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 #include <sys/cdefs.h>
14 #if defined(LIBM_SCCS) && !defined(lint)
15 __RCSID("$NetBSD: s_log1p.c,v 1.13 2024/07/16 14:52:50 riastradh Exp $");
16 #endif
17
18 /* double log1p(double x)
19 *
20 * Method :
21 * 1. Argument Reduction: find k and f such that
22 * 1+x = 2^k * (1+f),
23 * where sqrt(2)/2 < 1+f < sqrt(2) .
24 *
25 * Note. If k=0, then f=x is exact. However, if k!=0, then f
26 * may not be representable exactly. In that case, a correction
27 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
28 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
29 * and add back the correction term c/u.
30 * (Note: when x > 2**53, one can simply return log(x))
31 *
32 * 2. Approximation of log1p(f).
33 * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
34 * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
35 * = 2s + s*R
36 * We use a special Reme algorithm on [0,0.1716] to generate
37 * a polynomial of degree 14 to approximate R The maximum error
38 * of this polynomial approximation is bounded by 2**-58.45. In
39 * other words,
40 * 2 4 6 8 10 12 14
41 * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
42 * (the values of Lp1 to Lp7 are listed in the program)
43 * and
44 * | 2 14 | -58.45
45 * | Lp1*s +...+Lp7*s - R(z) | <= 2
46 * | |
47 * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
48 * In order to guarantee error in log below 1ulp, we compute log
49 * by
50 * log1p(f) = f - (hfsq - s*(hfsq+R)).
51 *
52 * 3. Finally, log1p(x) = k*ln2 + log1p(f).
53 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
54 * Here ln2 is split into two floating point number:
55 * ln2_hi + ln2_lo,
56 * where n*ln2_hi is always exact for |n| < 2000.
57 *
58 * Special cases:
59 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
60 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
61 * log1p(NaN) is that NaN with no signal.
62 *
63 * Accuracy:
64 * according to an error analysis, the error is always less than
65 * 1 ulp (unit in the last place).
66 *
67 * Constants:
68 * The hexadecimal values are the intended ones for the following
69 * constants. The decimal values may be used, provided that the
70 * compiler will convert from decimal to binary accurately enough
71 * to produce the hexadecimal values shown.
72 *
73 * Note: Assuming log() return accurate answer, the following
74 * algorithm can be used to compute log1p(x) to within a few ULP:
75 *
76 * u = 1+x;
77 * if(u==1.0) return x ; else
78 * return log(u)*(x/(u-1.0));
79 *
80 * See HP-15C Advanced Functions Handbook, p.193.
81 */
82
83 #include "namespace.h"
84
85 #include "math.h"
86 #include "math_private.h"
87
88 #ifndef __HAVE_LONG_DOUBLE
89 __weak_alias(log1pl, _log1pl)
90 __strong_alias(_log1pl, _log1p)
91 #endif
92
93 static const double
94 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
95 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
96 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
97 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
98 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
99 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
100 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
101 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
102 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
103 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
104
105 static const double zero = 0.0;
106
__weak_alias(log1p,_log1p)107 __weak_alias(log1p, _log1p)
108 double
109 log1p(double x)
110 {
111 double hfsq,f,c,s,z,R,u;
112 int32_t k,hx,hu,ax;
113
114 f = c = 0;
115 hu = 0;
116 GET_HIGH_WORD(hx,x);
117 ax = hx&0x7fffffff;
118
119 k = 1;
120 if (hx < 0x3FDA827A) { /* x < 0.41422 */
121 if(ax>=0x3ff00000) { /* x <= -1.0 */
122 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
123 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
124 }
125 if(ax<0x3e200000) { /* |x| < 2**-29 */
126 if(two54+x>zero /* raise inexact */
127 &&ax<0x3c900000) /* |x| < 2**-54 */
128 return x;
129 else
130 return x - x*x*0.5;
131 }
132 if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
133 k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
134 }
135 if (hx >= 0x7ff00000) return x+x;
136 if(k!=0) {
137 if(hx<0x43400000) {
138 u = 1.0+x;
139 GET_HIGH_WORD(hu,u);
140 k = (hu>>20)-1023;
141 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
142 c /= u;
143 } else {
144 u = x;
145 GET_HIGH_WORD(hu,u);
146 k = (hu>>20)-1023;
147 c = 0;
148 }
149 hu &= 0x000fffff;
150 if(hu<0x6a09e) {
151 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
152 } else {
153 k += 1;
154 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
155 hu = (0x00100000-hu)>>2;
156 }
157 f = u-1.0;
158 }
159 hfsq=0.5*f*f;
160 if(hu==0) { /* |f| < 2**-20 */
161 if(f==zero) { if(k==0) return zero;
162 else {c += k*ln2_lo; return k*ln2_hi+c;}
163 }
164 R = hfsq*(1.0-0.66666666666666666*f);
165 if(k==0) return f-R; else
166 return k*ln2_hi-((R-(k*ln2_lo+c))-f);
167 }
168 s = f/(2.0+f);
169 z = s*s;
170 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
171 if(k==0) return f-(hfsq-s*(hfsq+R)); else
172 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
173 }
174