1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /* double log1p(double x)
13 *
14 * Method :
15 * 1. Argument Reduction: find k and f such that
16 * 1+x = 2^k * (1+f),
17 * where sqrt(2)/2 < 1+f < sqrt(2) .
18 *
19 * Note. If k=0, then f=x is exact. However, if k!=0, then f
20 * may not be representable exactly. In that case, a correction
21 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
22 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
23 * and add back the correction term c/u.
24 * (Note: when x > 2**53, one can simply return log(x))
25 *
26 * 2. Approximation of log1p(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) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
36 * (the values of Lp1 to Lp7 are listed in the program)
37 * and
38 * | 2 14 | -58.45
39 * | Lp1*s +...+Lp7*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 * log1p(f) = f - (hfsq - s*(hfsq+R)).
45 *
46 * 3. Finally, log1p(x) = k*ln2 + log1p(f).
47 * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
48 * Here ln2 is split into two floating point number:
49 * ln2_hi + ln2_lo,
50 * where n*ln2_hi is always exact for |n| < 2000.
51 *
52 * Special cases:
53 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
54 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
55 * log1p(NaN) is that NaN with no signal.
56 *
57 * Accuracy:
58 * according to an error analysis, the error is always less than
59 * 1 ulp (unit in the last place).
60 *
61 * Constants:
62 * The hexadecimal values are the intended ones for the following
63 * constants. The decimal values may be used, provided that the
64 * compiler will convert from decimal to binary accurately enough
65 * to produce the hexadecimal values shown.
66 *
67 * Note: Assuming log() return accurate answer, the following
68 * algorithm can be used to compute log1p(x) to within a few ULP:
69 *
70 * u = 1+x;
71 * if(u==1.0) return x ; else
72 * return log(u)*(x/(u-1.0));
73 *
74 * See HP-15C Advanced Functions Handbook, p.193.
75 */
76
77 #include "math.h"
78 #include "math_private.h"
79
80 static const double
81 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
82 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
83 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
84 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
85 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
86 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
87 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
88 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
89 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
90 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
91
92 static const double zero = 0.0;
93
log1p(double x)94 double log1p(double x)
95 {
96 double hfsq,f=0,c=0,s,z,R,u;
97 int32_t k,hx,hu=0,ax;
98
99 GET_HIGH_WORD(hx,x);
100 ax = hx&0x7fffffff;
101
102 k = 1;
103 if (hx < 0x3FDA827A) { /* x < 0.41422 */
104 if(ax>=0x3ff00000) { /* x <= -1.0 */
105 if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
106 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
107 }
108 if(ax<0x3e200000) { /* |x| < 2**-29 */
109 if(two54+x>zero /* raise inexact */
110 &&ax<0x3c900000) /* |x| < 2**-54 */
111 return x;
112 else
113 return x - x*x*0.5;
114 }
115 if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
116 k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
117 }
118 if (hx >= 0x7ff00000) return x+x;
119 if(k!=0) {
120 if(hx<0x43400000) {
121 u = 1.0+x;
122 GET_HIGH_WORD(hu,u);
123 k = (hu>>20)-1023;
124 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
125 c /= u;
126 } else {
127 u = x;
128 GET_HIGH_WORD(hu,u);
129 k = (hu>>20)-1023;
130 c = 0;
131 }
132 hu &= 0x000fffff;
133 if(hu<0x6a09e) {
134 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
135 } else {
136 k += 1;
137 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
138 hu = (0x00100000-hu)>>2;
139 }
140 f = u-1.0;
141 }
142 hfsq=0.5*f*f;
143 if(hu==0) { /* |f| < 2**-20 */
144 if(f==zero) {if(k==0) return zero;
145 else {c += k*ln2_lo; return k*ln2_hi+c;}
146 }
147 R = hfsq*(1.0-0.66666666666666666*f);
148 if(k==0) return f-R; else
149 return k*ln2_hi-((R-(k*ln2_lo+c))-f);
150 }
151 s = f/(2.0+f);
152 z = s*s;
153 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
154 if(k==0) return f-(hfsq-s*(hfsq+R)); else
155 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
156 }
157 libm_hidden_def(log1p)
158