Line data Source code
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 : //__FBSDID("$FreeBSD$");
15 :
16 : /* double log1p(double x)
17 : *
18 : * Method :
19 : * 1. Argument Reduction: find k and f such that
20 : * 1+x = 2^k * (1+f),
21 : * where sqrt(2)/2 < 1+f < sqrt(2) .
22 : *
23 : * Note. If k=0, then f=x is exact. However, if k!=0, then f
24 : * may not be representable exactly. In that case, a correction
25 : * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
26 : * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
27 : * and add back the correction term c/u.
28 : * (Note: when x > 2**53, one can simply return log(x))
29 : *
30 : * 2. Approximation of log1p(f).
31 : * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
32 : * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
33 : * = 2s + s*R
34 : * We use a special Reme algorithm on [0,0.1716] to generate
35 : * a polynomial of degree 14 to approximate R The maximum error
36 : * of this polynomial approximation is bounded by 2**-58.45. In
37 : * other words,
38 : * 2 4 6 8 10 12 14
39 : * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
40 : * (the values of Lp1 to Lp7 are listed in the program)
41 : * and
42 : * | 2 14 | -58.45
43 : * | Lp1*s +...+Lp7*s - R(z) | <= 2
44 : * | |
45 : * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
46 : * In order to guarantee error in log below 1ulp, we compute log
47 : * by
48 : * log1p(f) = f - (hfsq - s*(hfsq+R)).
49 : *
50 : * 3. Finally, log1p(x) = k*ln2 + log1p(f).
51 : * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
52 : * Here ln2 is split into two floating point number:
53 : * ln2_hi + ln2_lo,
54 : * where n*ln2_hi is always exact for |n| < 2000.
55 : *
56 : * Special cases:
57 : * log1p(x) is NaN with signal if x < -1 (including -INF) ;
58 : * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
59 : * log1p(NaN) is that NaN with no signal.
60 : *
61 : * Accuracy:
62 : * according to an error analysis, the error is always less than
63 : * 1 ulp (unit in the last place).
64 : *
65 : * Constants:
66 : * The hexadecimal values are the intended ones for the following
67 : * constants. The decimal values may be used, provided that the
68 : * compiler will convert from decimal to binary accurately enough
69 : * to produce the hexadecimal values shown.
70 : *
71 : * Note: Assuming log() return accurate answer, the following
72 : * algorithm can be used to compute log1p(x) to within a few ULP:
73 : *
74 : * u = 1+x;
75 : * if(u==1.0) return x ; else
76 : * return log(u)*(x/(u-1.0));
77 : *
78 : * See HP-15C Advanced Functions Handbook, p.193.
79 : */
80 :
81 : #include <float.h>
82 :
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 : static volatile double vzero = 0.0;
99 :
100 : double
101 0 : log1p(double x)
102 : {
103 : double hfsq,f,c,s,z,R,u;
104 : int32_t k,hx,hu,ax;
105 :
106 0 : GET_HIGH_WORD(hx,x);
107 0 : ax = hx&0x7fffffff;
108 :
109 0 : k = 1;
110 0 : if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
111 0 : if(ax>=0x3ff00000) { /* x <= -1.0 */
112 0 : if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */
113 0 : else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
114 : }
115 0 : if(ax<0x3e200000) { /* |x| < 2**-29 */
116 0 : if(two54+x>zero /* raise inexact */
117 0 : &&ax<0x3c900000) /* |x| < 2**-54 */
118 0 : return x;
119 : else
120 0 : return x - x*x*0.5;
121 : }
122 0 : if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
123 0 : k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
124 : }
125 0 : if (hx >= 0x7ff00000) return x+x;
126 0 : if(k!=0) {
127 0 : if(hx<0x43400000) {
128 0 : STRICT_ASSIGN(double,u,1.0+x);
129 0 : GET_HIGH_WORD(hu,u);
130 0 : k = (hu>>20)-1023;
131 0 : c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
132 0 : c /= u;
133 : } else {
134 0 : u = x;
135 0 : GET_HIGH_WORD(hu,u);
136 0 : k = (hu>>20)-1023;
137 0 : c = 0;
138 : }
139 0 : hu &= 0x000fffff;
140 : /*
141 : * The approximation to sqrt(2) used in thresholds is not
142 : * critical. However, the ones used above must give less
143 : * strict bounds than the one here so that the k==0 case is
144 : * never reached from here, since here we have committed to
145 : * using the correction term but don't use it if k==0.
146 : */
147 0 : if(hu<0x6a09e) { /* u ~< sqrt(2) */
148 0 : SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
149 : } else {
150 0 : k += 1;
151 0 : SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
152 0 : hu = (0x00100000-hu)>>2;
153 : }
154 0 : f = u-1.0;
155 : }
156 0 : hfsq=0.5*f*f;
157 0 : if(hu==0) { /* |f| < 2**-20 */
158 0 : if(f==zero) {
159 0 : if(k==0) {
160 0 : return zero;
161 : } else {
162 0 : c += k*ln2_lo;
163 0 : return k*ln2_hi+c;
164 : }
165 : }
166 0 : R = hfsq*(1.0-0.66666666666666666*f);
167 0 : if(k==0) return f-R; else
168 0 : return k*ln2_hi-((R-(k*ln2_lo+c))-f);
169 : }
170 0 : s = f/(2.0+f);
171 0 : z = s*s;
172 0 : R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
173 0 : if(k==0) return f-(hfsq-s*(hfsq+R)); else
174 0 : return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
175 : }
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