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/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13 for performance improvement on pipelined processors.
14 */
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 <math.h>
83#include <math-barriers.h>
84#include <math_private.h>
85#include <math-underflow.h>
86#include <libc-diag.h>
87
88static const double
89 ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
90 ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
91 two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
92 Lp[] = { 0.0, 6.666666666666735130e-01, /* 3FE55555 55555593 */
93 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
94 2.857142874366239149e-01, /* 3FD24924 94229359 */
95 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
96 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
97 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
98 1.479819860511658591e-01 }; /* 3FC2F112 DF3E5244 */
99
100static const double zero = 0.0;
101
102double
103__log1p (double x)
104{
105 double hfsq, f, c, s, z, R, u, z2, z4, z6, R1, R2, R3, R4;
106 int32_t k, hx, hu, ax;
107
108 GET_HIGH_WORD (hx, x);
109 ax = hx & 0x7fffffff;
110
111 k = 1;
112 if (hx < 0x3FDA827A) /* x < 0.41422 */
113 {
114 if (__glibc_unlikely (ax >= 0x3ff00000)) /* x <= -1.0 */
115 {
116 if (x == -1.0)
117 return -two54 / zero; /* log1p(-1)=-inf */
118 else
119 return (x - x) / (x - x); /* log1p(x<-1)=NaN */
120 }
121 if (__glibc_unlikely (ax < 0x3e200000)) /* |x| < 2**-29 */
122 {
123 math_force_eval (two54 + x); /* raise inexact */
124 if (ax < 0x3c900000) /* |x| < 2**-54 */
125 {
126 math_check_force_underflow (x);
127 return x;
128 }
129 else
130 return x - x * x * 0.5;
131 }
132 if (hx > 0 || hx <= ((int32_t) 0xbfd2bec3))
133 {
134 k = 0; f = x; hu = 1;
135 } /* -0.2929<x<0.41422 */
136 }
137 else if (__glibc_unlikely (hx >= 0x7ff00000))
138 return x + x;
139 if (k != 0)
140 {
141 if (hx < 0x43400000)
142 {
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 }
149 else
150 {
151 u = x;
152 GET_HIGH_WORD (hu, u);
153 k = (hu >> 20) - 1023;
154 c = 0;
155 }
156 hu &= 0x000fffff;
157 if (hu < 0x6a09e)
158 {
159 SET_HIGH_WORD (u, hu | 0x3ff00000); /* normalize u */
160 }
161 else
162 {
163 k += 1;
164 SET_HIGH_WORD (u, hu | 0x3fe00000); /* normalize u/2 */
165 hu = (0x00100000 - hu) >> 2;
166 }
167 f = u - 1.0;
168 }
169 hfsq = 0.5 * f * f;
170 if (hu == 0) /* |f| < 2**-20 */
171 {
172 if (f == zero)
173 {
174 if (k == 0)
175 return zero;
176 else
177 {
178 c += k * ln2_lo; return k * ln2_hi + c;
179 }
180 }
181 R = hfsq * (1.0 - 0.66666666666666666 * f);
182 if (k == 0)
183 return f - R;
184 else
185 return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
186 }
187 s = f / (2.0 + f);
188 z = s * s;
189 R1 = z * Lp[1]; z2 = z * z;
190 R2 = Lp[2] + z * Lp[3]; z4 = z2 * z2;
191 R3 = Lp[4] + z * Lp[5]; z6 = z4 * z2;
192 R4 = Lp[6] + z * Lp[7];
193 R = R1 + z2 * R2 + z4 * R3 + z6 * R4;
194 if (k == 0)
195 return f - (hfsq - s * (hfsq + R));
196 else
197 {
198 /* With GCC 7 when compiling with -Os the compiler warns that c
199 might be used uninitialized. This can't be true because k
200 must be 0 for c to be uninitialized and we handled that
201 computation earlier without using c. */
202 DIAG_PUSH_NEEDS_COMMENT;
203 DIAG_IGNORE_Os_NEEDS_COMMENT (7, "-Wmaybe-uninitialized");
204 return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
205 DIAG_POP_NEEDS_COMMENT;
206 }
207}
208