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 <float.h>
78
79 #include "math.h"
80 #include "math_private.h"
81
82 static const double
83 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
84 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
85 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
86 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
87 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
88 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
89 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
90 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
91 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
92 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
93
94 static const double zero = 0.0;
95 static volatile double vzero = 0.0;
96
97 double
log1p(double x)98 log1p(double x)
99 {
100 double hfsq,f,c,s,z,R,u;
101 int32_t k,hx,hu,ax;
102
103 GET_HIGH_WORD(hx,x);
104 ax = hx&0x7fffffff;
105
106 k = 1;
107 if (hx < 0x3FDA827A) { /* 1+x < sqrt(2)+ */
108 if(ax>=0x3ff00000) { /* x <= -1.0 */
109 if(x==-1.0) return -two54/vzero; /* log1p(-1)=+inf */
110 else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
111 }
112 if(ax<0x3e200000) { /* |x| < 2**-29 */
113 if(two54+x>zero /* raise inexact */
114 &&ax<0x3c900000) /* |x| < 2**-54 */
115 return x;
116 else
117 return x - x*x*0.5;
118 }
119 if(hx>0||hx<=((int32_t)0xbfd2bec4)) {
120 k=0;f=x;hu=1;} /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
121 }
122 if (hx >= 0x7ff00000) return x+x;
123 if(k!=0) {
124 if(hx<0x43400000) {
125 STRICT_ASSIGN(double,u,1.0+x);
126 GET_HIGH_WORD(hu,u);
127 k = (hu>>20)-1023;
128 c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
129 c /= u;
130 } else {
131 u = x;
132 GET_HIGH_WORD(hu,u);
133 k = (hu>>20)-1023;
134 c = 0;
135 }
136 hu &= 0x000fffff;
137 /*
138 * The approximation to sqrt(2) used in thresholds is not
139 * critical. However, the ones used above must give less
140 * strict bounds than the one here so that the k==0 case is
141 * never reached from here, since here we have committed to
142 * using the correction term but don't use it if k==0.
143 */
144 if(hu<0x6a09e) { /* u ~< sqrt(2) */
145 SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
146 } else {
147 k += 1;
148 SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
149 hu = (0x00100000-hu)>>2;
150 }
151 f = u-1.0;
152 }
153 hfsq=0.5*f*f;
154 if(hu==0) { /* |f| < 2**-20 */
155 if(f==zero) {
156 if(k==0) {
157 return zero;
158 } else {
159 c += k*ln2_lo;
160 return k*ln2_hi+c;
161 }
162 }
163 R = hfsq*(1.0-0.66666666666666666*f);
164 if(k==0) return f-R; else
165 return k*ln2_hi-((R-(k*ln2_lo+c))-f);
166 }
167 s = f/(2.0+f);
168 z = s*s;
169 R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
170 if(k==0) return f-(hfsq-s*(hfsq+R)); else
171 return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
172 }
173
174 #if (LDBL_MANT_DIG == 53)
175 __weak_reference(log1p, log1pl);
176 #endif
177