1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
|
/*
* M_APM - mapm_lg3.c
*
* Copyright (C) 2003 - 2007 Michael C. Ring
*
* Permission to use, copy, and distribute this software and its
* documentation for any purpose with or without fee is hereby granted,
* provided that the above copyright notice appear in all copies and
* that both that copyright notice and this permission notice appear
* in supporting documentation.
*
* Permission to modify the software is granted. Permission to distribute
* the modified code is granted. Modifications are to be distributed by
* using the file 'license.txt' as a template to modify the file header.
* 'license.txt' is available in the official MAPM distribution.
*
* This software is provided "as is" without express or implied warranty.
*/
/*
*
* This file contains the function to compute log(2), log(10),
* and 1/log(10) to the desired precision using an AGM algorithm.
*
*/
#include "pgAdmin3.h"
#include "pgscript/utilities/mapm-lib/m_apm_lc.h"
/*
* using the 'R' function (defined below) for 'N' decimal places :
*
*
* -N -N
* log(2) = R(1, 0.5 * 10 ) - R(1, 10 )
*
*
* -N -N
* log(10) = R(1, 0.1 * 10 ) - R(1, 10 )
*
*
* In general:
*
* -N -N
* log(x) = R(1, 10 / x) - R(1, 10 )
*
*
* I found this on a web site which went into considerable detail
* on the history of log(2). This formula is algebraically identical
* to the formula specified in J. Borwein and P. Borwein's book
* "PI and the AGM". (reference algorithm 7.2)
*/
/****************************************************************************/
/*
* check if our local copy of log(2) & log(10) is precise
* enough for our purpose. if not, calculate them so it's
* as precise as desired, accurate to at least 'places'.
*/
void M_check_log_places(int places)
{
M_APM tmp6, tmp7, tmp8, tmp9;
int dplaces;
dplaces = places + 4;
if (dplaces > MM_lc_log_digits)
{
MM_lc_log_digits = dplaces + 4;
tmp6 = M_get_stack_var();
tmp7 = M_get_stack_var();
tmp8 = M_get_stack_var();
tmp9 = M_get_stack_var();
dplaces += 6 + (int)log10((double)places);
m_apm_copy(tmp7, MM_One);
tmp7->m_apm_exponent = -places;
M_log_AGM_R_func(tmp8, dplaces, MM_One, tmp7);
m_apm_multiply(tmp6, tmp7, MM_0_5);
M_log_AGM_R_func(tmp9, dplaces, MM_One, tmp6);
m_apm_subtract(MM_lc_log2, tmp9, tmp8); /* log(2) */
tmp7->m_apm_exponent -= 1; /* divide by 10 */
M_log_AGM_R_func(tmp9, dplaces, MM_One, tmp7);
m_apm_subtract(MM_lc_log10, tmp9, tmp8); /* log(10) */
m_apm_reciprocal(MM_lc_log10R, dplaces, MM_lc_log10); /* 1 / log(10) */
M_restore_stack(4);
}
}
/****************************************************************************/
/*
* define a notation for a function 'R' :
*
*
*
* 1
* R (a0, b0) = ------------------------------
*
* ----
* \
* \ n-1 2 2
* 1 - | 2 * (a - b )
* / n n
* /
* ----
* n >= 0
*
*
* where a, b are the classic AGM iteration :
*
*
* a = 0.5 * (a + b )
* n+1 n n
*
*
* b = sqrt(a * b )
* n+1 n n
*
*
*
* define a variable 'c' for more efficient computation :
*
* 2 2 2
* c = 0.5 * (a - b ) , c = a - b
* n+1 n n n n n
*
*/
/****************************************************************************/
void M_log_AGM_R_func(M_APM rr, int places, M_APM aa, M_APM bb)
{
M_APM tmp1, tmp2, tmp3, tmp4, tmpC2, sum, pow_2, tmpA0, tmpB0;
int tolerance, dplaces;
tmpA0 = M_get_stack_var();
tmpB0 = M_get_stack_var();
tmpC2 = M_get_stack_var();
tmp1 = M_get_stack_var();
tmp2 = M_get_stack_var();
tmp3 = M_get_stack_var();
tmp4 = M_get_stack_var();
sum = M_get_stack_var();
pow_2 = M_get_stack_var();
tolerance = places + 8;
dplaces = places + 16;
m_apm_copy(tmpA0, aa);
m_apm_copy(tmpB0, bb);
m_apm_copy(pow_2, MM_0_5);
m_apm_multiply(tmp1, aa, aa); /* 0.5 * [ a ^ 2 - b ^ 2 ] */
m_apm_multiply(tmp2, bb, bb);
m_apm_subtract(tmp3, tmp1, tmp2);
m_apm_multiply(sum, MM_0_5, tmp3);
while (TRUE)
{
m_apm_subtract(tmp1, tmpA0, tmpB0); /* C n+1 = 0.5 * [ An - Bn ] */
m_apm_multiply(tmp4, MM_0_5, tmp1); /* C n+1 */
m_apm_multiply(tmpC2, tmp4, tmp4); /* C n+1 ^ 2 */
/* do the AGM */
m_apm_add(tmp1, tmpA0, tmpB0);
m_apm_multiply(tmp3, MM_0_5, tmp1);
m_apm_multiply(tmp2, tmpA0, tmpB0);
m_apm_sqrt(tmpB0, dplaces, tmp2);
m_apm_round(tmpA0, dplaces, tmp3);
/* end AGM */
m_apm_multiply(tmp2, MM_Two, pow_2);
m_apm_copy(pow_2, tmp2);
m_apm_multiply(tmp1, tmpC2, pow_2);
m_apm_add(tmp3, sum, tmp1);
if ((tmp1->m_apm_sign == 0) ||
((-2 * tmp1->m_apm_exponent) > tolerance))
break;
m_apm_round(sum, dplaces, tmp3);
}
m_apm_subtract(tmp4, MM_One, tmp3);
m_apm_reciprocal(rr, places, tmp4);
M_restore_stack(9);
}
/****************************************************************************/
|
