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.h"
84 #include "math_private.h"
85
86 static const double
87 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
88 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
89 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
90 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
91 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
92 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
93 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
94 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
95 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
96 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
97
98 static const double zero = 0.0;
99 static volatile double vzero = 0.0;
100
101 double
log1p(double x)102 log1p(double x)
103 {
104 double hfsq,f,c,s,z,R,u;
105 int32_t k,hx,hu,ax;
106
107 GET_HIGH_WORD(hx,x);
108 ax = hx&0x7fffffff;
109
110 k = 1;
111 if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
112 if(ax>=0x3ff00000) { /* x <= -1.0 */
113 if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */
114 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
115 }
116 if(ax<0x3e200000) { /* |x| < 2**-29 */
117 if(two54+x>zero /* raise inexact */
118 &&ax<0x3c900000) /* |x| < 2**-54 */
119 return x;
120 else
121 return x - x*x*0.5;
122 }
123 if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
124 k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
125 }
126 if (hx >= 0x7ff00000) return x+x;
127 if(k!=0) {
128 if(hx<0x43400000) {
129 STRICT_ASSIGN(double,u,1.0+x);
130 GET_HIGH_WORD(hu,u);
131 k = (hu>>20)-1023;
132 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
133 c /= u;
134 } else {
135 u = x;
136 GET_HIGH_WORD(hu,u);
137 k = (hu>>20)-1023;
138 c = 0;
139 }
140 hu &= 0x000fffff;
141 /*
142 * The approximation to sqrt(2) used in thresholds is not
143 * critical. However, the ones used above must give less
144 * strict bounds than the one here so that the k==0 case is
145 * never reached from here, since here we have committed to
146 * using the correction term but don't use it if k==0.
147 */
148 if(hu<0x6a09e) { /* u ~< sqrt(2) */
149 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
150 } else {
151 k += 1;
152 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
153 hu = (0x00100000-hu)>>2;
154 }
155 f = u-1.0;
156 }
157 hfsq=0.5*f*f;
158 if(hu==0) { /* |f| < 2**-20 */
159 if(f==zero) {
160 if(k==0) {
161 return zero;
162 } else {
163 c += k*ln2_lo;
164 return k*ln2_hi+c;
165 }
166 }
167 R = hfsq*(1.0-0.66666666666666666*f);
168 if(k==0) return f-R; else
169 return k*ln2_hi-((R-(k*ln2_lo+c))-f);
170 }
171 s = f/(2.0+f);
172 z = s*s;
173 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
174 if(k==0) return f-(hfsq-s*(hfsq+R)); else
175 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
176 }
177
178 #if (LDBL_MANT_DIG == 53)
179 __weak_reference(log1p, log1pl);
180 #endif
181