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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 | |
cccda09f | 8 | * software is freely granted, provided that this notice |
f7eac6eb RM |
9 | * is preserved. |
10 | * ==================================================== | |
11 | */ | |
923609d1 UD |
12 | /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, |
13 | for performance improvement on pipelined processors. | |
c5d5d574 | 14 | */ |
f7eac6eb | 15 | |
f7eac6eb RM |
16 | /* double log1p(double x) |
17 | * | |
cccda09f UD |
18 | * Method : |
19 | * 1. Argument Reduction: find k and f such that | |
20 | * 1+x = 2^k * (1+f), | |
f7eac6eb RM |
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 + ....., | |
d7826aa1 | 33 | * = 2s + s*R |
cccda09f | 34 | * We use a special Reme algorithm on [0,0.1716] to generate |
d7826aa1 | 35 | * a polynomial of degree 14 to approximate R The maximum error |
f7eac6eb RM |
36 | * of this polynomial approximation is bounded by 2**-58.45. In |
37 | * other words, | |
d7826aa1 | 38 | * 2 4 6 8 10 12 14 |
f7eac6eb | 39 | * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s |
d7826aa1 | 40 | * (the values of Lp1 to Lp7 are listed in the program) |
f7eac6eb RM |
41 | * and |
42 | * | 2 14 | -58.45 | |
cccda09f | 43 | * | Lp1*s +...+Lp7*s - R(z) | <= 2 |
f7eac6eb RM |
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)). | |
cccda09f UD |
49 | * |
50 | * 3. Finally, log1p(x) = k*ln2 + log1p(f). | |
d7826aa1 | 51 | * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo))) |
cccda09f | 52 | * Here ln2 is split into two floating point number: |
f7eac6eb RM |
53 | * ln2_hi + ln2_lo, |
54 | * where n*ln2_hi is always exact for |n| < 2000. | |
55 | * | |
56 | * Special cases: | |
cccda09f | 57 | * log1p(x) is NaN with signal if x < -1 (including -INF) ; |
f7eac6eb RM |
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: | |
cccda09f UD |
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 | |
f7eac6eb RM |
69 | * to produce the hexadecimal values shown. |
70 | * | |
71 | * Note: Assuming log() return accurate answer, the following | |
d7826aa1 | 72 | * algorithm can be used to compute log1p(x) to within a few ULP: |
cccda09f | 73 | * |
f7eac6eb RM |
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 | ||
1ed0291c RH |
81 | #include <math.h> |
82 | #include <math_private.h> | |
f7eac6eb | 83 | |
f7eac6eb | 84 | static const double |
c5d5d574 OB |
85 | ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */ |
86 | ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */ | |
87 | two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */ | |
88 | Lp[] = { 0.0, 6.666666666666735130e-01, /* 3FE55555 55555593 */ | |
89 | 3.999999999940941908e-01, /* 3FD99999 9997FA04 */ | |
90 | 2.857142874366239149e-01, /* 3FD24924 94229359 */ | |
91 | 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */ | |
92 | 1.818357216161805012e-01, /* 3FC74664 96CB03DE */ | |
93 | 1.531383769920937332e-01, /* 3FC39A09 D078C69F */ | |
94 | 1.479819860511658591e-01 }; /* 3FC2F112 DF3E5244 */ | |
f7eac6eb | 95 | |
f7eac6eb | 96 | static const double zero = 0.0; |
f7eac6eb | 97 | |
d7826aa1 | 98 | double |
c5d5d574 | 99 | __log1p (double x) |
f7eac6eb | 100 | { |
c5d5d574 OB |
101 | double hfsq, f, c, s, z, R, u, z2, z4, z6, R1, R2, R3, R4; |
102 | int32_t k, hx, hu, ax; | |
f7eac6eb | 103 | |
c5d5d574 OB |
104 | GET_HIGH_WORD (hx, x); |
105 | ax = hx & 0x7fffffff; | |
f7eac6eb | 106 | |
c5d5d574 OB |
107 | k = 1; |
108 | if (hx < 0x3FDA827A) /* x < 0.41422 */ | |
109 | { | |
a1ffb40e | 110 | if (__glibc_unlikely (ax >= 0x3ff00000)) /* x <= -1.0 */ |
c5d5d574 OB |
111 | { |
112 | if (x == -1.0) | |
113 | return -two54 / (x - x); /* log1p(-1)=+inf */ | |
114 | else | |
115 | return (x - x) / (x - x); /* log1p(x<-1)=NaN */ | |
cccda09f | 116 | } |
a1ffb40e | 117 | if (__glibc_unlikely (ax < 0x3e200000)) /* |x| < 2**-29 */ |
c5d5d574 OB |
118 | { |
119 | math_force_eval (two54 + x); /* raise inexact */ | |
120 | if (ax < 0x3c900000) /* |x| < 2**-54 */ | |
121 | return x; | |
122 | else | |
123 | return x - x * x * 0.5; | |
124 | } | |
125 | if (hx > 0 || hx <= ((int32_t) 0xbfd2bec3)) | |
126 | { | |
127 | k = 0; f = x; hu = 1; | |
128 | } /* -0.2929<x<0.41422 */ | |
129 | } | |
a1ffb40e | 130 | else if (__glibc_unlikely (hx >= 0x7ff00000)) |
c5d5d574 OB |
131 | return x + x; |
132 | if (k != 0) | |
133 | { | |
134 | if (hx < 0x43400000) | |
135 | { | |
136 | u = 1.0 + x; | |
137 | GET_HIGH_WORD (hu, u); | |
138 | k = (hu >> 20) - 1023; | |
139 | c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */ | |
140 | c /= u; | |
141 | } | |
142 | else | |
143 | { | |
144 | u = x; | |
145 | GET_HIGH_WORD (hu, u); | |
146 | k = (hu >> 20) - 1023; | |
147 | c = 0; | |
148 | } | |
149 | hu &= 0x000fffff; | |
150 | if (hu < 0x6a09e) | |
151 | { | |
152 | SET_HIGH_WORD (u, hu | 0x3ff00000); /* normalize u */ | |
153 | } | |
154 | else | |
155 | { | |
156 | k += 1; | |
157 | SET_HIGH_WORD (u, hu | 0x3fe00000); /* normalize u/2 */ | |
158 | hu = (0x00100000 - hu) >> 2; | |
f7eac6eb | 159 | } |
c5d5d574 OB |
160 | f = u - 1.0; |
161 | } | |
162 | hfsq = 0.5 * f * f; | |
163 | if (hu == 0) /* |f| < 2**-20 */ | |
164 | { | |
165 | if (f == zero) | |
166 | { | |
167 | if (k == 0) | |
168 | return zero; | |
169 | else | |
170 | { | |
171 | c += k * ln2_lo; return k * ln2_hi + c; | |
3bb266e0 | 172 | } |
f7eac6eb | 173 | } |
c5d5d574 OB |
174 | R = hfsq * (1.0 - 0.66666666666666666 * f); |
175 | if (k == 0) | |
176 | return f - R; | |
177 | else | |
178 | return k * ln2_hi - ((R - (k * ln2_lo + c)) - f); | |
179 | } | |
180 | s = f / (2.0 + f); | |
181 | z = s * s; | |
182 | R1 = z * Lp[1]; z2 = z * z; | |
183 | R2 = Lp[2] + z * Lp[3]; z4 = z2 * z2; | |
184 | R3 = Lp[4] + z * Lp[5]; z6 = z4 * z2; | |
185 | R4 = Lp[6] + z * Lp[7]; | |
186 | R = R1 + z2 * R2 + z4 * R3 + z6 * R4; | |
187 | if (k == 0) | |
188 | return f - (hfsq - s * (hfsq + R)); | |
189 | else | |
190 | return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f); | |
f7eac6eb RM |
191 | } |
192 | weak_alias (__log1p, log1p) | |
cccda09f UD |
193 | #ifdef NO_LONG_DOUBLE |
194 | strong_alias (__log1p, __log1pl) | |
195 | weak_alias (__log1p, log1pl) | |
196 | #endif |