1 /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
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 /* double log1p(double x)
13 * Return the natural logarithm of 1+x.
14 *
15 * Method :
16 * 1. Argument Reduction: find k and f such that
17 * 1+x = 2^k * (1+f),
18 * where sqrt(2)/2 < 1+f < sqrt(2) .
19 *
20 * Note. If k=0, then f=x is exact. However, if k!=0, then f
21 * may not be representable exactly. In that case, a correction
22 * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
23 * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
24 * and add back the correction term c/u.
25 * (Note: when x > 2**53, one can simply return log(x))
26 *
27 * 2. Approximation of log(1+f): See log.c
28 *
29 * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
30 *
31 * Special cases:
32 * log1p(x) is NaN with signal if x < -1 (including -INF) ;
33 * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
34 * log1p(NaN) is that NaN with no signal.
35 *
36 * Accuracy:
37 * according to an error analysis, the error is always less than
38 * 1 ulp (unit in the last place).
39 *
40 * Constants:
41 * The hexadecimal values are the intended ones for the following
42 * constants. The decimal values may be used, provided that the
43 * compiler will convert from decimal to binary accurately enough
44 * to produce the hexadecimal values shown.
45 *
46 * Note: Assuming log() return accurate answer, the following
47 * algorithm can be used to compute log1p(x) to within a few ULP:
48 *
49 * u = 1+x;
50 * if(u==1.0) return x ; else
51 * return log(u)*(x/(u-1.0));
52 *
53 * See HP-15C Advanced Functions Handbook, p.193.
54 */
55 56 #include "libm.h"
57 58 static const double
59 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
60 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
61 Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
62 Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
63 Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
64 Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
65 Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
66 Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
67 Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
68 69 double log1p(double x)
70 {
71 union {double f; uint64_t i;} u = {x};
72 double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
73 uint32_t hx,hu;
74 int k;
75 76 hx = u.i>>32;
77 k = 1;
78 if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */
79 if (hx >= 0xbff00000) { /* x <= -1.0 */
80 if (x == -1)
81 return x/0.0; /* log1p(-1) = -inf */
82 return (x-x)/0.0; /* log1p(x<-1) = NaN */
83 }
84 if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */
85 /* underflow if subnormal */
86 if ((hx&0x7ff00000) == 0)
87 FORCE_EVAL((float)x);
88 return x;
89 }
90 if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
91 k = 0;
92 c = 0;
93 f = x;
94 }
95 } else if (hx >= 0x7ff00000)
96 return x;
97 if (k) {
98 u.f = 1 + x;
99 hu = u.i>>32;
100 hu += 0x3ff00000 - 0x3fe6a09e;
101 k = (int)(hu>>20) - 0x3ff;
102 /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
103 if (k < 54) {
104 c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
105 c /= u.f;
106 } else
107 c = 0;
108 /* reduce u into [sqrt(2)/2, sqrt(2)] */
109 hu = (hu&0x000fffff) + 0x3fe6a09e;
110 u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
111 f = u.f - 1;
112 }
113 hfsq = 0.5*f*f;
114 s = f/(2.0+f);
115 z = s*s;
116 w = z*z;
117 t1 = w*(Lg2+w*(Lg4+w*Lg6));
118 t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
119 R = t2 + t1;
120 dk = k;
121 return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
122 }
123