 1+x = 2^k * (1+f),
22 * where sqrt(2)/2 < 1+f < sqrt(2) .
23 *
24 * Note. If k=0, then f=x is exact. However, if k!=0, then f
25 * may not be representable exactly. In that case, a correction
26 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
27 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
28 * and add back the correction term c/u.
29 * (Note: when x > 2**53, one can simply return log(x))
30 *
31 * 2. Approximation of log1p(f).
32 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
33 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
34 * = 2s + s*R
35 * We use a special Reme algorithm on [0,0.1716] to generate
36 * a polynomial of degree 14 to approximate R The maximum error
37 * of this polynomial approximation is bounded by 2**-58.45. In
38 * other words,
39 * 2 4 6 8 10 12 14
40 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
41 * (the values of Lp1 to Lp7 are listed in the program)
42 * and
43 * | 2 14 | -58.45
44 * | Lp1*s +...+Lp7*s - R(z) | <= 2
45 * | |
46 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
47 * In order to guarantee error in log below 1ulp, we compute log
48 * by
49 * log1p(f) = f - (hfsq - s*(hfsq+R)).
50 *
51 * 3. Finally, log1p(x) = k*ln2 + log1p(f).
52 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
53 * Here ln2 is split into two floating point number:
54 * ln2_hi + ln2_lo,
55 * where n*ln2_hi is always exact for |n| < 2000.
56 *
57 * Special cases:
58 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
59 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
60 * log1p(NaN) is that NaN with no signal.
61 *
62 * Accuracy:
63 * according to an error analysis, the error is always less than
64 * 1 ulp (unit in the last place).
65 *
66 * Constants:
67 * The hexadecimal values are the intended ones for the following
68 * constants. The decimal values may be used, provided that the
69 * compiler will convert from decimal to binary accurately enough
70 * to produce the hexadecimal values shown.
71 *
72 * Note: Assuming log() return accurate answer, the following
73 * algorithm can be used to compute log1p(x) to within a few ULP:
74 *
75 * u = 1+x;
76 * if(u==1.0) return x ; else
77 * return log(u)*(x/(u-1.0));
78 *
79 * See HP-15C Advanced Functions Handbook, p.193.
80 */
81
82 #include "math.h"
83 #include "math_private.h"
84
85 static const double
86 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
87 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
88 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
89 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
90 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
91 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
92 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
93 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
94 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
95 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
96
97 static const double zero = 0.0;
98
99 double
100 log1p(double x)
101 {
102 double hfsq,f,c,s,z,R,u;
103 int32_t k,hx,hu,ax;
104
105 GET_HIGH_WORD(hx,x);
106 ax = hx&0x7fffffff;
107
108 k = 1;
109 if (hx < 0x3FDA827A) { /* x < 0.41422 */
110 if(ax>=0x3ff00000) { /* x <= -1.0 */
111 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
112 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
113 }
114 if(ax<0x3e200000) { /* |x| < 2**-29 */
115 if(two54+x>zero /* raise inexact */
116 &&ax<0x3c900000) /* |x| < 2**-54 */
117 return x;
118 else
119 return x - x*x*0.5;
120 }
121 if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
122 k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
123 }
124 if (hx >= 0x7ff00000) return x+x;
125 if(k!=0) {
126 if(hx<0x43400000) {
127 u = 1.0+x;
128 GET_HIGH_WORD(hu,u);
129 k = (hu>>20)-1023;
130 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
131 c /= u;
132 } else {
133 u = x;
134 GET_HIGH_WORD(hu,u);
135 k = (hu>>20)-1023;
136 c = 0;
137 }
138 hu &= 0x000fffff;
139 if(hu<0x6a09e) {
140 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
141 } else {
142 k += 1;
143 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
144 hu = (0x00100000-hu)>>2;
145 }
146 f = u-1.0;
147 }
148 hfsq=0.5*f*f;
149 if(hu==0) { /* |f| < 2**-20 */
150 if(f==zero) if(k==0) return zero;
151 else {c += k*ln2_lo; return k*ln2_hi+c;}
152 R = hfsq*(1.0-0.66666666666666666*f);
153 if(k==0) return f-R; else
154 return k*ln2_hi-((R-(k*ln2_lo+c))-f);
155 }
156 s = f/(2.0+f);
157 z = s*s;
158 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
159 if(k==0) return f-(hfsq-s*(hfsq+R)); else
160 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
161 }