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
|
;; Copyright (C) 1991, 1993, 1994, 1995 Free Software Foundation, Inc.
;;
;; This program is free software; you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation; either version 2, or (at your option)
;; any later version.
;;
;; This program is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
;; GNU General Public License for more details.
;;
;; You should have received a copy of the GNU General Public License
;; along with this software; see the file COPYING. If not, write to
;; the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA.
;;
;; As a special exception, the Free Software Foundation gives permission
;; for additional uses of the text contained in its release of GUILE.
;;
;; The exception is that, if you link the GUILE library with other files
;; to produce an executable, this does not by itself cause the
;; resulting executable to be covered by the GNU General Public License.
;; Your use of that executable is in no way restricted on account of
;; linking the GUILE library code into it.
;;
;; This exception does not however invalidate any other reasons why
;; the executable file might be covered by the GNU General Public License.
;;
;; This exception applies only to the code released by the
;; Free Software Foundation under the name GUILE. If you copy
;; code from other Free Software Foundation releases into a copy of
;; GUILE, as the General Public License permits, the exception does
;; not apply to the code that you add in this way. To avoid misleading
;; anyone as to the status of such modified files, you must delete
;; this exception notice from them.
;;
;; If you write modifications of your own for GUILE, it is your choice
;; whether to permit this exception to apply to your modifications.
;; If you do not wish that, delete this exception notice.
;;;; "pi.scm", program for computing digits of numerical value of PI.
;;;; "bigpi.scm", program for computing digits of numerical value of PI.
;;;; "e.scm", program for computing digits of numerical value of 'e'.
;;; Authors: Aubrey Jaffer & Jerry D. Hedden
;;; (pi <n> <d>) prints out <n> digits of pi in groups of <d> digits.
;;; 'Spigot' algorithm origionally due to Stanly Rabinowitz.
;;; This algorithm takes time proportional to the square of <n>/<d>.
;;; This fact can make comparisons of computational speed between systems
;;; of vastly differring performances quicker and more accurate.
;;; Try (pi 100 5)
;;; The digit size <d> will have to be reduced for larger <n> or an
;;; overflow error will occur (on systems lacking bignums).
;;; It your Scheme has bignums try (pi 1000).
(define (pi n . args)
(if (null? args) (bigpi n)
(let* ((d (car args))
(r (do ((s 1 (* 10 s))
(i d (- i 1)))
((zero? i) s)))
(n (+ (quotient n d) 1))
(m (quotient (* n d 3322) 1000))
(a (make-vector (+ 1 m) 2)))
(vector-set! a m 4)
(do ((j 1 (+ 1 j))
(q 0 0)
(b 2 (remainder q r)))
((> j n))
(do ((k m (- k 1)))
((zero? k))
(set! q (+ q (* (vector-ref a k) r)))
(let ((t (+ 1 (* 2 k))))
(vector-set! a k (remainder q t))
(set! q (* k (quotient q t)))))
(let ((s (number->string (+ b (quotient q r)))))
(do ((l (string-length s) (+ 1 l)))
((>= l d) (display s))
(display #\0)))
(if (zero? (modulo j 10)) (newline) (display #\ )))
(newline))))
;;; (pi <n>) prints out <n> digits of pi.
;;; 'Spigot' algorithm originally due to Stanly Rabinowitz:
;;;
;;; PI = 2+(1/3)*(2+(2/5)*(2+(3/7)*(2+ ... *(2+(k/(2k+1))*(4)) ... )))
;;;
;;; where 'k' is approximately equal to the desired precision of 'n'
;;; places times 'log2(10)'.
;;;
;;; This version takes advantage of "bignums" in SCM to compute all
;;; of the requested digits in one pass! Basically, it calculates
;;; the truncated portion of (PI * 10^n), and then displays it in a
;;; nice format.
(define (bigpi digits)
(let* ((n (* 10 (quotient (+ digits 9) 10))) ; digits in multiples of 10
(z (inexact->exact (truncate ; z = number of terms
(/ (* n (log 10)) (log 2)))))
(q (do ((x 2 (* 10000000000 x)) ; q = 2 * 10^n
(i (/ n 10) (- i 1)))
((zero? i) x)))
(_pi (number->string ; _pi = PI * 10^n
;; do the calculations in one pass!!!
(let pi_calc ((j z) (k (+ z z 1)) (p (+ q q)))
(if (zero? j)
p
(pi_calc (- j 1) (- k 2) (+ q (quotient (* p j) k))))))))
;; print out the result ("3." followed by 5 groups of 10 digits per line)
(display (substring _pi 0 1)) (display #\.) (newline)
(do ((i 0 (+ i 10)))
((>= i n))
(display (substring _pi (+ i 1) (+ i 11)))
(display (if (zero? (modulo (+ i 10) 50)) #\newline #\ )))
(if (not (zero? (modulo n 50))) (newline))))
;;; (e <n>) prints out <n> digits of 'e'.
;;; Uses the formula:
;;;
;;; 1 1 1 1 1
;;; e = 1 + -- + -- + -- + -- + ... + --
;;; 1! 2! 3! 4! k!
;;;
;;; where 'k' is determined using the desired precision 'n' in:
;;;
;;; n < ((k * (ln(k) - 1)) / ln(10))
;;;
;;; which uses Stirling's formula for approximating ln(k!)
;;;
;;; This program takes advantage of "bignums" in SCM to compute all
;;; the requested digits at once! Basically, it calculates the
;;; fractional part of 'e' (i.e., e-2) as a fraction of two bignums
;;; 'e_n' and 'e_d', determines the integer part of (e_n * 10^n)/e_d,
;;; and then displays it in a nice format.
(define (e digits)
(let* ((n (* 10 (quotient (+ digits 9) 10))) ; digits in multiples of 10
(k (do ((i 15 (+ i 1))) ; k = number of terms
((< n (/ (* i (- (log i) 1)) (log 10))) i)))
(q (do ((x 1 (* 10000000000 x)) ; q = 10^n
(i (/ n 10) (- i 1)))
((zero? i) x)))
(_e (let ((ee
; do calculations
(let e_calc ((i k) (e_d 1) (e_n 0))
(if (= i 1)
(cons (* q e_n) e_d)
(e_calc (- i 1) (* e_d i) (+ e_n e_d))))))
(number->string (+ (quotient (car ee) (cdr ee))
; rounding
(if (< (remainder (car ee) (cdr ee))
(quotient (cdr ee) 2))
0 1))))))
;; print out the result ("2." followed by 5 groups of 10 digits per line)
(display "2.") (newline)
(do ((i 0 (+ i 10)))
((>= i n))
(display (substring _e i (+ i 10)))
(display (if (zero? (modulo (+ i 10) 50)) #\newline #\ )))
(if (not (zero? (modulo n 50))) (newline))))
|
