145
|
1 /* log1pq.c
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
2 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
3 * Relative error logarithm
|
145
|
4 * Natural logarithm of 1+x, 128-bit long double precision
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
5 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
6 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
7 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
8 * SYNOPSIS:
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
9 *
|
145
|
10 * long double x, y, log1pq();
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
11 *
|
111
|
12 * y = log1pq( x );
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
13 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
14 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
15 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
16 * DESCRIPTION:
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
17 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
18 * Returns the base e (2.718...) logarithm of 1+x.
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
19 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
20 * The argument 1+x is separated into its exponent and fractional
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
21 * parts. If the exponent is between -1 and +1, the logarithm
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
22 * of the fraction is approximated by
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
23 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
24 * log(1+x) = x - 0.5 x^2 + x^3 P(x)/Q(x).
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
25 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
26 * Otherwise, setting z = 2(w-1)/(w+1),
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
27 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
28 * log(w) = z + z^3 P(z)/Q(z).
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
29 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
30 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
31 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
32 * ACCURACY:
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
33 *
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
34 * Relative error:
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
35 * arithmetic domain # trials peak rms
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
36 * IEEE -1, 8 100000 1.9e-34 4.3e-35
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
37 */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
38
|
111
|
39 /* Copyright 2001 by Stephen L. Moshier
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
40
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
41 This library is free software; you can redistribute it and/or
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
42 modify it under the terms of the GNU Lesser General Public
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
43 License as published by the Free Software Foundation; either
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
44 version 2.1 of the License, or (at your option) any later version.
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
45
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
46 This library is distributed in the hope that it will be useful,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
47 but WITHOUT ANY WARRANTY; without even the implied warranty of
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
48 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
49 Lesser General Public License for more details.
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
50
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
51 You should have received a copy of the GNU Lesser General Public
|
145
|
52 License along with this library; if not, see
|
|
53 <http://www.gnu.org/licenses/>. */
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
54
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
55 #include "quadmath-imp.h"
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
56
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
57 /* Coefficients for log(1+x) = x - x^2 / 2 + x^3 P(x)/Q(x)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
58 * 1/sqrt(2) <= 1+x < sqrt(2)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
59 * Theoretical peak relative error = 5.3e-37,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
60 * relative peak error spread = 2.3e-14
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
61 */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
62 static const __float128
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
63 P12 = 1.538612243596254322971797716843006400388E-6Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
64 P11 = 4.998469661968096229986658302195402690910E-1Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
65 P10 = 2.321125933898420063925789532045674660756E1Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
66 P9 = 4.114517881637811823002128927449878962058E2Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
67 P8 = 3.824952356185897735160588078446136783779E3Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
68 P7 = 2.128857716871515081352991964243375186031E4Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
69 P6 = 7.594356839258970405033155585486712125861E4Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
70 P5 = 1.797628303815655343403735250238293741397E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
71 P4 = 2.854829159639697837788887080758954924001E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
72 P3 = 3.007007295140399532324943111654767187848E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
73 P2 = 2.014652742082537582487669938141683759923E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
74 P1 = 7.771154681358524243729929227226708890930E4Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
75 P0 = 1.313572404063446165910279910527789794488E4Q,
|
145
|
76 /* Q12 = 1.000000000000000000000000000000000000000E0L, */
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
77 Q11 = 4.839208193348159620282142911143429644326E1Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
78 Q10 = 9.104928120962988414618126155557301584078E2Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
79 Q9 = 9.147150349299596453976674231612674085381E3Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
80 Q8 = 5.605842085972455027590989944010492125825E4Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
81 Q7 = 2.248234257620569139969141618556349415120E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
82 Q6 = 6.132189329546557743179177159925690841200E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
83 Q5 = 1.158019977462989115839826904108208787040E6Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
84 Q4 = 1.514882452993549494932585972882995548426E6Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
85 Q3 = 1.347518538384329112529391120390701166528E6Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
86 Q2 = 7.777690340007566932935753241556479363645E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
87 Q1 = 2.626900195321832660448791748036714883242E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
88 Q0 = 3.940717212190338497730839731583397586124E4Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
89
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
90 /* Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2),
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
91 * where z = 2(x-1)/(x+1)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
92 * 1/sqrt(2) <= x < sqrt(2)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
93 * Theoretical peak relative error = 1.1e-35,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
94 * relative peak error spread 1.1e-9
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
95 */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
96 static const __float128
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
97 R5 = -8.828896441624934385266096344596648080902E-1Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
98 R4 = 8.057002716646055371965756206836056074715E1Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
99 R3 = -2.024301798136027039250415126250455056397E3Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
100 R2 = 2.048819892795278657810231591630928516206E4Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
101 R1 = -8.977257995689735303686582344659576526998E4Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
102 R0 = 1.418134209872192732479751274970992665513E5Q,
|
145
|
103 /* S6 = 1.000000000000000000000000000000000000000E0L, */
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
104 S5 = -1.186359407982897997337150403816839480438E2Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
105 S4 = 3.998526750980007367835804959888064681098E3Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
106 S3 = -5.748542087379434595104154610899551484314E4Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
107 S2 = 4.001557694070773974936904547424676279307E5Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
108 S1 = -1.332535117259762928288745111081235577029E6Q,
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
109 S0 = 1.701761051846631278975701529965589676574E6Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
110
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
111 /* C1 + C2 = ln 2 */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
112 static const __float128 C1 = 6.93145751953125E-1Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
113 static const __float128 C2 = 1.428606820309417232121458176568075500134E-6Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
114
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
115 static const __float128 sqrth = 0.7071067811865475244008443621048490392848Q;
|
145
|
116 /* ln (2^16384 * (1 - 2^-113)) */
|
|
117 static const __float128 zero = 0;
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
118
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
119 __float128
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
120 log1pq (__float128 xm1)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
121 {
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
122 __float128 x, y, z, r, s;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
123 ieee854_float128 u;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
124 int32_t hx;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
125 int e;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
126
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
127 /* Test for NaN or infinity input. */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
128 u.value = xm1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
129 hx = u.words32.w0;
|
111
|
130 if ((hx & 0x7fffffff) >= 0x7fff0000)
|
|
131 return xm1 + fabsq (xm1);
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
132
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
133 /* log1p(+- 0) = +- 0. */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
134 if (((hx & 0x7fffffff) == 0)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
135 && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
136 return xm1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
137
|
111
|
138 if ((hx & 0x7fffffff) < 0x3f8e0000)
|
|
139 {
|
|
140 math_check_force_underflow (xm1);
|
|
141 if ((int) xm1 == 0)
|
145
|
142 return xm1;
|
111
|
143 }
|
|
144
|
|
145 if (xm1 >= 0x1p113Q)
|
|
146 x = xm1;
|
|
147 else
|
145
|
148 x = xm1 + 1;
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
149
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
150 /* log1p(-1) = -inf */
|
145
|
151 if (x <= 0)
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
152 {
|
145
|
153 if (x == 0)
|
|
154 return (-1 / zero); /* log1p(-1) = -inf */
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
155 else
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
156 return (zero / (x - x));
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
157 }
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
158
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
159 /* Separate mantissa from exponent. */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
160
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
161 /* Use frexp used so that denormal numbers will be handled properly. */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
162 x = frexpq (x, &e);
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
163
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
164 /* Logarithm using log(x) = z + z^3 P(z^2)/Q(z^2),
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
165 where z = 2(x-1)/x+1). */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
166 if ((e > 2) || (e < -2))
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
167 {
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
168 if (x < sqrth)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
169 { /* 2( 2x-1 )/( 2x+1 ) */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
170 e -= 1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
171 z = x - 0.5Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
172 y = 0.5Q * z + 0.5Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
173 }
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
174 else
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
175 { /* 2 (x-1)/(x+1) */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
176 z = x - 0.5Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
177 z -= 0.5Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
178 y = 0.5Q * x + 0.5Q;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
179 }
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
180 x = z / y;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
181 z = x * x;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
182 r = ((((R5 * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
183 + R4) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
184 + R3) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
185 + R2) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
186 + R1) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
187 + R0;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
188 s = (((((z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
189 + S5) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
190 + S4) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
191 + S3) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
192 + S2) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
193 + S1) * z
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
194 + S0;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
195 z = x * (z * r / s);
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
196 z = z + e * C2;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
197 z = z + x;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
198 z = z + e * C1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
199 return (z);
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
200 }
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
201
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
202
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
203 /* Logarithm using log(1+x) = x - .5x^2 + x^3 P(x)/Q(x). */
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
204
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
205 if (x < sqrth)
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
206 {
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
207 e -= 1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
208 if (e != 0)
|
145
|
209 x = 2 * x - 1; /* 2x - 1 */
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
210 else
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
211 x = xm1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
212 }
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
213 else
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
214 {
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
215 if (e != 0)
|
145
|
216 x = x - 1;
|
68
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
217 else
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
218 x = xm1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
219 }
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
220 z = x * x;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
221 r = (((((((((((P12 * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
222 + P11) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
223 + P10) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
224 + P9) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
225 + P8) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
226 + P7) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
227 + P6) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
228 + P5) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
229 + P4) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
230 + P3) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
231 + P2) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
232 + P1) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
233 + P0;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
234 s = (((((((((((x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
235 + Q11) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
236 + Q10) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
237 + Q9) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
238 + Q8) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
239 + Q7) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
240 + Q6) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
241 + Q5) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
242 + Q4) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
243 + Q3) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
244 + Q2) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
245 + Q1) * x
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
246 + Q0;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
247 y = x * (z * r / s);
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
248 y = y + e * C2;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
249 z = y - 0.5Q * z;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
250 z = z + x;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
251 z = z + e * C1;
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
252 return (z);
|
Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
parents:
diff
changeset
|
253 }
|