Mercurial > hg > CbC > CbC_gcc
comparison libquadmath/math/log2q.c @ 68:561a7518be6b
update gcc-4.6
author | Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp> |
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date | Sun, 21 Aug 2011 07:07:55 +0900 |
parents | |
children | 04ced10e8804 |
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67:f6334be47118 | 68:561a7518be6b |
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1 /* log2l.c | |
2 * Base 2 logarithm, 128-bit long double precision | |
3 * | |
4 * | |
5 * | |
6 * SYNOPSIS: | |
7 * | |
8 * long double x, y, log2l(); | |
9 * | |
10 * y = log2l( x ); | |
11 * | |
12 * | |
13 * | |
14 * DESCRIPTION: | |
15 * | |
16 * Returns the base 2 logarithm of x. | |
17 * | |
18 * The argument is separated into its exponent and fractional | |
19 * parts. If the exponent is between -1 and +1, the (natural) | |
20 * logarithm of the fraction is approximated by | |
21 * | |
22 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x). | |
23 * | |
24 * Otherwise, setting z = 2(x-1)/x+1), | |
25 * | |
26 * log(x) = z + z^3 P(z)/Q(z). | |
27 * | |
28 * | |
29 * | |
30 * ACCURACY: | |
31 * | |
32 * Relative error: | |
33 * arithmetic domain # trials peak rms | |
34 * IEEE 0.5, 2.0 100,000 2.6e-34 4.9e-35 | |
35 * IEEE exp(+-10000) 100,000 9.6e-35 4.0e-35 | |
36 * | |
37 * In the tests over the interval exp(+-10000), the logarithms | |
38 * of the random arguments were uniformly distributed over | |
39 * [-10000, +10000]. | |
40 * | |
41 */ | |
42 | |
43 /* | |
44 Cephes Math Library Release 2.2: January, 1991 | |
45 Copyright 1984, 1991 by Stephen L. Moshier | |
46 Adapted for glibc November, 2001 | |
47 | |
48 This library is free software; you can redistribute it and/or | |
49 modify it under the terms of the GNU Lesser General Public | |
50 License as published by the Free Software Foundation; either | |
51 version 2.1 of the License, or (at your option) any later version. | |
52 | |
53 This library is distributed in the hope that it will be useful, | |
54 but WITHOUT ANY WARRANTY; without even the implied warranty of | |
55 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
56 Lesser General Public License for more details. | |
57 | |
58 You should have received a copy of the GNU Lesser General Public | |
59 License along with this library; if not, write to the Free Software | |
60 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA | |
61 */ | |
62 | |
63 #include "quadmath-imp.h" | |
64 | |
65 /* Coefficients for ln(1+x) = x - x**2/2 + x**3 P(x)/Q(x) | |
66 * 1/sqrt(2) <= x < sqrt(2) | |
67 * Theoretical peak relative error = 5.3e-37, | |
68 * relative peak error spread = 2.3e-14 | |
69 */ | |
70 static const __float128 P[13] = | |
71 { | |
72 1.313572404063446165910279910527789794488E4Q, | |
73 7.771154681358524243729929227226708890930E4Q, | |
74 2.014652742082537582487669938141683759923E5Q, | |
75 3.007007295140399532324943111654767187848E5Q, | |
76 2.854829159639697837788887080758954924001E5Q, | |
77 1.797628303815655343403735250238293741397E5Q, | |
78 7.594356839258970405033155585486712125861E4Q, | |
79 2.128857716871515081352991964243375186031E4Q, | |
80 3.824952356185897735160588078446136783779E3Q, | |
81 4.114517881637811823002128927449878962058E2Q, | |
82 2.321125933898420063925789532045674660756E1Q, | |
83 4.998469661968096229986658302195402690910E-1Q, | |
84 1.538612243596254322971797716843006400388E-6Q | |
85 }; | |
86 static const __float128 Q[12] = | |
87 { | |
88 3.940717212190338497730839731583397586124E4Q, | |
89 2.626900195321832660448791748036714883242E5Q, | |
90 7.777690340007566932935753241556479363645E5Q, | |
91 1.347518538384329112529391120390701166528E6Q, | |
92 1.514882452993549494932585972882995548426E6Q, | |
93 1.158019977462989115839826904108208787040E6Q, | |
94 6.132189329546557743179177159925690841200E5Q, | |
95 2.248234257620569139969141618556349415120E5Q, | |
96 5.605842085972455027590989944010492125825E4Q, | |
97 9.147150349299596453976674231612674085381E3Q, | |
98 9.104928120962988414618126155557301584078E2Q, | |
99 4.839208193348159620282142911143429644326E1Q | |
100 /* 1.000000000000000000000000000000000000000E0Q, */ | |
101 }; | |
102 | |
103 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2), | |
104 * where z = 2(x-1)/(x+1) | |
105 * 1/sqrt(2) <= x < sqrt(2) | |
106 * Theoretical peak relative error = 1.1e-35, | |
107 * relative peak error spread 1.1e-9 | |
108 */ | |
109 static const __float128 R[6] = | |
110 { | |
111 1.418134209872192732479751274970992665513E5Q, | |
112 -8.977257995689735303686582344659576526998E4Q, | |
113 2.048819892795278657810231591630928516206E4Q, | |
114 -2.024301798136027039250415126250455056397E3Q, | |
115 8.057002716646055371965756206836056074715E1Q, | |
116 -8.828896441624934385266096344596648080902E-1Q | |
117 }; | |
118 static const __float128 S[6] = | |
119 { | |
120 1.701761051846631278975701529965589676574E6Q, | |
121 -1.332535117259762928288745111081235577029E6Q, | |
122 4.001557694070773974936904547424676279307E5Q, | |
123 -5.748542087379434595104154610899551484314E4Q, | |
124 3.998526750980007367835804959888064681098E3Q, | |
125 -1.186359407982897997337150403816839480438E2Q | |
126 /* 1.000000000000000000000000000000000000000E0Q, */ | |
127 }; | |
128 | |
129 static const __float128 | |
130 /* log2(e) - 1 */ | |
131 LOG2EA = 4.4269504088896340735992468100189213742664595E-1Q, | |
132 /* sqrt(2)/2 */ | |
133 SQRTH = 7.071067811865475244008443621048490392848359E-1Q; | |
134 | |
135 | |
136 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ | |
137 | |
138 static __float128 | |
139 neval (__float128 x, const __float128 *p, int n) | |
140 { | |
141 __float128 y; | |
142 | |
143 p += n; | |
144 y = *p--; | |
145 do | |
146 { | |
147 y = y * x + *p--; | |
148 } | |
149 while (--n > 0); | |
150 return y; | |
151 } | |
152 | |
153 | |
154 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ | |
155 | |
156 static __float128 | |
157 deval (__float128 x, const __float128 *p, int n) | |
158 { | |
159 __float128 y; | |
160 | |
161 p += n; | |
162 y = x + *p--; | |
163 do | |
164 { | |
165 y = y * x + *p--; | |
166 } | |
167 while (--n > 0); | |
168 return y; | |
169 } | |
170 | |
171 | |
172 | |
173 __float128 | |
174 log2q (__float128 x) | |
175 { | |
176 __float128 z; | |
177 __float128 y; | |
178 int e; | |
179 int64_t hx, lx; | |
180 | |
181 /* Test for domain */ | |
182 GET_FLT128_WORDS64 (hx, lx, x); | |
183 if (((hx & 0x7fffffffffffffffLL) | lx) == 0) | |
184 return (-1.0Q / (x - x)); | |
185 if (hx < 0) | |
186 return (x - x) / (x - x); | |
187 if (hx >= 0x7fff000000000000LL) | |
188 return (x + x); | |
189 | |
190 /* separate mantissa from exponent */ | |
191 | |
192 /* Note, frexp is used so that denormal numbers | |
193 * will be handled properly. | |
194 */ | |
195 x = frexpq (x, &e); | |
196 | |
197 | |
198 /* logarithm using log(x) = z + z**3 P(z)/Q(z), | |
199 * where z = 2(x-1)/x+1) | |
200 */ | |
201 if ((e > 2) || (e < -2)) | |
202 { | |
203 if (x < SQRTH) | |
204 { /* 2( 2x-1 )/( 2x+1 ) */ | |
205 e -= 1; | |
206 z = x - 0.5Q; | |
207 y = 0.5Q * z + 0.5Q; | |
208 } | |
209 else | |
210 { /* 2 (x-1)/(x+1) */ | |
211 z = x - 0.5Q; | |
212 z -= 0.5Q; | |
213 y = 0.5Q * x + 0.5Q; | |
214 } | |
215 x = z / y; | |
216 z = x * x; | |
217 y = x * (z * neval (z, R, 5) / deval (z, S, 5)); | |
218 goto done; | |
219 } | |
220 | |
221 | |
222 /* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ | |
223 | |
224 if (x < SQRTH) | |
225 { | |
226 e -= 1; | |
227 x = 2.0 * x - 1.0Q; /* 2x - 1 */ | |
228 } | |
229 else | |
230 { | |
231 x = x - 1.0Q; | |
232 } | |
233 z = x * x; | |
234 y = x * (z * neval (x, P, 12) / deval (x, Q, 11)); | |
235 y = y - 0.5 * z; | |
236 | |
237 done: | |
238 | |
239 /* Multiply log of fraction by log2(e) | |
240 * and base 2 exponent by 1 | |
241 */ | |
242 z = y * LOG2EA; | |
243 z += x * LOG2EA; | |
244 z += y; | |
245 z += x; | |
246 z += e; | |
247 return (z); | |
248 } |