POK
log1p.c
1 /*
2  * POK header
3  *
4  * The following file is a part of the POK project. Any modification should
5  * made according to the POK licence. You CANNOT use this file or a part of
6  * this file is this part of a file for your own project
7  *
8  * For more information on the POK licence, please see our LICENCE FILE
9  *
10  * Please follow the coding guidelines described in doc/CODING_GUIDELINES
11  *
12  * Copyright (c) 2007-2009 POK team
13  *
14  * Created by julien on Fri Jan 30 14:41:34 2009
15  */
16 
17 /* @(#)s_log1p.c 5.1 93/09/24 */
18 /*
19  * ====================================================
20  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
21  *
22  * Developed at SunPro, a Sun Microsystems, Inc. business.
23  * Permission to use, copy, modify, and distribute this
24  * software is freely granted, provided that this notice
25  * is preserved.
26  * ====================================================
27  */
28 
29 /* double log1p(double x)
30  *
31  * Method :
32  * 1. Argument Reduction: find k and f such that
33  * 1+x = 2^k * (1+f),
34  * where sqrt(2)/2 < 1+f < sqrt(2) .
35  *
36  * Note. If k=0, then f=x is exact. However, if k!=0, then f
37  * may not be representable exactly. In that case, a correction
38  * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
39  * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
40  * and add back the correction term c/u.
41  * (Note: when x > 2**53, one can simply return log(x))
42  *
43  * 2. Approximation of log1p(f).
44  * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
45  * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
46  * = 2s + s*R
47  * We use a special Reme algorithm on [0,0.1716] to generate
48  * a polynomial of degree 14 to approximate R The maximum error
49  * of this polynomial approximation is bounded by 2**-58.45. In
50  * other words,
51  * 2 4 6 8 10 12 14
52  * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
53  * (the values of Lp1 to Lp7 are listed in the program)
54  * and
55  * | 2 14 | -58.45
56  * | Lp1*s +...+Lp7*s - R(z) | <= 2
57  * | |
58  * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
59  * In order to guarantee error in log below 1ulp, we compute log
60  * by
61  * log1p(f) = f - (hfsq - s*(hfsq+R)).
62  *
63  * 3. Finally, log1p(x) = k*ln2 + log1p(f).
64  * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
65  * Here ln2 is split into two floating point number:
66  * ln2_hi + ln2_lo,
67  * where n*ln2_hi is always exact for |n| < 2000.
68  *
69  * Special cases:
70  * log1p(x) is NaN with signal if x < -1 (including -INF) ;
71  * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
72  * log1p(NaN) is that NaN with no signal.
73  *
74  * Accuracy:
75  * according to an error analysis, the error is always less than
76  * 1 ulp (unit in the last place).
77  *
78  * Constants:
79  * The hexadecimal values are the intended ones for the following
80  * constants. The decimal values may be used, provided that the
81  * compiler will convert from decimal to binary accurately enough
82  * to produce the hexadecimal values shown.
83  *
84  * Note: Assuming log() return accurate answer, the following
85  * algorithm can be used to compute log1p(x) to within a few ULP:
86  *
87  * u = 1+x;
88  * if(u==1.0) return x ; else
89  * return log(u)*(x/(u-1.0));
90  *
91  * See HP-15C Advanced Functions Handbook, p.193.
92  */
93 
94 #ifdef POK_NEEDS_LIBMATH
95 
96 #include <types.h>
97 #include "math_private.h"
98 
99 static const double
100 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
101 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
102 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
103 Lp1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
104 Lp2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
105 Lp3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
106 Lp4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
107 Lp5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
108 Lp6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
109 Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
110 
111 static const double zero = 0.0;
112 
113 double
114 log1p(double x)
115 {
116  double hfsq,f,c,s,z,R,u;
117  int32_t k,hx,hu,ax;
118 
119  f = c = 0;
120  hu = 0;
121  GET_HIGH_WORD(hx,x);
122  ax = hx&0x7fffffff;
123 
124  k = 1;
125  if (hx < 0x3FDA827A) { /* x < 0.41422 */
126  if(ax>=0x3ff00000) { /* x <= -1.0 */
127  if(x==-1.0) return -two54/zero; /* log1p(-1)=+inf */
128  else return (x-x)/(x-x); /* log1p(x<-1)=NaN */
129  }
130  if(ax<0x3e200000) { /* |x| < 2**-29 */
131  if(two54+x>zero /* raise inexact */
132  &&ax<0x3c900000) /* |x| < 2**-54 */
133  return x;
134  else
135  return x - x*x*0.5;
136  }
137  if(hx>0||hx<=((int32_t)0xbfd2bec3)) {
138  k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
139  }
140  if (hx >= 0x7ff00000) return x+x;
141  if(k!=0) {
142  if(hx<0x43400000) {
143  u = 1.0+x;
144  GET_HIGH_WORD(hu,u);
145  k = (hu>>20)-1023;
146  c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
147  c /= u;
148  } else {
149  u = x;
150  GET_HIGH_WORD(hu,u);
151  k = (hu>>20)-1023;
152  c = 0;
153  }
154  hu &= 0x000fffff;
155  if(hu<0x6a09e) {
156  SET_HIGH_WORD(u,hu|0x3ff00000); /* normalize u */
157  } else {
158  k += 1;
159  SET_HIGH_WORD(u,hu|0x3fe00000); /* normalize u/2 */
160  hu = (0x00100000-hu)>>2;
161  }
162  f = u-1.0;
163  }
164  hfsq=0.5*f*f;
165  if(hu==0) { /* |f| < 2**-20 */
166  if(f==zero) { if(k==0) return zero;
167  else {c += k*ln2_lo; return k*ln2_hi+c;}
168  }
169  R = hfsq*(1.0-0.66666666666666666*f);
170  if(k==0) return f-R; else
171  return k*ln2_hi-((R-(k*ln2_lo+c))-f);
172  }
173  s = f/(2.0+f);
174  z = s*s;
175  R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
176  if(k==0) return f-(hfsq-s*(hfsq+R)); else
177  return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
178 }
179 #endif
180