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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 "cdefs-compat.h"
14 : : //__FBSDID("$FreeBSD: src/lib/msun/src/s_log1p.c,v 1.10 2008/03/29 16:37:59 das Exp $");
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 : : #include <openlibm_math.h>
83 : :
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 : :
100 : : OLM_DLLEXPORT double
101 : 13 : log1p(double x)
102 : : {
103 : : double hfsq,f,c,s,z,R,u;
104 : : int32_t k,hx,hu,ax;
105 : :
106 : 13 : GET_HIGH_WORD(hx,x);
107 : 13 : ax = hx&0x7fffffff;
108 : :
109 : 13 : k = 1;
110 [ + + ]: 13 : if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
111 [ + + ]: 7 : if(ax>=0x3ff00000) { /* x <= -1.0 */
112 [ + + ]: 3 : if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
113 : 2 : else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
114 : : }
115 [ + + ]: 4 : if(ax<0x3e200000) { /* |x| < 2**-29 */
116 [ + - ]: 3 : if(two54+x>zero /* raise inexact */
117 [ + - ]: 3 : &&ax<0x3c900000) /* |x| < 2**-54 */
118 : 3 : return x;
119 : : else
120 : 0 : return x - x*x*0.5;
121 : : }
122 [ + - - + ]: 1 : if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
123 : 0 : k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
124 : : }
125 [ + + ]: 7 : if (hx >= 0x7ff00000) return x+x;
126 [ + - ]: 6 : if(k!=0) {
127 [ + - ]: 6 : if(hx<0x43400000) {
128 : 6 : STRICT_ASSIGN(double,u,1.0+x);
129 : 6 : GET_HIGH_WORD(hu,u);
130 : 6 : k = (hu>>20)-1023;
131 [ + + ]: 6 : c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
132 : 6 : 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 : 6 : 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 [ + + ]: 6 : if(hu<0x6a09e) { /* u ~< sqrt(2) */
148 : 2 : SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
149 : : } else {
150 : 4 : k += 1;
151 : 4 : SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
152 : 4 : hu = (0x00100000-hu)>>2;
153 : : }
154 : 6 : f = u-1.0;
155 : : }
156 : 6 : hfsq=0.5*f*f;
157 [ - + ]: 6 : 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 : 6 : s = f/(2.0+f);
171 : 6 : z = s*s;
172 : 6 : R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
173 [ - + ]: 6 : if(k==0) return f-(hfsq-s*(hfsq+R)); else
174 : 6 : return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
175 : : }
176 : :
177 : : #if (LDBL_MANT_DIG == 53)
178 : : openlibm_weak_reference(log1p, log1pl);
179 : : #endif
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