1*c9945492SAndroid Build Coastguard Worker /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
2*c9945492SAndroid Build Coastguard Worker /*
3*c9945492SAndroid Build Coastguard Worker * ====================================================
4*c9945492SAndroid Build Coastguard Worker * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*c9945492SAndroid Build Coastguard Worker *
6*c9945492SAndroid Build Coastguard Worker * Developed at SunPro, a Sun Microsystems, Inc. business.
7*c9945492SAndroid Build Coastguard Worker * Permission to use, copy, modify, and distribute this
8*c9945492SAndroid Build Coastguard Worker * software is freely granted, provided that this notice
9*c9945492SAndroid Build Coastguard Worker * is preserved.
10*c9945492SAndroid Build Coastguard Worker * ====================================================
11*c9945492SAndroid Build Coastguard Worker */
12*c9945492SAndroid Build Coastguard Worker /* double log1p(double x)
13*c9945492SAndroid Build Coastguard Worker * Return the natural logarithm of 1+x.
14*c9945492SAndroid Build Coastguard Worker *
15*c9945492SAndroid Build Coastguard Worker * Method :
16*c9945492SAndroid Build Coastguard Worker * 1. Argument Reduction: find k and f such that
17*c9945492SAndroid Build Coastguard Worker * 1+x = 2^k * (1+f),
18*c9945492SAndroid Build Coastguard Worker * where sqrt(2)/2 < 1+f < sqrt(2) .
19*c9945492SAndroid Build Coastguard Worker *
20*c9945492SAndroid Build Coastguard Worker * Note. If k=0, then f=x is exact. However, if k!=0, then f
21*c9945492SAndroid Build Coastguard Worker * may not be representable exactly. In that case, a correction
22*c9945492SAndroid Build Coastguard Worker * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
23*c9945492SAndroid Build Coastguard Worker * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
24*c9945492SAndroid Build Coastguard Worker * and add back the correction term c/u.
25*c9945492SAndroid Build Coastguard Worker * (Note: when x > 2**53, one can simply return log(x))
26*c9945492SAndroid Build Coastguard Worker *
27*c9945492SAndroid Build Coastguard Worker * 2. Approximation of log(1+f): See log.c
28*c9945492SAndroid Build Coastguard Worker *
29*c9945492SAndroid Build Coastguard Worker * 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
30*c9945492SAndroid Build Coastguard Worker *
31*c9945492SAndroid Build Coastguard Worker * Special cases:
32*c9945492SAndroid Build Coastguard Worker * log1p(x) is NaN with signal if x < -1 (including -INF) ;
33*c9945492SAndroid Build Coastguard Worker * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
34*c9945492SAndroid Build Coastguard Worker * log1p(NaN) is that NaN with no signal.
35*c9945492SAndroid Build Coastguard Worker *
36*c9945492SAndroid Build Coastguard Worker * Accuracy:
37*c9945492SAndroid Build Coastguard Worker * according to an error analysis, the error is always less than
38*c9945492SAndroid Build Coastguard Worker * 1 ulp (unit in the last place).
39*c9945492SAndroid Build Coastguard Worker *
40*c9945492SAndroid Build Coastguard Worker * Constants:
41*c9945492SAndroid Build Coastguard Worker * The hexadecimal values are the intended ones for the following
42*c9945492SAndroid Build Coastguard Worker * constants. The decimal values may be used, provided that the
43*c9945492SAndroid Build Coastguard Worker * compiler will convert from decimal to binary accurately enough
44*c9945492SAndroid Build Coastguard Worker * to produce the hexadecimal values shown.
45*c9945492SAndroid Build Coastguard Worker *
46*c9945492SAndroid Build Coastguard Worker * Note: Assuming log() return accurate answer, the following
47*c9945492SAndroid Build Coastguard Worker * algorithm can be used to compute log1p(x) to within a few ULP:
48*c9945492SAndroid Build Coastguard Worker *
49*c9945492SAndroid Build Coastguard Worker * u = 1+x;
50*c9945492SAndroid Build Coastguard Worker * if(u==1.0) return x ; else
51*c9945492SAndroid Build Coastguard Worker * return log(u)*(x/(u-1.0));
52*c9945492SAndroid Build Coastguard Worker *
53*c9945492SAndroid Build Coastguard Worker * See HP-15C Advanced Functions Handbook, p.193.
54*c9945492SAndroid Build Coastguard Worker */
55*c9945492SAndroid Build Coastguard Worker
56*c9945492SAndroid Build Coastguard Worker #include "libm.h"
57*c9945492SAndroid Build Coastguard Worker
58*c9945492SAndroid Build Coastguard Worker static const double
59*c9945492SAndroid Build Coastguard Worker ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
60*c9945492SAndroid Build Coastguard Worker ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
61*c9945492SAndroid Build Coastguard Worker Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
62*c9945492SAndroid Build Coastguard Worker Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
63*c9945492SAndroid Build Coastguard Worker Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
64*c9945492SAndroid Build Coastguard Worker Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
65*c9945492SAndroid Build Coastguard Worker Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
66*c9945492SAndroid Build Coastguard Worker Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
67*c9945492SAndroid Build Coastguard Worker Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
68*c9945492SAndroid Build Coastguard Worker
log1p(double x)69*c9945492SAndroid Build Coastguard Worker double log1p(double x)
70*c9945492SAndroid Build Coastguard Worker {
71*c9945492SAndroid Build Coastguard Worker union {double f; uint64_t i;} u = {x};
72*c9945492SAndroid Build Coastguard Worker double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
73*c9945492SAndroid Build Coastguard Worker uint32_t hx,hu;
74*c9945492SAndroid Build Coastguard Worker int k;
75*c9945492SAndroid Build Coastguard Worker
76*c9945492SAndroid Build Coastguard Worker hx = u.i>>32;
77*c9945492SAndroid Build Coastguard Worker k = 1;
78*c9945492SAndroid Build Coastguard Worker if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */
79*c9945492SAndroid Build Coastguard Worker if (hx >= 0xbff00000) { /* x <= -1.0 */
80*c9945492SAndroid Build Coastguard Worker if (x == -1)
81*c9945492SAndroid Build Coastguard Worker return x/0.0; /* log1p(-1) = -inf */
82*c9945492SAndroid Build Coastguard Worker return (x-x)/0.0; /* log1p(x<-1) = NaN */
83*c9945492SAndroid Build Coastguard Worker }
84*c9945492SAndroid Build Coastguard Worker if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */
85*c9945492SAndroid Build Coastguard Worker /* underflow if subnormal */
86*c9945492SAndroid Build Coastguard Worker if ((hx&0x7ff00000) == 0)
87*c9945492SAndroid Build Coastguard Worker FORCE_EVAL((float)x);
88*c9945492SAndroid Build Coastguard Worker return x;
89*c9945492SAndroid Build Coastguard Worker }
90*c9945492SAndroid Build Coastguard Worker if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
91*c9945492SAndroid Build Coastguard Worker k = 0;
92*c9945492SAndroid Build Coastguard Worker c = 0;
93*c9945492SAndroid Build Coastguard Worker f = x;
94*c9945492SAndroid Build Coastguard Worker }
95*c9945492SAndroid Build Coastguard Worker } else if (hx >= 0x7ff00000)
96*c9945492SAndroid Build Coastguard Worker return x;
97*c9945492SAndroid Build Coastguard Worker if (k) {
98*c9945492SAndroid Build Coastguard Worker u.f = 1 + x;
99*c9945492SAndroid Build Coastguard Worker hu = u.i>>32;
100*c9945492SAndroid Build Coastguard Worker hu += 0x3ff00000 - 0x3fe6a09e;
101*c9945492SAndroid Build Coastguard Worker k = (int)(hu>>20) - 0x3ff;
102*c9945492SAndroid Build Coastguard Worker /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
103*c9945492SAndroid Build Coastguard Worker if (k < 54) {
104*c9945492SAndroid Build Coastguard Worker c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
105*c9945492SAndroid Build Coastguard Worker c /= u.f;
106*c9945492SAndroid Build Coastguard Worker } else
107*c9945492SAndroid Build Coastguard Worker c = 0;
108*c9945492SAndroid Build Coastguard Worker /* reduce u into [sqrt(2)/2, sqrt(2)] */
109*c9945492SAndroid Build Coastguard Worker hu = (hu&0x000fffff) + 0x3fe6a09e;
110*c9945492SAndroid Build Coastguard Worker u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
111*c9945492SAndroid Build Coastguard Worker f = u.f - 1;
112*c9945492SAndroid Build Coastguard Worker }
113*c9945492SAndroid Build Coastguard Worker hfsq = 0.5*f*f;
114*c9945492SAndroid Build Coastguard Worker s = f/(2.0+f);
115*c9945492SAndroid Build Coastguard Worker z = s*s;
116*c9945492SAndroid Build Coastguard Worker w = z*z;
117*c9945492SAndroid Build Coastguard Worker t1 = w*(Lg2+w*(Lg4+w*Lg6));
118*c9945492SAndroid Build Coastguard Worker t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
119*c9945492SAndroid Build Coastguard Worker R = t2 + t1;
120*c9945492SAndroid Build Coastguard Worker dk = k;
121*c9945492SAndroid Build Coastguard Worker return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
122*c9945492SAndroid Build Coastguard Worker }
123