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3132f345 CW |
1 | ;;; calc-math.el --- mathematical functions for Calc |
2 | ||
491c3062 | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc. |
3132f345 CW |
4 | |
5 | ;; Author: David Gillespie <daveg@synaptics.com> | |
a1506d29 | 6 | ;; Maintainers: D. Goel <deego@gnufans.org> |
6e1c888a | 7 | ;; Colin Walters <walters@debian.org> |
136211a9 EZ |
8 | |
9 | ;; This file is part of GNU Emacs. | |
10 | ||
11 | ;; GNU Emacs is distributed in the hope that it will be useful, | |
12 | ;; but WITHOUT ANY WARRANTY. No author or distributor | |
13 | ;; accepts responsibility to anyone for the consequences of using it | |
14 | ;; or for whether it serves any particular purpose or works at all, | |
15 | ;; unless he says so in writing. Refer to the GNU Emacs General Public | |
16 | ;; License for full details. | |
17 | ||
18 | ;; Everyone is granted permission to copy, modify and redistribute | |
19 | ;; GNU Emacs, but only under the conditions described in the | |
20 | ;; GNU Emacs General Public License. A copy of this license is | |
21 | ;; supposed to have been given to you along with GNU Emacs so you | |
22 | ;; can know your rights and responsibilities. It should be in a | |
23 | ;; file named COPYING. Among other things, the copyright notice | |
24 | ;; and this notice must be preserved on all copies. | |
25 | ||
3132f345 | 26 | ;;; Commentary: |
136211a9 | 27 | |
3132f345 | 28 | ;;; Code: |
136211a9 EZ |
29 | |
30 | ;; This file is autoloaded from calc-ext.el. | |
31 | (require 'calc-ext) | |
32 | ||
33 | (require 'calc-macs) | |
34 | ||
35 | (defun calc-Need-calc-math () nil) | |
36 | ||
37 | ||
38 | (defun calc-sqrt (arg) | |
39 | (interactive "P") | |
40 | (calc-slow-wrapper | |
41 | (if (calc-is-inverse) | |
42 | (calc-unary-op "^2" 'calcFunc-sqr arg) | |
491c3062 | 43 | (calc-unary-op "sqrt" 'calcFunc-sqrt arg)))) |
136211a9 EZ |
44 | |
45 | (defun calc-isqrt (arg) | |
46 | (interactive "P") | |
47 | (calc-slow-wrapper | |
48 | (if (calc-is-inverse) | |
49 | (calc-unary-op "^2" 'calcFunc-sqr arg) | |
491c3062 | 50 | (calc-unary-op "isqt" 'calcFunc-isqrt arg)))) |
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51 | |
52 | ||
53 | (defun calc-hypot (arg) | |
54 | (interactive "P") | |
55 | (calc-slow-wrapper | |
491c3062 | 56 | (calc-binary-op "hypt" 'calcFunc-hypot arg))) |
136211a9 EZ |
57 | |
58 | (defun calc-ln (arg) | |
59 | (interactive "P") | |
60 | (calc-invert-func) | |
491c3062 | 61 | (calc-exp arg)) |
136211a9 EZ |
62 | |
63 | (defun calc-log10 (arg) | |
64 | (interactive "P") | |
65 | (calc-hyperbolic-func) | |
491c3062 | 66 | (calc-ln arg)) |
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67 | |
68 | (defun calc-log (arg) | |
69 | (interactive "P") | |
70 | (calc-slow-wrapper | |
71 | (if (calc-is-inverse) | |
72 | (calc-binary-op "alog" 'calcFunc-alog arg) | |
491c3062 | 73 | (calc-binary-op "log" 'calcFunc-log arg)))) |
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74 | |
75 | (defun calc-ilog (arg) | |
76 | (interactive "P") | |
77 | (calc-slow-wrapper | |
78 | (if (calc-is-inverse) | |
79 | (calc-binary-op "alog" 'calcFunc-alog arg) | |
491c3062 | 80 | (calc-binary-op "ilog" 'calcFunc-ilog arg)))) |
136211a9 EZ |
81 | |
82 | (defun calc-lnp1 (arg) | |
83 | (interactive "P") | |
84 | (calc-invert-func) | |
491c3062 | 85 | (calc-expm1 arg)) |
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86 | |
87 | (defun calc-exp (arg) | |
88 | (interactive "P") | |
89 | (calc-slow-wrapper | |
90 | (if (calc-is-hyperbolic) | |
91 | (if (calc-is-inverse) | |
92 | (calc-unary-op "lg10" 'calcFunc-log10 arg) | |
93 | (calc-unary-op "10^" 'calcFunc-exp10 arg)) | |
94 | (if (calc-is-inverse) | |
95 | (calc-unary-op "ln" 'calcFunc-ln arg) | |
491c3062 | 96 | (calc-unary-op "exp" 'calcFunc-exp arg))))) |
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97 | |
98 | (defun calc-expm1 (arg) | |
99 | (interactive "P") | |
100 | (calc-slow-wrapper | |
101 | (if (calc-is-inverse) | |
102 | (calc-unary-op "ln+1" 'calcFunc-lnp1 arg) | |
491c3062 | 103 | (calc-unary-op "ex-1" 'calcFunc-expm1 arg)))) |
136211a9 EZ |
104 | |
105 | (defun calc-pi () | |
106 | (interactive) | |
107 | (calc-slow-wrapper | |
108 | (if (calc-is-inverse) | |
109 | (if (calc-is-hyperbolic) | |
110 | (if calc-symbolic-mode | |
111 | (calc-pop-push-record 0 "phi" '(var phi var-phi)) | |
112 | (calc-pop-push-record 0 "phi" (math-phi))) | |
113 | (if calc-symbolic-mode | |
114 | (calc-pop-push-record 0 "gmma" '(var gamma var-gamma)) | |
115 | (calc-pop-push-record 0 "gmma" (math-gamma-const)))) | |
116 | (if (calc-is-hyperbolic) | |
117 | (if calc-symbolic-mode | |
118 | (calc-pop-push-record 0 "e" '(var e var-e)) | |
119 | (calc-pop-push-record 0 "e" (math-e))) | |
120 | (if calc-symbolic-mode | |
121 | (calc-pop-push-record 0 "pi" '(var pi var-pi)) | |
491c3062 | 122 | (calc-pop-push-record 0 "pi" (math-pi))))))) |
136211a9 EZ |
123 | |
124 | (defun calc-sin (arg) | |
125 | (interactive "P") | |
126 | (calc-slow-wrapper | |
127 | (if (calc-is-hyperbolic) | |
128 | (if (calc-is-inverse) | |
129 | (calc-unary-op "asnh" 'calcFunc-arcsinh arg) | |
130 | (calc-unary-op "sinh" 'calcFunc-sinh arg)) | |
131 | (if (calc-is-inverse) | |
132 | (calc-unary-op "asin" 'calcFunc-arcsin arg) | |
491c3062 | 133 | (calc-unary-op "sin" 'calcFunc-sin arg))))) |
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134 | |
135 | (defun calc-arcsin (arg) | |
136 | (interactive "P") | |
137 | (calc-invert-func) | |
491c3062 | 138 | (calc-sin arg)) |
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139 | |
140 | (defun calc-sinh (arg) | |
141 | (interactive "P") | |
142 | (calc-hyperbolic-func) | |
491c3062 | 143 | (calc-sin arg)) |
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144 | |
145 | (defun calc-arcsinh (arg) | |
146 | (interactive "P") | |
147 | (calc-invert-func) | |
148 | (calc-hyperbolic-func) | |
491c3062 | 149 | (calc-sin arg)) |
136211a9 EZ |
150 | |
151 | (defun calc-cos (arg) | |
152 | (interactive "P") | |
153 | (calc-slow-wrapper | |
154 | (if (calc-is-hyperbolic) | |
155 | (if (calc-is-inverse) | |
156 | (calc-unary-op "acsh" 'calcFunc-arccosh arg) | |
157 | (calc-unary-op "cosh" 'calcFunc-cosh arg)) | |
158 | (if (calc-is-inverse) | |
159 | (calc-unary-op "acos" 'calcFunc-arccos arg) | |
491c3062 | 160 | (calc-unary-op "cos" 'calcFunc-cos arg))))) |
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161 | |
162 | (defun calc-arccos (arg) | |
163 | (interactive "P") | |
164 | (calc-invert-func) | |
491c3062 | 165 | (calc-cos arg)) |
136211a9 EZ |
166 | |
167 | (defun calc-cosh (arg) | |
168 | (interactive "P") | |
169 | (calc-hyperbolic-func) | |
491c3062 | 170 | (calc-cos arg)) |
136211a9 EZ |
171 | |
172 | (defun calc-arccosh (arg) | |
173 | (interactive "P") | |
174 | (calc-invert-func) | |
175 | (calc-hyperbolic-func) | |
491c3062 | 176 | (calc-cos arg)) |
136211a9 EZ |
177 | |
178 | (defun calc-sincos () | |
179 | (interactive) | |
180 | (calc-slow-wrapper | |
181 | (if (calc-is-inverse) | |
182 | (calc-enter-result 1 "asnc" (list 'calcFunc-arcsincos (calc-top-n 1))) | |
491c3062 | 183 | (calc-enter-result 1 "sncs" (list 'calcFunc-sincos (calc-top-n 1)))))) |
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184 | |
185 | (defun calc-tan (arg) | |
186 | (interactive "P") | |
187 | (calc-slow-wrapper | |
188 | (if (calc-is-hyperbolic) | |
189 | (if (calc-is-inverse) | |
190 | (calc-unary-op "atnh" 'calcFunc-arctanh arg) | |
191 | (calc-unary-op "tanh" 'calcFunc-tanh arg)) | |
192 | (if (calc-is-inverse) | |
193 | (calc-unary-op "atan" 'calcFunc-arctan arg) | |
491c3062 | 194 | (calc-unary-op "tan" 'calcFunc-tan arg))))) |
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195 | |
196 | (defun calc-arctan (arg) | |
197 | (interactive "P") | |
198 | (calc-invert-func) | |
491c3062 | 199 | (calc-tan arg)) |
136211a9 EZ |
200 | |
201 | (defun calc-tanh (arg) | |
202 | (interactive "P") | |
203 | (calc-hyperbolic-func) | |
491c3062 | 204 | (calc-tan arg)) |
136211a9 EZ |
205 | |
206 | (defun calc-arctanh (arg) | |
207 | (interactive "P") | |
208 | (calc-invert-func) | |
209 | (calc-hyperbolic-func) | |
491c3062 | 210 | (calc-tan arg)) |
136211a9 EZ |
211 | |
212 | (defun calc-arctan2 () | |
213 | (interactive) | |
214 | (calc-slow-wrapper | |
491c3062 | 215 | (calc-enter-result 2 "atn2" (cons 'calcFunc-arctan2 (calc-top-list-n 2))))) |
136211a9 EZ |
216 | |
217 | (defun calc-conj (arg) | |
218 | (interactive "P") | |
219 | (calc-wrapper | |
491c3062 | 220 | (calc-unary-op "conj" 'calcFunc-conj arg))) |
136211a9 EZ |
221 | |
222 | (defun calc-imaginary () | |
223 | (interactive) | |
224 | (calc-slow-wrapper | |
491c3062 | 225 | (calc-pop-push-record 1 "i*" (math-imaginary (calc-top-n 1))))) |
136211a9 EZ |
226 | |
227 | ||
228 | ||
229 | (defun calc-to-degrees (arg) | |
230 | (interactive "P") | |
231 | (calc-wrapper | |
491c3062 | 232 | (calc-unary-op ">deg" 'calcFunc-deg arg))) |
136211a9 EZ |
233 | |
234 | (defun calc-to-radians (arg) | |
235 | (interactive "P") | |
236 | (calc-wrapper | |
491c3062 | 237 | (calc-unary-op ">rad" 'calcFunc-rad arg))) |
136211a9 EZ |
238 | |
239 | ||
240 | (defun calc-degrees-mode (arg) | |
241 | (interactive "p") | |
242 | (cond ((= arg 1) | |
243 | (calc-wrapper | |
244 | (calc-change-mode 'calc-angle-mode 'deg) | |
3132f345 | 245 | (message "Angles measured in degrees"))) |
136211a9 EZ |
246 | ((= arg 2) (calc-radians-mode)) |
247 | ((= arg 3) (calc-hms-mode)) | |
491c3062 | 248 | (t (error "Prefix argument out of range")))) |
136211a9 EZ |
249 | |
250 | (defun calc-radians-mode () | |
251 | (interactive) | |
252 | (calc-wrapper | |
253 | (calc-change-mode 'calc-angle-mode 'rad) | |
3132f345 | 254 | (message "Angles measured in radians"))) |
136211a9 EZ |
255 | |
256 | ||
257 | ;;; Compute the integer square-root floor(sqrt(A)). A > 0. [I I] [Public] | |
258 | ;;; This method takes advantage of the fact that Newton's method starting | |
259 | ;;; with an overestimate always works, even using truncating integer division! | |
260 | (defun math-isqrt (a) | |
261 | (cond ((Math-zerop a) a) | |
262 | ((not (math-natnump a)) | |
263 | (math-reject-arg a 'natnump)) | |
264 | ((integerp a) | |
265 | (math-isqrt-small a)) | |
266 | (t | |
491c3062 | 267 | (math-normalize (cons 'bigpos (cdr (math-isqrt-bignum (cdr a)))))))) |
136211a9 EZ |
268 | |
269 | (defun calcFunc-isqrt (a) | |
270 | (if (math-realp a) | |
271 | (math-isqrt (math-floor a)) | |
491c3062 | 272 | (math-floor (math-sqrt a)))) |
136211a9 EZ |
273 | |
274 | ||
f0529b5b | 275 | ;;; This returns (flag . result) where the flag is t if A is a perfect square. |
136211a9 EZ |
276 | (defun math-isqrt-bignum (a) ; [P.l L] |
277 | (let ((len (length a))) | |
278 | (if (= (% len 2) 0) | |
279 | (let* ((top (nthcdr (- len 2) a))) | |
280 | (math-isqrt-bignum-iter | |
281 | a | |
282 | (math-scale-bignum-3 | |
283 | (math-bignum-big | |
284 | (1+ (math-isqrt-small | |
285 | (+ (* (nth 1 top) 1000) (car top))))) | |
286 | (1- (/ len 2))))) | |
287 | (let* ((top (nth (1- len) a))) | |
288 | (math-isqrt-bignum-iter | |
289 | a | |
290 | (math-scale-bignum-3 | |
291 | (list (1+ (math-isqrt-small top))) | |
491c3062 | 292 | (/ len 2))))))) |
136211a9 EZ |
293 | |
294 | (defun math-isqrt-bignum-iter (a guess) ; [l L l] | |
295 | (math-working "isqrt" (cons 'bigpos guess)) | |
296 | (let* ((q (math-div-bignum a guess)) | |
297 | (s (math-add-bignum (car q) guess)) | |
298 | (g2 (math-div2-bignum s)) | |
299 | (comp (math-compare-bignum g2 guess))) | |
300 | (if (< comp 0) | |
301 | (math-isqrt-bignum-iter a g2) | |
302 | (cons (and (= comp 0) | |
303 | (math-zerop-bignum (cdr q)) | |
304 | (= (% (car s) 2) 0)) | |
491c3062 | 305 | guess)))) |
136211a9 EZ |
306 | |
307 | (defun math-zerop-bignum (a) | |
308 | (and (eq (car a) 0) | |
309 | (progn | |
310 | (while (eq (car (setq a (cdr a))) 0)) | |
491c3062 | 311 | (null a)))) |
136211a9 EZ |
312 | |
313 | (defun math-scale-bignum-3 (a n) ; [L L S] | |
314 | (while (> n 0) | |
315 | (setq a (cons 0 a) | |
316 | n (1- n))) | |
491c3062 | 317 | a) |
136211a9 EZ |
318 | |
319 | (defun math-isqrt-small (a) ; A > 0. [S S] | |
320 | (let ((g (cond ((>= a 10000) 1000) | |
321 | ((>= a 100) 100) | |
322 | (t 10))) | |
323 | g2) | |
324 | (while (< (setq g2 (/ (+ g (/ a g)) 2)) g) | |
325 | (setq g g2)) | |
491c3062 | 326 | g)) |
136211a9 EZ |
327 | |
328 | ||
329 | ||
330 | ||
331 | ;;; Compute the square root of a number. | |
332 | ;;; [T N] if possible, else [F N] if possible, else [C N]. [Public] | |
333 | (defun math-sqrt (a) | |
334 | (or | |
335 | (and (Math-zerop a) a) | |
336 | (and (math-known-nonposp a) | |
337 | (math-imaginary (math-sqrt (math-neg a)))) | |
338 | (and (integerp a) | |
339 | (let ((sqrt (math-isqrt-small a))) | |
340 | (if (= (* sqrt sqrt) a) | |
341 | sqrt | |
342 | (if calc-symbolic-mode | |
343 | (list 'calcFunc-sqrt a) | |
344 | (math-sqrt-float (math-float a) (math-float sqrt)))))) | |
345 | (and (eq (car-safe a) 'bigpos) | |
346 | (let* ((res (math-isqrt-bignum (cdr a))) | |
347 | (sqrt (math-normalize (cons 'bigpos (cdr res))))) | |
348 | (if (car res) | |
349 | sqrt | |
350 | (if calc-symbolic-mode | |
351 | (list 'calcFunc-sqrt a) | |
352 | (math-sqrt-float (math-float a) (math-float sqrt)))))) | |
353 | (and (eq (car-safe a) 'frac) | |
354 | (let* ((num-res (math-isqrt-bignum (cdr (Math-bignum-test (nth 1 a))))) | |
355 | (num-sqrt (math-normalize (cons 'bigpos (cdr num-res)))) | |
356 | (den-res (math-isqrt-bignum (cdr (Math-bignum-test (nth 2 a))))) | |
357 | (den-sqrt (math-normalize (cons 'bigpos (cdr den-res))))) | |
358 | (if (and (car num-res) (car den-res)) | |
359 | (list 'frac num-sqrt den-sqrt) | |
360 | (if calc-symbolic-mode | |
361 | (if (or (car num-res) (car den-res)) | |
362 | (math-div (if (car num-res) | |
363 | num-sqrt (list 'calcFunc-sqrt (nth 1 a))) | |
364 | (if (car den-res) | |
365 | den-sqrt (list 'calcFunc-sqrt (nth 2 a)))) | |
366 | (list 'calcFunc-sqrt a)) | |
367 | (math-sqrt-float (math-float a) | |
368 | (math-div (math-float num-sqrt) den-sqrt)))))) | |
369 | (and (eq (car-safe a) 'float) | |
370 | (if calc-symbolic-mode | |
371 | (if (= (% (nth 2 a) 2) 0) | |
372 | (let ((res (math-isqrt-bignum | |
373 | (cdr (Math-bignum-test (nth 1 a)))))) | |
374 | (if (car res) | |
375 | (math-make-float (math-normalize | |
376 | (cons 'bigpos (cdr res))) | |
377 | (/ (nth 2 a) 2)) | |
378 | (signal 'inexact-result nil))) | |
379 | (signal 'inexact-result nil)) | |
380 | (math-sqrt-float a))) | |
381 | (and (eq (car-safe a) 'cplx) | |
382 | (math-with-extra-prec 2 | |
383 | (let* ((d (math-abs a)) | |
384 | (imag (math-sqrt (math-mul (math-sub d (nth 1 a)) | |
385 | '(float 5 -1))))) | |
386 | (list 'cplx | |
387 | (math-sqrt (math-mul (math-add d (nth 1 a)) '(float 5 -1))) | |
388 | (if (math-negp (nth 2 a)) (math-neg imag) imag))))) | |
389 | (and (eq (car-safe a) 'polar) | |
390 | (list 'polar | |
391 | (math-sqrt (nth 1 a)) | |
392 | (math-mul (nth 2 a) '(float 5 -1)))) | |
393 | (and (eq (car-safe a) 'sdev) | |
394 | (let ((sqrt (math-sqrt (nth 1 a)))) | |
395 | (math-make-sdev sqrt | |
396 | (math-div (nth 2 a) (math-mul sqrt 2))))) | |
397 | (and (eq (car-safe a) 'intv) | |
398 | (not (math-negp (nth 2 a))) | |
399 | (math-make-intv (nth 1 a) (math-sqrt (nth 2 a)) (math-sqrt (nth 3 a)))) | |
400 | (and (eq (car-safe a) '*) | |
401 | (or (math-known-nonnegp (nth 1 a)) | |
402 | (math-known-nonnegp (nth 2 a))) | |
403 | (math-mul (math-sqrt (nth 1 a)) (math-sqrt (nth 2 a)))) | |
404 | (and (eq (car-safe a) '/) | |
405 | (or (and (math-known-nonnegp (nth 2 a)) | |
406 | (math-div (math-sqrt (nth 1 a)) (math-sqrt (nth 2 a)))) | |
407 | (and (math-known-nonnegp (nth 1 a)) | |
408 | (not (math-equal-int (nth 1 a) 1)) | |
409 | (math-mul (math-sqrt (nth 1 a)) | |
410 | (math-sqrt (math-div 1 (nth 2 a))))))) | |
411 | (and (eq (car-safe a) '^) | |
412 | (math-known-evenp (nth 2 a)) | |
413 | (math-known-realp (nth 1 a)) | |
414 | (math-abs (math-pow (nth 1 a) (math-div (nth 2 a) 2)))) | |
415 | (let ((inf (math-infinitep a))) | |
416 | (and inf | |
417 | (math-mul (math-sqrt (math-infinite-dir a inf)) inf))) | |
418 | (progn | |
419 | (calc-record-why 'numberp a) | |
491c3062 | 420 | (list 'calcFunc-sqrt a)))) |
3132f345 | 421 | (defalias 'calcFunc-sqrt 'math-sqrt) |
136211a9 EZ |
422 | |
423 | (defun math-infinite-dir (a &optional inf) | |
424 | (or inf (setq inf (math-infinitep a))) | |
491c3062 | 425 | (math-normalize (math-expr-subst a inf 1))) |
136211a9 EZ |
426 | |
427 | (defun math-sqrt-float (a &optional guess) ; [F F F] | |
428 | (if calc-symbolic-mode | |
429 | (signal 'inexact-result nil) | |
491c3062 | 430 | (math-with-extra-prec 1 (math-sqrt-raw a guess)))) |
136211a9 EZ |
431 | |
432 | (defun math-sqrt-raw (a &optional guess) ; [F F F] | |
433 | (if (not (Math-posp a)) | |
434 | (math-sqrt a) | |
435 | (if (null guess) | |
436 | (let ((ldiff (- (math-numdigs (nth 1 a)) 6))) | |
437 | (or (= (% (+ (nth 2 a) ldiff) 2) 0) (setq ldiff (1+ ldiff))) | |
438 | (setq guess (math-make-float (math-isqrt-small | |
439 | (math-scale-int (nth 1 a) (- ldiff))) | |
440 | (/ (+ (nth 2 a) ldiff) 2))))) | |
491c3062 | 441 | (math-sqrt-float-iter a guess))) |
136211a9 EZ |
442 | |
443 | (defun math-sqrt-float-iter (a guess) ; [F F F] | |
444 | (math-working "sqrt" guess) | |
445 | (let ((g2 (math-mul-float (math-add-float guess (math-div-float a guess)) | |
446 | '(float 5 -1)))) | |
447 | (if (math-nearly-equal-float g2 guess) | |
448 | g2 | |
491c3062 | 449 | (math-sqrt-float-iter a g2)))) |
136211a9 EZ |
450 | |
451 | ;;; True if A and B differ only in the last digit of precision. [P F F] | |
452 | (defun math-nearly-equal-float (a b) | |
453 | (let ((ediff (- (nth 2 a) (nth 2 b)))) | |
454 | (cond ((= ediff 0) ;; Expanded out for speed | |
455 | (setq ediff (math-add (Math-integer-neg (nth 1 a)) (nth 1 b))) | |
456 | (or (eq ediff 0) | |
457 | (and (not (consp ediff)) | |
458 | (< ediff 10) | |
459 | (> ediff -10) | |
460 | (= (math-numdigs (nth 1 a)) calc-internal-prec)))) | |
461 | ((= ediff 1) | |
462 | (setq ediff (math-add (Math-integer-neg (nth 1 b)) | |
463 | (math-scale-int (nth 1 a) 1))) | |
464 | (and (not (consp ediff)) | |
465 | (< ediff 10) | |
466 | (> ediff -10) | |
467 | (= (math-numdigs (nth 1 b)) calc-internal-prec))) | |
468 | ((= ediff -1) | |
469 | (setq ediff (math-add (Math-integer-neg (nth 1 a)) | |
470 | (math-scale-int (nth 1 b) 1))) | |
471 | (and (not (consp ediff)) | |
472 | (< ediff 10) | |
473 | (> ediff -10) | |
491c3062 | 474 | (= (math-numdigs (nth 1 a)) calc-internal-prec)))))) |
136211a9 EZ |
475 | |
476 | (defun math-nearly-equal (a b) ; [P N N] [Public] | |
477 | (setq a (math-float a)) | |
478 | (setq b (math-float b)) | |
479 | (if (eq (car a) 'polar) (setq a (math-complex a))) | |
480 | (if (eq (car b) 'polar) (setq b (math-complex b))) | |
481 | (if (eq (car a) 'cplx) | |
482 | (if (eq (car b) 'cplx) | |
483 | (and (or (math-nearly-equal-float (nth 1 a) (nth 1 b)) | |
484 | (and (math-nearly-zerop-float (nth 1 a) (nth 2 a)) | |
485 | (math-nearly-zerop-float (nth 1 b) (nth 2 b)))) | |
486 | (or (math-nearly-equal-float (nth 2 a) (nth 2 b)) | |
487 | (and (math-nearly-zerop-float (nth 2 a) (nth 1 a)) | |
488 | (math-nearly-zerop-float (nth 2 b) (nth 1 b))))) | |
489 | (and (math-nearly-equal-float (nth 1 a) b) | |
490 | (math-nearly-zerop-float (nth 2 a) b))) | |
491 | (if (eq (car b) 'cplx) | |
492 | (and (math-nearly-equal-float a (nth 1 b)) | |
493 | (math-nearly-zerop-float a (nth 2 b))) | |
491c3062 | 494 | (math-nearly-equal-float a b)))) |
136211a9 EZ |
495 | |
496 | ;;; True if A is nearly zero compared to B. [P F F] | |
497 | (defun math-nearly-zerop-float (a b) | |
498 | (or (eq (nth 1 a) 0) | |
499 | (<= (+ (math-numdigs (nth 1 a)) (nth 2 a)) | |
491c3062 | 500 | (1+ (- (+ (math-numdigs (nth 1 b)) (nth 2 b)) calc-internal-prec))))) |
136211a9 EZ |
501 | |
502 | (defun math-nearly-zerop (a b) ; [P N R] [Public] | |
503 | (setq a (math-float a)) | |
504 | (setq b (math-float b)) | |
505 | (if (eq (car a) 'cplx) | |
506 | (and (math-nearly-zerop-float (nth 1 a) b) | |
507 | (math-nearly-zerop-float (nth 2 a) b)) | |
508 | (if (eq (car a) 'polar) | |
509 | (math-nearly-zerop-float (nth 1 a) b) | |
491c3062 | 510 | (math-nearly-zerop-float a b)))) |
136211a9 EZ |
511 | |
512 | ;;; This implementation could be improved, accuracy-wise. | |
513 | (defun math-hypot (a b) | |
514 | (cond ((Math-zerop a) (math-abs b)) | |
515 | ((Math-zerop b) (math-abs a)) | |
516 | ((not (Math-scalarp a)) | |
517 | (if (math-infinitep a) | |
518 | (if (math-infinitep b) | |
519 | (if (equal a b) | |
520 | a | |
521 | '(var nan var-nan)) | |
522 | a) | |
523 | (calc-record-why 'scalarp a) | |
524 | (list 'calcFunc-hypot a b))) | |
525 | ((not (Math-scalarp b)) | |
526 | (if (math-infinitep b) | |
527 | b | |
528 | (calc-record-why 'scalarp b) | |
529 | (list 'calcFunc-hypot a b))) | |
530 | ((and (Math-numberp a) (Math-numberp b)) | |
531 | (math-with-extra-prec 1 | |
532 | (math-sqrt (math-add (calcFunc-abssqr a) (calcFunc-abssqr b))))) | |
533 | ((eq (car-safe a) 'hms) | |
534 | (if (eq (car-safe b) 'hms) ; this helps sdev's of hms forms | |
535 | (math-to-hms (math-hypot (math-from-hms a 'deg) | |
536 | (math-from-hms b 'deg))) | |
537 | (math-to-hms (math-hypot (math-from-hms a 'deg) b)))) | |
538 | ((eq (car-safe b) 'hms) | |
539 | (math-to-hms (math-hypot a (math-from-hms b 'deg)))) | |
491c3062 | 540 | (t nil))) |
3132f345 | 541 | (defalias 'calcFunc-hypot 'math-hypot) |
136211a9 EZ |
542 | |
543 | (defun calcFunc-sqr (x) | |
491c3062 | 544 | (math-pow x 2)) |
136211a9 EZ |
545 | |
546 | ||
547 | ||
548 | (defun math-nth-root (a n) | |
549 | (cond ((= n 2) (math-sqrt a)) | |
550 | ((Math-zerop a) a) | |
551 | ((Math-negp a) nil) | |
552 | ((Math-integerp a) | |
553 | (let ((root (math-nth-root-integer a n))) | |
554 | (if (car root) | |
555 | (cdr root) | |
556 | (and (not calc-symbolic-mode) | |
557 | (math-nth-root-float (math-float a) n | |
558 | (math-float (cdr root))))))) | |
559 | ((eq (car-safe a) 'frac) | |
560 | (let* ((num-root (math-nth-root-integer (nth 1 a) n)) | |
561 | (den-root (math-nth-root-integer (nth 2 a) n))) | |
562 | (if (and (car num-root) (car den-root)) | |
563 | (list 'frac (cdr num-root) (cdr den-root)) | |
564 | (and (not calc-symbolic-mode) | |
565 | (math-nth-root-float | |
566 | (math-float a) n | |
567 | (math-div-float (math-float (cdr num-root)) | |
568 | (math-float (cdr den-root)))))))) | |
569 | ((eq (car-safe a) 'float) | |
570 | (and (not calc-symbolic-mode) | |
571 | (math-nth-root-float a n))) | |
572 | ((eq (car-safe a) 'polar) | |
573 | (let ((root (math-nth-root (nth 1 a) n))) | |
574 | (and root (list 'polar root (math-div (nth 2 a) n))))) | |
491c3062 | 575 | (t nil))) |
136211a9 | 576 | |
86498823 JB |
577 | ;; The variables math-nrf-n, math-nrf-nf and math-nrf-nfm1 are local |
578 | ;; to math-nth-root-float, but are used by math-nth-root-float-iter, | |
579 | ;; which is called by math-nth-root-float. | |
580 | (defvar math-nrf-n) | |
581 | (defvar math-nrf-nf) | |
582 | (defvar math-nrf-nfm1) | |
583 | ||
584 | (defun math-nth-root-float (a math-nrf-n &optional guess) | |
136211a9 EZ |
585 | (math-inexact-result) |
586 | (math-with-extra-prec 1 | |
86498823 JB |
587 | (let ((math-nrf-nf (math-float math-nrf-n)) |
588 | (math-nrf-nfm1 (math-float (1- math-nrf-n)))) | |
136211a9 EZ |
589 | (math-nth-root-float-iter a (or guess |
590 | (math-make-float | |
591 | 1 (/ (+ (math-numdigs (nth 1 a)) | |
592 | (nth 2 a) | |
86498823 JB |
593 | (/ math-nrf-n 2)) |
594 | math-nrf-n))))))) | |
136211a9 | 595 | |
86498823 | 596 | (defun math-nth-root-float-iter (a guess) |
136211a9 | 597 | (math-working "root" guess) |
86498823 | 598 | (let ((g2 (math-div-float (math-add-float (math-mul math-nrf-nfm1 guess) |
136211a9 | 599 | (math-div-float |
86498823 JB |
600 | a (math-ipow guess (1- math-nrf-n)))) |
601 | math-nrf-nf))) | |
136211a9 EZ |
602 | (if (math-nearly-equal-float g2 guess) |
603 | g2 | |
491c3062 | 604 | (math-nth-root-float-iter a g2)))) |
136211a9 | 605 | |
86498823 JB |
606 | ;; The variable math-nri-n is local to math-nth-root-integer, but |
607 | ;; is used by math-nth-root-int-iter, which is called by | |
608 | ;; math-nth-root-int. | |
609 | (defvar math-nri-n) | |
610 | ||
611 | (defun math-nth-root-integer (a math-nri-n &optional guess) ; [I I S] | |
136211a9 EZ |
612 | (math-nth-root-int-iter a (or guess |
613 | (math-scale-int 1 (/ (+ (math-numdigs a) | |
86498823 JB |
614 | (1- math-nri-n)) |
615 | math-nri-n))))) | |
136211a9 | 616 | |
86498823 | 617 | (defun math-nth-root-int-iter (a guess) |
136211a9 | 618 | (math-working "root" guess) |
86498823 JB |
619 | (let* ((q (math-idivmod a (math-ipow guess (1- math-nri-n)))) |
620 | (s (math-add (car q) (math-mul (1- math-nri-n) guess))) | |
621 | (g2 (math-idivmod s math-nri-n))) | |
136211a9 EZ |
622 | (if (Math-natnum-lessp (car g2) guess) |
623 | (math-nth-root-int-iter a (car g2)) | |
624 | (cons (and (equal (car g2) guess) | |
625 | (eq (cdr q) 0) | |
626 | (eq (cdr g2) 0)) | |
491c3062 | 627 | guess)))) |
136211a9 EZ |
628 | |
629 | (defun calcFunc-nroot (x n) | |
630 | (calcFunc-pow x (if (integerp n) | |
631 | (math-make-frac 1 n) | |
491c3062 | 632 | (math-div 1 n)))) |
136211a9 EZ |
633 | |
634 | ||
635 | ||
636 | ||
637 | ;;;; Transcendental functions. | |
638 | ||
639 | ;;; All of these functions are defined on the complex plane. | |
640 | ;;; (Branch cuts, etc. follow Steele's Common Lisp book.) | |
641 | ||
642 | ;;; Most functions increase calc-internal-prec by 2 digits, then round | |
643 | ;;; down afterward. "-raw" functions use the current precision, require | |
644 | ;;; their arguments to be in float (or complex float) format, and always | |
645 | ;;; work in radians (where applicable). | |
646 | ||
647 | (defun math-to-radians (a) ; [N N] | |
648 | (cond ((eq (car-safe a) 'hms) | |
649 | (math-from-hms a 'rad)) | |
650 | ((memq calc-angle-mode '(deg hms)) | |
651 | (math-mul a (math-pi-over-180))) | |
491c3062 | 652 | (t a))) |
136211a9 EZ |
653 | |
654 | (defun math-from-radians (a) ; [N N] | |
655 | (cond ((eq calc-angle-mode 'deg) | |
656 | (if (math-constp a) | |
657 | (math-div a (math-pi-over-180)) | |
658 | (list 'calcFunc-deg a))) | |
659 | ((eq calc-angle-mode 'hms) | |
660 | (math-to-hms a 'rad)) | |
491c3062 | 661 | (t a))) |
136211a9 EZ |
662 | |
663 | (defun math-to-radians-2 (a) ; [N N] | |
664 | (cond ((eq (car-safe a) 'hms) | |
665 | (math-from-hms a 'rad)) | |
666 | ((memq calc-angle-mode '(deg hms)) | |
667 | (if calc-symbolic-mode | |
668 | (math-div (math-mul a '(var pi var-pi)) 180) | |
669 | (math-mul a (math-pi-over-180)))) | |
491c3062 | 670 | (t a))) |
136211a9 EZ |
671 | |
672 | (defun math-from-radians-2 (a) ; [N N] | |
673 | (cond ((memq calc-angle-mode '(deg hms)) | |
674 | (if calc-symbolic-mode | |
675 | (math-div (math-mul 180 a) '(var pi var-pi)) | |
676 | (math-div a (math-pi-over-180)))) | |
491c3062 | 677 | (t a))) |
136211a9 EZ |
678 | |
679 | ||
680 | ||
681 | ;;; Sine, cosine, and tangent. | |
682 | ||
683 | (defun calcFunc-sin (x) ; [N N] [Public] | |
684 | (cond ((and (integerp x) | |
685 | (if (eq calc-angle-mode 'deg) | |
686 | (= (% x 90) 0) | |
687 | (= x 0))) | |
688 | (aref [0 1 0 -1] (math-mod (/ x 90) 4))) | |
689 | ((Math-scalarp x) | |
690 | (math-with-extra-prec 2 | |
691 | (math-sin-raw (math-to-radians (math-float x))))) | |
692 | ((eq (car x) 'sdev) | |
693 | (if (math-constp x) | |
694 | (math-with-extra-prec 2 | |
695 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
696 | (xs (math-to-radians (math-float (nth 2 x)))) | |
697 | (sc (math-sin-cos-raw xx))) | |
698 | (math-make-sdev (car sc) (math-mul xs (cdr sc))))) | |
699 | (math-make-sdev (calcFunc-sin (nth 1 x)) | |
700 | (math-mul (nth 2 x) (calcFunc-cos (nth 1 x)))))) | |
701 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
702 | (calcFunc-cos (math-sub x (math-quarter-circle nil)))) | |
703 | ((equal x '(var nan var-nan)) | |
704 | x) | |
705 | (t (calc-record-why 'scalarp x) | |
491c3062 | 706 | (list 'calcFunc-sin x)))) |
136211a9 EZ |
707 | |
708 | (defun calcFunc-cos (x) ; [N N] [Public] | |
709 | (cond ((and (integerp x) | |
710 | (if (eq calc-angle-mode 'deg) | |
711 | (= (% x 90) 0) | |
712 | (= x 0))) | |
713 | (aref [1 0 -1 0] (math-mod (/ x 90) 4))) | |
714 | ((Math-scalarp x) | |
715 | (math-with-extra-prec 2 | |
716 | (math-cos-raw (math-to-radians (math-float x))))) | |
717 | ((eq (car x) 'sdev) | |
718 | (if (math-constp x) | |
719 | (math-with-extra-prec 2 | |
720 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
721 | (xs (math-to-radians (math-float (nth 2 x)))) | |
722 | (sc (math-sin-cos-raw xx))) | |
723 | (math-make-sdev (cdr sc) (math-mul xs (car sc))))) | |
724 | (math-make-sdev (calcFunc-cos (nth 1 x)) | |
725 | (math-mul (nth 2 x) (calcFunc-sin (nth 1 x)))))) | |
726 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
727 | (math-with-extra-prec 2 | |
728 | (let* ((xx (math-to-radians (math-float x))) | |
729 | (na (math-floor (math-div (nth 2 xx) (math-pi)))) | |
730 | (nb (math-floor (math-div (nth 3 xx) (math-pi)))) | |
731 | (span (math-sub nb na))) | |
732 | (if (memq span '(0 1)) | |
733 | (let ((int (math-sort-intv (nth 1 x) | |
734 | (math-cos-raw (nth 2 xx)) | |
735 | (math-cos-raw (nth 3 xx))))) | |
736 | (if (eq span 1) | |
737 | (if (math-evenp na) | |
738 | (math-make-intv (logior (nth 1 x) 2) | |
739 | -1 | |
740 | (nth 3 int)) | |
741 | (math-make-intv (logior (nth 1 x) 1) | |
742 | (nth 2 int) | |
743 | 1)) | |
744 | int)) | |
745 | (list 'intv 3 -1 1))))) | |
746 | ((equal x '(var nan var-nan)) | |
747 | x) | |
748 | (t (calc-record-why 'scalarp x) | |
491c3062 | 749 | (list 'calcFunc-cos x)))) |
136211a9 EZ |
750 | |
751 | (defun calcFunc-sincos (x) ; [V N] [Public] | |
752 | (if (Math-scalarp x) | |
753 | (math-with-extra-prec 2 | |
754 | (let ((sc (math-sin-cos-raw (math-to-radians (math-float x))))) | |
755 | (list 'vec (cdr sc) (car sc)))) ; the vector [cos, sin] | |
491c3062 | 756 | (list 'vec (calcFunc-sin x) (calcFunc-cos x)))) |
136211a9 EZ |
757 | |
758 | (defun calcFunc-tan (x) ; [N N] [Public] | |
759 | (cond ((and (integerp x) | |
760 | (if (eq calc-angle-mode 'deg) | |
761 | (= (% x 180) 0) | |
762 | (= x 0))) | |
763 | 0) | |
764 | ((Math-scalarp x) | |
765 | (math-with-extra-prec 2 | |
766 | (math-tan-raw (math-to-radians (math-float x))))) | |
767 | ((eq (car x) 'sdev) | |
768 | (if (math-constp x) | |
769 | (math-with-extra-prec 2 | |
770 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
771 | (xs (math-to-radians (math-float (nth 2 x)))) | |
772 | (sc (math-sin-cos-raw xx))) | |
773 | (if (and (math-zerop (cdr sc)) (not calc-infinite-mode)) | |
774 | (progn | |
775 | (calc-record-why "*Division by zero") | |
776 | (list 'calcFunc-tan x)) | |
777 | (math-make-sdev (math-div-float (car sc) (cdr sc)) | |
778 | (math-div-float xs (math-sqr (cdr sc))))))) | |
779 | (math-make-sdev (calcFunc-tan (nth 1 x)) | |
780 | (math-div (nth 2 x) | |
781 | (math-sqr (calcFunc-cos (nth 1 x))))))) | |
782 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
783 | (or (math-with-extra-prec 2 | |
784 | (let* ((xx (math-to-radians (math-float x))) | |
785 | (na (math-floor (math-div (math-sub (nth 2 xx) | |
786 | (math-pi-over-2)) | |
787 | (math-pi)))) | |
788 | (nb (math-floor (math-div (math-sub (nth 3 xx) | |
789 | (math-pi-over-2)) | |
790 | (math-pi))))) | |
791 | (and (equal na nb) | |
792 | (math-sort-intv (nth 1 x) | |
793 | (math-tan-raw (nth 2 xx)) | |
794 | (math-tan-raw (nth 3 xx)))))) | |
795 | '(intv 3 (neg (var inf var-inf)) (var inf var-inf)))) | |
796 | ((equal x '(var nan var-nan)) | |
797 | x) | |
798 | (t (calc-record-why 'scalarp x) | |
491c3062 | 799 | (list 'calcFunc-tan x)))) |
136211a9 EZ |
800 | |
801 | (defun math-sin-raw (x) ; [N N] | |
802 | (cond ((eq (car x) 'cplx) | |
803 | (let* ((expx (math-exp-raw (nth 2 x))) | |
804 | (expmx (math-div-float '(float 1 0) expx)) | |
805 | (sc (math-sin-cos-raw (nth 1 x)))) | |
806 | (list 'cplx | |
807 | (math-mul-float (car sc) | |
808 | (math-mul-float (math-add-float expx expmx) | |
809 | '(float 5 -1))) | |
810 | (math-mul-float (cdr sc) | |
811 | (math-mul-float (math-sub-float expx expmx) | |
812 | '(float 5 -1)))))) | |
813 | ((eq (car x) 'polar) | |
814 | (math-polar (math-sin-raw (math-complex x)))) | |
815 | ((Math-integer-negp (nth 1 x)) | |
816 | (math-neg-float (math-sin-raw (math-neg-float x)))) | |
817 | ((math-lessp-float '(float 7 0) x) ; avoid inf loops due to roundoff | |
818 | (math-sin-raw (math-mod x (math-two-pi)))) | |
491c3062 | 819 | (t (math-sin-raw-2 x x)))) |
136211a9 EZ |
820 | |
821 | (defun math-cos-raw (x) ; [N N] | |
822 | (if (eq (car-safe x) 'polar) | |
823 | (math-polar (math-cos-raw (math-complex x))) | |
491c3062 | 824 | (math-sin-raw (math-sub (math-pi-over-2) x)))) |
136211a9 EZ |
825 | |
826 | ;;; This could use a smarter method: Reduce x as in math-sin-raw, then | |
827 | ;;; compute either sin(x) or cos(x), whichever is smaller, and compute | |
828 | ;;; the other using the identity sin(x)^2 + cos(x)^2 = 1. | |
829 | (defun math-sin-cos-raw (x) ; [F.F F] (result is (sin x . cos x)) | |
491c3062 | 830 | (cons (math-sin-raw x) (math-cos-raw x))) |
136211a9 EZ |
831 | |
832 | (defun math-tan-raw (x) ; [N N] | |
833 | (cond ((eq (car x) 'cplx) | |
834 | (let* ((x (math-mul x '(float 2 0))) | |
835 | (expx (math-exp-raw (nth 2 x))) | |
836 | (expmx (math-div-float '(float 1 0) expx)) | |
837 | (sc (math-sin-cos-raw (nth 1 x))) | |
838 | (d (math-add-float (cdr sc) | |
839 | (math-mul-float (math-add-float expx expmx) | |
840 | '(float 5 -1))))) | |
841 | (and (not (eq (nth 1 d) 0)) | |
842 | (list 'cplx | |
843 | (math-div-float (car sc) d) | |
844 | (math-div-float (math-mul-float (math-sub-float expx | |
845 | expmx) | |
846 | '(float 5 -1)) d))))) | |
847 | ((eq (car x) 'polar) | |
848 | (math-polar (math-tan-raw (math-complex x)))) | |
849 | (t | |
850 | (let ((sc (math-sin-cos-raw x))) | |
851 | (if (eq (nth 1 (cdr sc)) 0) | |
852 | (math-div (car sc) 0) | |
491c3062 | 853 | (math-div-float (car sc) (cdr sc))))))) |
136211a9 EZ |
854 | |
855 | (defun math-sin-raw-2 (x orgx) ; This avoids poss of inf recursion. [F F] | |
856 | (let ((xmpo2 (math-sub-float (math-pi-over-2) x))) | |
857 | (cond ((Math-integer-negp (nth 1 xmpo2)) | |
858 | (math-neg-float (math-sin-raw-2 (math-sub-float x (math-pi)) | |
859 | orgx))) | |
860 | ((math-lessp-float (math-pi-over-4) x) | |
861 | (math-cos-raw-2 xmpo2 orgx)) | |
862 | ((math-lessp-float x (math-neg (math-pi-over-4))) | |
863 | (math-neg (math-cos-raw-2 (math-add (math-pi-over-2) x) orgx))) | |
864 | ((math-nearly-zerop-float x orgx) '(float 0 0)) | |
865 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
491c3062 | 866 | (t (math-sin-series x 6 4 x (math-neg-float (math-sqr-float x))))))) |
136211a9 EZ |
867 | |
868 | (defun math-cos-raw-2 (x orgx) ; [F F] | |
869 | (cond ((math-nearly-zerop-float x orgx) '(float 1 0)) | |
870 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
871 | (t (let ((xnegsqr (math-neg-float (math-sqr-float x)))) | |
872 | (math-sin-series | |
873 | (math-add-float '(float 1 0) | |
874 | (math-mul-float xnegsqr '(float 5 -1))) | |
491c3062 | 875 | 24 5 xnegsqr xnegsqr))))) |
136211a9 EZ |
876 | |
877 | (defun math-sin-series (sum nfac n x xnegsqr) | |
878 | (math-working "sin" sum) | |
879 | (let* ((nextx (math-mul-float x xnegsqr)) | |
880 | (nextsum (math-add-float sum (math-div-float nextx | |
881 | (math-float nfac))))) | |
882 | (if (math-nearly-equal-float sum nextsum) | |
883 | sum | |
884 | (math-sin-series nextsum (math-mul nfac (* n (1+ n))) | |
491c3062 | 885 | (+ n 2) nextx xnegsqr)))) |
136211a9 EZ |
886 | |
887 | ||
888 | ;;; Inverse sine, cosine, tangent. | |
889 | ||
890 | (defun calcFunc-arcsin (x) ; [N N] [Public] | |
891 | (cond ((eq x 0) 0) | |
892 | ((and (eq x 1) (eq calc-angle-mode 'deg)) 90) | |
893 | ((and (eq x -1) (eq calc-angle-mode 'deg)) -90) | |
894 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
895 | ((Math-numberp x) | |
896 | (math-with-extra-prec 2 | |
897 | (math-from-radians (math-arcsin-raw (math-float x))))) | |
898 | ((eq (car x) 'sdev) | |
899 | (math-make-sdev (calcFunc-arcsin (nth 1 x)) | |
900 | (math-from-radians | |
901 | (math-div (nth 2 x) | |
902 | (math-sqrt | |
903 | (math-sub 1 (math-sqr (nth 1 x)))))))) | |
904 | ((eq (car x) 'intv) | |
905 | (math-sort-intv (nth 1 x) | |
906 | (calcFunc-arcsin (nth 2 x)) | |
907 | (calcFunc-arcsin (nth 3 x)))) | |
908 | ((equal x '(var nan var-nan)) | |
909 | x) | |
910 | (t (calc-record-why 'numberp x) | |
491c3062 | 911 | (list 'calcFunc-arcsin x)))) |
136211a9 EZ |
912 | |
913 | (defun calcFunc-arccos (x) ; [N N] [Public] | |
914 | (cond ((eq x 1) 0) | |
915 | ((and (eq x 0) (eq calc-angle-mode 'deg)) 90) | |
916 | ((and (eq x -1) (eq calc-angle-mode 'deg)) 180) | |
917 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
918 | ((Math-numberp x) | |
919 | (math-with-extra-prec 2 | |
920 | (math-from-radians (math-arccos-raw (math-float x))))) | |
921 | ((eq (car x) 'sdev) | |
922 | (math-make-sdev (calcFunc-arccos (nth 1 x)) | |
923 | (math-from-radians | |
924 | (math-div (nth 2 x) | |
925 | (math-sqrt | |
926 | (math-sub 1 (math-sqr (nth 1 x)))))))) | |
927 | ((eq (car x) 'intv) | |
928 | (math-sort-intv (nth 1 x) | |
929 | (calcFunc-arccos (nth 2 x)) | |
930 | (calcFunc-arccos (nth 3 x)))) | |
931 | ((equal x '(var nan var-nan)) | |
932 | x) | |
933 | (t (calc-record-why 'numberp x) | |
491c3062 | 934 | (list 'calcFunc-arccos x)))) |
136211a9 EZ |
935 | |
936 | (defun calcFunc-arctan (x) ; [N N] [Public] | |
937 | (cond ((eq x 0) 0) | |
938 | ((and (eq x 1) (eq calc-angle-mode 'deg)) 45) | |
939 | ((and (eq x -1) (eq calc-angle-mode 'deg)) -45) | |
940 | ((Math-numberp x) | |
941 | (math-with-extra-prec 2 | |
942 | (math-from-radians (math-arctan-raw (math-float x))))) | |
943 | ((eq (car x) 'sdev) | |
944 | (math-make-sdev (calcFunc-arctan (nth 1 x)) | |
945 | (math-from-radians | |
946 | (math-div (nth 2 x) | |
947 | (math-add 1 (math-sqr (nth 1 x))))))) | |
948 | ((eq (car x) 'intv) | |
949 | (math-sort-intv (nth 1 x) | |
950 | (calcFunc-arctan (nth 2 x)) | |
951 | (calcFunc-arctan (nth 3 x)))) | |
952 | ((equal x '(var inf var-inf)) | |
953 | (math-quarter-circle t)) | |
954 | ((equal x '(neg (var inf var-inf))) | |
955 | (math-neg (math-quarter-circle t))) | |
956 | ((equal x '(var nan var-nan)) | |
957 | x) | |
958 | (t (calc-record-why 'numberp x) | |
491c3062 | 959 | (list 'calcFunc-arctan x)))) |
136211a9 EZ |
960 | |
961 | (defun math-arcsin-raw (x) ; [N N] | |
962 | (let ((a (math-sqrt-raw (math-sub '(float 1 0) (math-sqr x))))) | |
963 | (if (or (memq (car x) '(cplx polar)) | |
964 | (memq (car a) '(cplx polar))) | |
965 | (math-with-extra-prec 2 ; use extra precision for difficult case | |
966 | (math-mul '(cplx 0 -1) | |
967 | (math-ln-raw (math-add (math-mul '(cplx 0 1) x) a)))) | |
491c3062 | 968 | (math-arctan2-raw x a)))) |
136211a9 EZ |
969 | |
970 | (defun math-arccos-raw (x) ; [N N] | |
491c3062 | 971 | (math-sub (math-pi-over-2) (math-arcsin-raw x))) |
136211a9 EZ |
972 | |
973 | (defun math-arctan-raw (x) ; [N N] | |
974 | (cond ((memq (car x) '(cplx polar)) | |
975 | (math-with-extra-prec 2 ; extra-extra | |
976 | (math-div (math-sub | |
977 | (math-ln-raw (math-add 1 (math-mul '(cplx 0 1) x))) | |
978 | (math-ln-raw (math-add 1 (math-mul '(cplx 0 -1) x)))) | |
979 | '(cplx 0 2)))) | |
980 | ((Math-integer-negp (nth 1 x)) | |
981 | (math-neg-float (math-arctan-raw (math-neg-float x)))) | |
982 | ((math-zerop x) x) | |
983 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
984 | ((math-equal-int x 1) (math-pi-over-4)) | |
985 | ((math-equal-int x -1) (math-neg (math-pi-over-4))) | |
986 | ((math-lessp-float '(float 414214 -6) x) ; if x > sqrt(2) - 1, reduce | |
987 | (if (math-lessp-float '(float 1 0) x) | |
988 | (math-sub-float (math-mul-float (math-pi) '(float 5 -1)) | |
989 | (math-arctan-raw (math-div-float '(float 1 0) x))) | |
990 | (math-sub-float (math-mul-float (math-pi) '(float 25 -2)) | |
991 | (math-arctan-raw (math-div-float | |
992 | (math-sub-float '(float 1 0) x) | |
993 | (math-add-float '(float 1 0) | |
994 | x)))))) | |
491c3062 | 995 | (t (math-arctan-series x 3 x (math-neg-float (math-sqr-float x)))))) |
136211a9 EZ |
996 | |
997 | (defun math-arctan-series (sum n x xnegsqr) | |
998 | (math-working "arctan" sum) | |
999 | (let* ((nextx (math-mul-float x xnegsqr)) | |
1000 | (nextsum (math-add-float sum (math-div-float nextx (math-float n))))) | |
1001 | (if (math-nearly-equal-float sum nextsum) | |
1002 | sum | |
491c3062 | 1003 | (math-arctan-series nextsum (+ n 2) nextx xnegsqr)))) |
136211a9 EZ |
1004 | |
1005 | (defun calcFunc-arctan2 (y x) ; [F R R] [Public] | |
1006 | (if (Math-anglep y) | |
1007 | (if (Math-anglep x) | |
1008 | (math-with-extra-prec 2 | |
1009 | (math-from-radians (math-arctan2-raw (math-float y) | |
1010 | (math-float x)))) | |
1011 | (calc-record-why 'anglep x) | |
1012 | (list 'calcFunc-arctan2 y x)) | |
1013 | (if (and (or (math-infinitep x) (math-anglep x)) | |
1014 | (or (math-infinitep y) (math-anglep y))) | |
1015 | (progn | |
1016 | (if (math-posp x) | |
1017 | (setq x 1) | |
1018 | (if (math-negp x) | |
1019 | (setq x -1) | |
1020 | (or (math-zerop x) | |
1021 | (setq x nil)))) | |
1022 | (if (math-posp y) | |
1023 | (setq y 1) | |
1024 | (if (math-negp y) | |
1025 | (setq y -1) | |
1026 | (or (math-zerop y) | |
1027 | (setq y nil)))) | |
1028 | (if (and y x) | |
1029 | (calcFunc-arctan2 y x) | |
1030 | '(var nan var-nan))) | |
1031 | (calc-record-why 'anglep y) | |
491c3062 | 1032 | (list 'calcFunc-arctan2 y x)))) |
136211a9 EZ |
1033 | |
1034 | (defun math-arctan2-raw (y x) ; [F R R] | |
1035 | (cond ((math-zerop y) | |
1036 | (if (math-negp x) (math-pi) | |
1037 | (if (or (math-floatp x) (math-floatp y)) '(float 0 0) 0))) | |
1038 | ((math-zerop x) | |
1039 | (if (math-posp y) | |
1040 | (math-pi-over-2) | |
1041 | (math-neg (math-pi-over-2)))) | |
1042 | ((math-posp x) | |
1043 | (math-arctan-raw (math-div-float y x))) | |
1044 | ((math-posp y) | |
1045 | (math-add-float (math-arctan-raw (math-div-float y x)) | |
1046 | (math-pi))) | |
1047 | (t | |
1048 | (math-sub-float (math-arctan-raw (math-div-float y x)) | |
491c3062 | 1049 | (math-pi))))) |
136211a9 EZ |
1050 | |
1051 | (defun calcFunc-arcsincos (x) ; [V N] [Public] | |
1052 | (if (and (Math-vectorp x) | |
1053 | (= (length x) 3)) | |
1054 | (calcFunc-arctan2 (nth 2 x) (nth 1 x)) | |
491c3062 | 1055 | (math-reject-arg x "*Two-element vector expected"))) |
136211a9 EZ |
1056 | |
1057 | ||
1058 | ||
1059 | ;;; Exponential function. | |
1060 | ||
1061 | (defun calcFunc-exp (x) ; [N N] [Public] | |
1062 | (cond ((eq x 0) 1) | |
1063 | ((and (memq x '(1 -1)) calc-symbolic-mode) | |
1064 | (if (eq x 1) '(var e var-e) (math-div 1 '(var e var-e)))) | |
1065 | ((Math-numberp x) | |
1066 | (math-with-extra-prec 2 (math-exp-raw (math-float x)))) | |
1067 | ((eq (car-safe x) 'sdev) | |
1068 | (let ((ex (calcFunc-exp (nth 1 x)))) | |
1069 | (math-make-sdev ex (math-mul (nth 2 x) ex)))) | |
1070 | ((eq (car-safe x) 'intv) | |
1071 | (math-make-intv (nth 1 x) (calcFunc-exp (nth 2 x)) | |
1072 | (calcFunc-exp (nth 3 x)))) | |
1073 | ((equal x '(var inf var-inf)) | |
1074 | x) | |
1075 | ((equal x '(neg (var inf var-inf))) | |
1076 | 0) | |
1077 | ((equal x '(var nan var-nan)) | |
1078 | x) | |
1079 | (t (calc-record-why 'numberp x) | |
491c3062 | 1080 | (list 'calcFunc-exp x)))) |
136211a9 EZ |
1081 | |
1082 | (defun calcFunc-expm1 (x) ; [N N] [Public] | |
1083 | (cond ((eq x 0) 0) | |
1084 | ((math-zerop x) '(float 0 0)) | |
1085 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1086 | ((Math-numberp x) | |
1087 | (math-with-extra-prec 2 | |
1088 | (let ((x (math-float x))) | |
1089 | (if (and (eq (car x) 'float) | |
1090 | (math-lessp-float x '(float 1 0)) | |
1091 | (math-lessp-float '(float -1 0) x)) | |
1092 | (math-exp-minus-1-raw x) | |
1093 | (math-add (math-exp-raw x) -1))))) | |
1094 | ((eq (car-safe x) 'sdev) | |
1095 | (if (math-constp x) | |
1096 | (let ((ex (calcFunc-expm1 (nth 1 x)))) | |
1097 | (math-make-sdev ex (math-mul (nth 2 x) (math-add ex 1)))) | |
1098 | (math-make-sdev (calcFunc-expm1 (nth 1 x)) | |
1099 | (math-mul (nth 2 x) (calcFunc-exp (nth 1 x)))))) | |
1100 | ((eq (car-safe x) 'intv) | |
1101 | (math-make-intv (nth 1 x) | |
1102 | (calcFunc-expm1 (nth 2 x)) | |
1103 | (calcFunc-expm1 (nth 3 x)))) | |
1104 | ((equal x '(var inf var-inf)) | |
1105 | x) | |
1106 | ((equal x '(neg (var inf var-inf))) | |
1107 | -1) | |
1108 | ((equal x '(var nan var-nan)) | |
1109 | x) | |
1110 | (t (calc-record-why 'numberp x) | |
491c3062 | 1111 | (list 'calcFunc-expm1 x)))) |
136211a9 EZ |
1112 | |
1113 | (defun calcFunc-exp10 (x) ; [N N] [Public] | |
1114 | (if (eq x 0) | |
1115 | 1 | |
491c3062 | 1116 | (math-pow '(float 1 1) x))) |
136211a9 EZ |
1117 | |
1118 | (defun math-exp-raw (x) ; [N N] | |
1119 | (cond ((math-zerop x) '(float 1 0)) | |
1120 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1121 | ((eq (car x) 'cplx) | |
1122 | (let ((expx (math-exp-raw (nth 1 x))) | |
1123 | (sc (math-sin-cos-raw (nth 2 x)))) | |
1124 | (list 'cplx | |
1125 | (math-mul-float expx (cdr sc)) | |
1126 | (math-mul-float expx (car sc))))) | |
1127 | ((eq (car x) 'polar) | |
1128 | (let ((xc (math-complex x))) | |
1129 | (list 'polar | |
1130 | (math-exp-raw (nth 1 xc)) | |
1131 | (math-from-radians (nth 2 xc))))) | |
1132 | ((or (math-lessp-float '(float 5 -1) x) | |
1133 | (math-lessp-float x '(float -5 -1))) | |
1134 | (if (math-lessp-float '(float 921035 1) x) | |
1135 | (math-overflow) | |
1136 | (if (math-lessp-float x '(float -921035 1)) | |
1137 | (math-underflow))) | |
1138 | (let* ((two-x (math-mul-float x '(float 2 0))) | |
1139 | (hint (math-scale-int (nth 1 two-x) (nth 2 two-x))) | |
1140 | (hfrac (math-sub-float x (math-mul-float (math-float hint) | |
1141 | '(float 5 -1))))) | |
1142 | (math-mul-float (math-ipow (math-sqrt-e) hint) | |
1143 | (math-add-float '(float 1 0) | |
1144 | (math-exp-minus-1-raw hfrac))))) | |
491c3062 | 1145 | (t (math-add-float '(float 1 0) (math-exp-minus-1-raw x))))) |
136211a9 EZ |
1146 | |
1147 | (defun math-exp-minus-1-raw (x) ; [F F] | |
491c3062 | 1148 | (math-exp-series x 2 3 x x)) |
136211a9 EZ |
1149 | |
1150 | (defun math-exp-series (sum nfac n xpow x) | |
1151 | (math-working "exp" sum) | |
1152 | (let* ((nextx (math-mul-float xpow x)) | |
1153 | (nextsum (math-add-float sum (math-div-float nextx | |
1154 | (math-float nfac))))) | |
1155 | (if (math-nearly-equal-float sum nextsum) | |
1156 | sum | |
491c3062 | 1157 | (math-exp-series nextsum (math-mul nfac n) (1+ n) nextx x)))) |
136211a9 EZ |
1158 | |
1159 | ||
1160 | ||
1161 | ;;; Logarithms. | |
1162 | ||
1163 | (defun calcFunc-ln (x) ; [N N] [Public] | |
1164 | (cond ((math-zerop x) | |
1165 | (if calc-infinite-mode | |
1166 | '(neg (var inf var-inf)) | |
1167 | (math-reject-arg x "*Logarithm of zero"))) | |
1168 | ((eq x 1) 0) | |
1169 | ((Math-numberp x) | |
1170 | (math-with-extra-prec 2 (math-ln-raw (math-float x)))) | |
1171 | ((eq (car-safe x) 'sdev) | |
1172 | (math-make-sdev (calcFunc-ln (nth 1 x)) | |
1173 | (math-div (nth 2 x) (nth 1 x)))) | |
1174 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1175 | (Math-zerop (nth 2 x)) | |
1176 | (not (math-intv-constp x)))) | |
1177 | (let ((calc-infinite-mode t)) | |
1178 | (math-make-intv (nth 1 x) (calcFunc-ln (nth 2 x)) | |
1179 | (calcFunc-ln (nth 3 x))))) | |
1180 | ((equal x '(var e var-e)) | |
1181 | 1) | |
1182 | ((and (eq (car-safe x) '^) | |
1183 | (equal (nth 1 x) '(var e var-e)) | |
1184 | (math-known-realp (nth 2 x))) | |
1185 | (nth 2 x)) | |
1186 | ((math-infinitep x) | |
1187 | (if (equal x '(var nan var-nan)) | |
1188 | x | |
1189 | '(var inf var-inf))) | |
1190 | (t (calc-record-why 'numberp x) | |
491c3062 | 1191 | (list 'calcFunc-ln x)))) |
136211a9 EZ |
1192 | |
1193 | (defun calcFunc-log10 (x) ; [N N] [Public] | |
1194 | (cond ((math-equal-int x 1) | |
1195 | (if (math-floatp x) '(float 0 0) 0)) | |
1196 | ((and (Math-integerp x) | |
1197 | (math-posp x) | |
1198 | (let ((res (math-integer-log x 10))) | |
1199 | (and (car res) | |
1200 | (setq x (cdr res))))) | |
1201 | x) | |
1202 | ((and (eq (car-safe x) 'frac) | |
1203 | (eq (nth 1 x) 1) | |
1204 | (let ((res (math-integer-log (nth 2 x) 10))) | |
1205 | (and (car res) | |
1206 | (setq x (- (cdr res)))))) | |
1207 | x) | |
1208 | ((math-zerop x) | |
1209 | (if calc-infinite-mode | |
1210 | '(neg (var inf var-inf)) | |
1211 | (math-reject-arg x "*Logarithm of zero"))) | |
1212 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1213 | ((Math-numberp x) | |
1214 | (math-with-extra-prec 2 | |
1215 | (let ((xf (math-float x))) | |
1216 | (if (eq (nth 1 xf) 0) | |
1217 | (math-reject-arg x "*Logarithm of zero")) | |
1218 | (if (Math-integer-posp (nth 1 xf)) | |
1219 | (if (eq (nth 1 xf) 1) ; log10(1*10^n) = n | |
1220 | (math-float (nth 2 xf)) | |
1221 | (let ((xdigs (1- (math-numdigs (nth 1 xf))))) | |
1222 | (math-add-float | |
1223 | (math-div-float (math-ln-raw-2 | |
1224 | (list 'float (nth 1 xf) (- xdigs))) | |
1225 | (math-ln-10)) | |
1226 | (math-float (+ (nth 2 xf) xdigs))))) | |
1227 | (math-div (calcFunc-ln xf) (math-ln-10)))))) | |
1228 | ((eq (car-safe x) 'sdev) | |
1229 | (math-make-sdev (calcFunc-log10 (nth 1 x)) | |
1230 | (math-div (nth 2 x) | |
1231 | (math-mul (nth 1 x) (math-ln-10))))) | |
1232 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1233 | (not (math-intv-constp x)))) | |
1234 | (math-make-intv (nth 1 x) | |
1235 | (calcFunc-log10 (nth 2 x)) | |
1236 | (calcFunc-log10 (nth 3 x)))) | |
1237 | ((math-infinitep x) | |
1238 | (if (equal x '(var nan var-nan)) | |
1239 | x | |
1240 | '(var inf var-inf))) | |
1241 | (t (calc-record-why 'numberp x) | |
491c3062 | 1242 | (list 'calcFunc-log10 x)))) |
136211a9 EZ |
1243 | |
1244 | (defun calcFunc-log (x &optional b) ; [N N N] [Public] | |
1245 | (cond ((or (null b) (equal b '(var e var-e))) | |
1246 | (math-normalize (list 'calcFunc-ln x))) | |
1247 | ((or (eq b 10) (equal b '(float 1 1))) | |
1248 | (math-normalize (list 'calcFunc-log10 x))) | |
1249 | ((math-zerop x) | |
1250 | (if calc-infinite-mode | |
1251 | (math-div (calcFunc-ln x) (calcFunc-ln b)) | |
1252 | (math-reject-arg x "*Logarithm of zero"))) | |
1253 | ((math-zerop b) | |
1254 | (if calc-infinite-mode | |
1255 | (math-div (calcFunc-ln x) (calcFunc-ln b)) | |
1256 | (math-reject-arg b "*Logarithm of zero"))) | |
1257 | ((math-equal-int b 1) | |
1258 | (if calc-infinite-mode | |
1259 | (math-div (calcFunc-ln x) 0) | |
1260 | (math-reject-arg b "*Logarithm base one"))) | |
1261 | ((math-equal-int x 1) | |
86498823 | 1262 | (if (math-floatp b) '(float 0 0) 0)) |
136211a9 EZ |
1263 | ((and (Math-ratp x) (Math-ratp b) |
1264 | (math-posp x) (math-posp b) | |
1265 | (let* ((sign 1) (inv nil) | |
1266 | (xx (if (Math-lessp 1 x) | |
1267 | x | |
1268 | (setq sign -1) | |
1269 | (math-div 1 x))) | |
1270 | (bb (if (Math-lessp 1 b) | |
1271 | b | |
1272 | (setq sign (- sign)) | |
1273 | (math-div 1 b))) | |
1274 | (res (if (Math-lessp xx bb) | |
1275 | (setq inv (math-integer-log bb xx)) | |
1276 | (math-integer-log xx bb)))) | |
1277 | (and (car res) | |
1278 | (setq x (if inv | |
1279 | (math-div 1 (* sign (cdr res))) | |
1280 | (* sign (cdr res))))))) | |
1281 | x) | |
1282 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1283 | ((and (Math-numberp x) (Math-numberp b)) | |
1284 | (math-with-extra-prec 2 | |
1285 | (math-div (math-ln-raw (math-float x)) | |
1286 | (math-log-base-raw b)))) | |
1287 | ((and (eq (car-safe x) 'sdev) | |
1288 | (Math-numberp b)) | |
1289 | (math-make-sdev (calcFunc-log (nth 1 x) b) | |
1290 | (math-div (nth 2 x) | |
1291 | (math-mul (nth 1 x) | |
1292 | (math-log-base-raw b))))) | |
1293 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1294 | (not (math-intv-constp x))) | |
1295 | (math-realp b)) | |
1296 | (math-make-intv (nth 1 x) | |
1297 | (calcFunc-log (nth 2 x) b) | |
1298 | (calcFunc-log (nth 3 x) b))) | |
1299 | ((or (eq (car-safe x) 'intv) (eq (car-safe b) 'intv)) | |
1300 | (math-div (calcFunc-ln x) (calcFunc-ln b))) | |
1301 | ((or (math-infinitep x) | |
1302 | (math-infinitep b)) | |
1303 | (math-div (calcFunc-ln x) (calcFunc-ln b))) | |
1304 | (t (if (Math-numberp b) | |
1305 | (calc-record-why 'numberp x) | |
1306 | (calc-record-why 'numberp b)) | |
491c3062 | 1307 | (list 'calcFunc-log x b)))) |
136211a9 EZ |
1308 | |
1309 | (defun calcFunc-alog (x &optional b) | |
1310 | (cond ((or (null b) (equal b '(var e var-e))) | |
1311 | (math-normalize (list 'calcFunc-exp x))) | |
491c3062 | 1312 | (t (math-pow b x)))) |
136211a9 EZ |
1313 | |
1314 | (defun calcFunc-ilog (x b) | |
1315 | (if (and (math-natnump x) (not (eq x 0)) | |
1316 | (math-natnump b) (not (eq b 0))) | |
1317 | (if (eq b 1) | |
1318 | (math-reject-arg x "*Logarithm base one") | |
1319 | (if (Math-natnum-lessp x b) | |
1320 | 0 | |
1321 | (cdr (math-integer-log x b)))) | |
491c3062 | 1322 | (math-floor (calcFunc-log x b)))) |
136211a9 EZ |
1323 | |
1324 | (defun math-integer-log (x b) | |
1325 | (let ((pows (list b)) | |
1326 | (pow (math-sqr b)) | |
1327 | next | |
1328 | sum n) | |
1329 | (while (not (Math-lessp x pow)) | |
1330 | (setq pows (cons pow pows) | |
1331 | pow (math-sqr pow))) | |
1332 | (setq n (lsh 1 (1- (length pows))) | |
1333 | sum n | |
1334 | pow (car pows)) | |
1335 | (while (and (setq pows (cdr pows)) | |
1336 | (Math-lessp pow x)) | |
1337 | (setq n (/ n 2) | |
1338 | next (math-mul pow (car pows))) | |
1339 | (or (Math-lessp x next) | |
1340 | (setq pow next | |
1341 | sum (+ sum n)))) | |
491c3062 | 1342 | (cons (equal pow x) sum))) |
136211a9 EZ |
1343 | |
1344 | ||
3132f345 | 1345 | (defvar math-log-base-cache nil) |
136211a9 EZ |
1346 | (defun math-log-base-raw (b) ; [N N] |
1347 | (if (not (and (equal (car math-log-base-cache) b) | |
1348 | (eq (nth 1 math-log-base-cache) calc-internal-prec))) | |
1349 | (setq math-log-base-cache (list b calc-internal-prec | |
1350 | (math-ln-raw (math-float b))))) | |
491c3062 | 1351 | (nth 2 math-log-base-cache)) |
136211a9 EZ |
1352 | |
1353 | (defun calcFunc-lnp1 (x) ; [N N] [Public] | |
1354 | (cond ((Math-equal-int x -1) | |
1355 | (if calc-infinite-mode | |
1356 | '(neg (var inf var-inf)) | |
1357 | (math-reject-arg x "*Logarithm of zero"))) | |
1358 | ((eq x 0) 0) | |
1359 | ((math-zerop x) '(float 0 0)) | |
1360 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1361 | ((Math-numberp x) | |
1362 | (math-with-extra-prec 2 | |
1363 | (let ((x (math-float x))) | |
1364 | (if (and (eq (car x) 'float) | |
1365 | (math-lessp-float x '(float 5 -1)) | |
1366 | (math-lessp-float '(float -5 -1) x)) | |
1367 | (math-ln-plus-1-raw x) | |
1368 | (math-ln-raw (math-add-float x '(float 1 0))))))) | |
1369 | ((eq (car-safe x) 'sdev) | |
1370 | (math-make-sdev (calcFunc-lnp1 (nth 1 x)) | |
1371 | (math-div (nth 2 x) (math-add (nth 1 x) 1)))) | |
1372 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1373 | (not (math-intv-constp x)))) | |
1374 | (math-make-intv (nth 1 x) | |
1375 | (calcFunc-lnp1 (nth 2 x)) | |
1376 | (calcFunc-lnp1 (nth 3 x)))) | |
1377 | ((math-infinitep x) | |
1378 | (if (equal x '(var nan var-nan)) | |
1379 | x | |
1380 | '(var inf var-inf))) | |
1381 | (t (calc-record-why 'numberp x) | |
491c3062 | 1382 | (list 'calcFunc-lnp1 x)))) |
136211a9 EZ |
1383 | |
1384 | (defun math-ln-raw (x) ; [N N] --- must be float format! | |
1385 | (cond ((eq (car-safe x) 'cplx) | |
1386 | (list 'cplx | |
1387 | (math-mul-float (math-ln-raw | |
1388 | (math-add-float (math-sqr-float (nth 1 x)) | |
1389 | (math-sqr-float (nth 2 x)))) | |
1390 | '(float 5 -1)) | |
1391 | (math-arctan2-raw (nth 2 x) (nth 1 x)))) | |
1392 | ((eq (car x) 'polar) | |
1393 | (math-polar (list 'cplx | |
1394 | (math-ln-raw (nth 1 x)) | |
1395 | (math-to-radians (nth 2 x))))) | |
1396 | ((Math-equal-int x 1) | |
1397 | '(float 0 0)) | |
1398 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1399 | ((math-posp (nth 1 x)) ; positive and real | |
1400 | (let ((xdigs (1- (math-numdigs (nth 1 x))))) | |
1401 | (math-add-float (math-ln-raw-2 (list 'float (nth 1 x) (- xdigs))) | |
1402 | (math-mul-float (math-float (+ (nth 2 x) xdigs)) | |
1403 | (math-ln-10))))) | |
1404 | ((math-zerop x) | |
1405 | (math-reject-arg x "*Logarithm of zero")) | |
1406 | ((eq calc-complex-mode 'polar) ; negative and real | |
1407 | (math-polar | |
1408 | (list 'cplx ; negative and real | |
1409 | (math-ln-raw (math-neg-float x)) | |
1410 | (math-pi)))) | |
1411 | (t (list 'cplx ; negative and real | |
1412 | (math-ln-raw (math-neg-float x)) | |
491c3062 | 1413 | (math-pi))))) |
136211a9 EZ |
1414 | |
1415 | (defun math-ln-raw-2 (x) ; [F F] | |
1416 | (cond ((math-lessp-float '(float 14 -1) x) | |
1417 | (math-add-float (math-ln-raw-2 (math-mul-float x '(float 5 -1))) | |
1418 | (math-ln-2))) | |
1419 | (t ; now .7 < x <= 1.4 | |
1420 | (math-ln-raw-3 (math-div-float (math-sub-float x '(float 1 0)) | |
491c3062 | 1421 | (math-add-float x '(float 1 0))))))) |
136211a9 EZ |
1422 | |
1423 | (defun math-ln-raw-3 (x) ; [F F] | |
1424 | (math-mul-float (math-ln-raw-series x 3 x (math-sqr-float x)) | |
491c3062 | 1425 | '(float 2 0))) |
136211a9 EZ |
1426 | |
1427 | ;;; Compute ln((1+x)/(1-x)) | |
1428 | (defun math-ln-raw-series (sum n x xsqr) | |
1429 | (math-working "log" sum) | |
1430 | (let* ((nextx (math-mul-float x xsqr)) | |
1431 | (nextsum (math-add-float sum (math-div-float nextx (math-float n))))) | |
1432 | (if (math-nearly-equal-float sum nextsum) | |
1433 | sum | |
491c3062 | 1434 | (math-ln-raw-series nextsum (+ n 2) nextx xsqr)))) |
136211a9 EZ |
1435 | |
1436 | (defun math-ln-plus-1-raw (x) | |
491c3062 | 1437 | (math-lnp1-series x 2 x (math-neg x))) |
136211a9 EZ |
1438 | |
1439 | (defun math-lnp1-series (sum n xpow x) | |
1440 | (math-working "lnp1" sum) | |
1441 | (let* ((nextx (math-mul-float xpow x)) | |
1442 | (nextsum (math-add-float sum (math-div-float nextx (math-float n))))) | |
1443 | (if (math-nearly-equal-float sum nextsum) | |
1444 | sum | |
491c3062 | 1445 | (math-lnp1-series nextsum (1+ n) nextx x)))) |
136211a9 EZ |
1446 | |
1447 | (math-defcache math-ln-10 (float (bigpos 018 684 045 994 092 585 302 2) -21) | |
1448 | (math-ln-raw-2 '(float 1 1))) | |
1449 | ||
1450 | (math-defcache math-ln-2 (float (bigpos 417 309 945 559 180 147 693) -21) | |
1451 | (math-ln-raw-3 (math-float '(frac 1 3)))) | |
1452 | ||
1453 | ||
1454 | ||
1455 | ;;; Hyperbolic functions. | |
1456 | ||
1457 | (defun calcFunc-sinh (x) ; [N N] [Public] | |
1458 | (cond ((eq x 0) 0) | |
1459 | (math-expand-formulas | |
1460 | (math-normalize | |
1461 | (list '/ (list '- (list 'calcFunc-exp x) | |
1462 | (list 'calcFunc-exp (list 'neg x))) 2))) | |
1463 | ((Math-numberp x) | |
1464 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1465 | (math-with-extra-prec 2 | |
1466 | (let ((expx (math-exp-raw (math-float x)))) | |
1467 | (math-mul (math-add expx (math-div -1 expx)) '(float 5 -1))))) | |
1468 | ((eq (car-safe x) 'sdev) | |
1469 | (math-make-sdev (calcFunc-sinh (nth 1 x)) | |
1470 | (math-mul (nth 2 x) (calcFunc-cosh (nth 1 x))))) | |
1471 | ((eq (car x) 'intv) | |
1472 | (math-sort-intv (nth 1 x) | |
1473 | (calcFunc-sinh (nth 2 x)) | |
1474 | (calcFunc-sinh (nth 3 x)))) | |
1475 | ((or (equal x '(var inf var-inf)) | |
1476 | (equal x '(neg (var inf var-inf))) | |
1477 | (equal x '(var nan var-nan))) | |
1478 | x) | |
1479 | (t (calc-record-why 'numberp x) | |
491c3062 | 1480 | (list 'calcFunc-sinh x)))) |
136211a9 EZ |
1481 | (put 'calcFunc-sinh 'math-expandable t) |
1482 | ||
1483 | (defun calcFunc-cosh (x) ; [N N] [Public] | |
1484 | (cond ((eq x 0) 1) | |
1485 | (math-expand-formulas | |
1486 | (math-normalize | |
1487 | (list '/ (list '+ (list 'calcFunc-exp x) | |
1488 | (list 'calcFunc-exp (list 'neg x))) 2))) | |
1489 | ((Math-numberp x) | |
1490 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1491 | (math-with-extra-prec 2 | |
1492 | (let ((expx (math-exp-raw (math-float x)))) | |
1493 | (math-mul (math-add expx (math-div 1 expx)) '(float 5 -1))))) | |
1494 | ((eq (car-safe x) 'sdev) | |
1495 | (math-make-sdev (calcFunc-cosh (nth 1 x)) | |
1496 | (math-mul (nth 2 x) | |
1497 | (calcFunc-sinh (nth 1 x))))) | |
1498 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
1499 | (setq x (math-abs x)) | |
1500 | (math-sort-intv (nth 1 x) | |
1501 | (calcFunc-cosh (nth 2 x)) | |
1502 | (calcFunc-cosh (nth 3 x)))) | |
1503 | ((or (equal x '(var inf var-inf)) | |
1504 | (equal x '(neg (var inf var-inf))) | |
1505 | (equal x '(var nan var-nan))) | |
1506 | (math-abs x)) | |
1507 | (t (calc-record-why 'numberp x) | |
491c3062 | 1508 | (list 'calcFunc-cosh x)))) |
136211a9 EZ |
1509 | (put 'calcFunc-cosh 'math-expandable t) |
1510 | ||
1511 | (defun calcFunc-tanh (x) ; [N N] [Public] | |
1512 | (cond ((eq x 0) 0) | |
1513 | (math-expand-formulas | |
1514 | (math-normalize | |
1515 | (let ((expx (list 'calcFunc-exp x)) | |
1516 | (expmx (list 'calcFunc-exp (list 'neg x)))) | |
1517 | (math-normalize | |
1518 | (list '/ (list '- expx expmx) (list '+ expx expmx)))))) | |
1519 | ((Math-numberp x) | |
1520 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1521 | (math-with-extra-prec 2 | |
1522 | (let* ((expx (calcFunc-exp (math-float x))) | |
1523 | (expmx (math-div 1 expx))) | |
1524 | (math-div (math-sub expx expmx) | |
1525 | (math-add expx expmx))))) | |
1526 | ((eq (car-safe x) 'sdev) | |
1527 | (math-make-sdev (calcFunc-tanh (nth 1 x)) | |
1528 | (math-div (nth 2 x) | |
1529 | (math-sqr (calcFunc-cosh (nth 1 x)))))) | |
1530 | ((eq (car x) 'intv) | |
1531 | (math-sort-intv (nth 1 x) | |
1532 | (calcFunc-tanh (nth 2 x)) | |
1533 | (calcFunc-tanh (nth 3 x)))) | |
1534 | ((equal x '(var inf var-inf)) | |
1535 | 1) | |
1536 | ((equal x '(neg (var inf var-inf))) | |
1537 | -1) | |
1538 | ((equal x '(var nan var-nan)) | |
1539 | x) | |
1540 | (t (calc-record-why 'numberp x) | |
491c3062 | 1541 | (list 'calcFunc-tanh x)))) |
136211a9 EZ |
1542 | (put 'calcFunc-tanh 'math-expandable t) |
1543 | ||
1544 | (defun calcFunc-arcsinh (x) ; [N N] [Public] | |
1545 | (cond ((eq x 0) 0) | |
1546 | (math-expand-formulas | |
1547 | (math-normalize | |
1548 | (list 'calcFunc-ln (list '+ x (list 'calcFunc-sqrt | |
1549 | (list '+ (list '^ x 2) 1)))))) | |
1550 | ((Math-numberp x) | |
1551 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1552 | (math-with-extra-prec 2 | |
1553 | (math-ln-raw (math-add x (math-sqrt-raw (math-add (math-sqr x) | |
1554 | '(float 1 0))))))) | |
1555 | ((eq (car-safe x) 'sdev) | |
1556 | (math-make-sdev (calcFunc-arcsinh (nth 1 x)) | |
1557 | (math-div (nth 2 x) | |
1558 | (math-sqrt | |
1559 | (math-add (math-sqr (nth 1 x)) 1))))) | |
1560 | ((eq (car x) 'intv) | |
1561 | (math-sort-intv (nth 1 x) | |
1562 | (calcFunc-arcsinh (nth 2 x)) | |
1563 | (calcFunc-arcsinh (nth 3 x)))) | |
1564 | ((or (equal x '(var inf var-inf)) | |
1565 | (equal x '(neg (var inf var-inf))) | |
1566 | (equal x '(var nan var-nan))) | |
1567 | x) | |
1568 | (t (calc-record-why 'numberp x) | |
491c3062 | 1569 | (list 'calcFunc-arcsinh x)))) |
136211a9 EZ |
1570 | (put 'calcFunc-arcsinh 'math-expandable t) |
1571 | ||
1572 | (defun calcFunc-arccosh (x) ; [N N] [Public] | |
1573 | (cond ((eq x 1) 0) | |
1574 | ((and (eq x -1) calc-symbolic-mode) | |
1575 | '(var pi var-pi)) | |
1576 | ((and (eq x 0) calc-symbolic-mode) | |
1577 | (math-div (math-mul '(var pi var-pi) '(var i var-i)) 2)) | |
1578 | (math-expand-formulas | |
1579 | (math-normalize | |
1580 | (list 'calcFunc-ln (list '+ x (list 'calcFunc-sqrt | |
1581 | (list '- (list '^ x 2) 1)))))) | |
1582 | ((Math-numberp x) | |
1583 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1584 | (if (Math-equal-int x -1) | |
1585 | (math-imaginary (math-pi)) | |
1586 | (math-with-extra-prec 2 | |
1587 | (if (or t ; need to do this even in the real case! | |
1588 | (memq (car-safe x) '(cplx polar))) | |
1589 | (let ((xp1 (math-add 1 x))) ; this gets the branch cuts right | |
1590 | (math-ln-raw | |
1591 | (math-add x (math-mul xp1 | |
1592 | (math-sqrt-raw | |
1593 | (math-div (math-sub | |
1594 | x | |
1595 | '(float 1 0)) | |
1596 | xp1)))))) | |
1597 | (math-ln-raw | |
1598 | (math-add x (math-sqrt-raw (math-add (math-sqr x) | |
1599 | '(float -1 0))))))))) | |
1600 | ((eq (car-safe x) 'sdev) | |
1601 | (math-make-sdev (calcFunc-arccosh (nth 1 x)) | |
1602 | (math-div (nth 2 x) | |
1603 | (math-sqrt | |
1604 | (math-add (math-sqr (nth 1 x)) -1))))) | |
1605 | ((eq (car x) 'intv) | |
1606 | (math-sort-intv (nth 1 x) | |
1607 | (calcFunc-arccosh (nth 2 x)) | |
1608 | (calcFunc-arccosh (nth 3 x)))) | |
1609 | ((or (equal x '(var inf var-inf)) | |
1610 | (equal x '(neg (var inf var-inf))) | |
1611 | (equal x '(var nan var-nan))) | |
1612 | x) | |
1613 | (t (calc-record-why 'numberp x) | |
491c3062 | 1614 | (list 'calcFunc-arccosh x)))) |
136211a9 EZ |
1615 | (put 'calcFunc-arccosh 'math-expandable t) |
1616 | ||
1617 | (defun calcFunc-arctanh (x) ; [N N] [Public] | |
1618 | (cond ((eq x 0) 0) | |
1619 | ((and (Math-equal-int x 1) calc-infinite-mode) | |
1620 | '(var inf var-inf)) | |
1621 | ((and (Math-equal-int x -1) calc-infinite-mode) | |
1622 | '(neg (var inf var-inf))) | |
1623 | (math-expand-formulas | |
1624 | (list '/ (list '- | |
1625 | (list 'calcFunc-ln (list '+ 1 x)) | |
1626 | (list 'calcFunc-ln (list '- 1 x))) 2)) | |
1627 | ((Math-numberp x) | |
1628 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1629 | (math-with-extra-prec 2 | |
1630 | (if (or (memq (car-safe x) '(cplx polar)) | |
1631 | (Math-lessp 1 x)) | |
1632 | (math-mul (math-sub (math-ln-raw (math-add '(float 1 0) x)) | |
1633 | (math-ln-raw (math-sub '(float 1 0) x))) | |
1634 | '(float 5 -1)) | |
1635 | (if (and (math-equal-int x 1) calc-infinite-mode) | |
1636 | '(var inf var-inf) | |
1637 | (if (and (math-equal-int x -1) calc-infinite-mode) | |
1638 | '(neg (var inf var-inf)) | |
1639 | (math-mul (math-ln-raw (math-div (math-add '(float 1 0) x) | |
1640 | (math-sub 1 x))) | |
1641 | '(float 5 -1))))))) | |
1642 | ((eq (car-safe x) 'sdev) | |
1643 | (math-make-sdev (calcFunc-arctanh (nth 1 x)) | |
1644 | (math-div (nth 2 x) | |
1645 | (math-sub 1 (math-sqr (nth 1 x)))))) | |
1646 | ((eq (car x) 'intv) | |
1647 | (math-sort-intv (nth 1 x) | |
1648 | (calcFunc-arctanh (nth 2 x)) | |
1649 | (calcFunc-arctanh (nth 3 x)))) | |
1650 | ((equal x '(var nan var-nan)) | |
1651 | x) | |
1652 | (t (calc-record-why 'numberp x) | |
491c3062 | 1653 | (list 'calcFunc-arctanh x)))) |
136211a9 EZ |
1654 | (put 'calcFunc-arctanh 'math-expandable t) |
1655 | ||
1656 | ||
1657 | ;;; Convert A from HMS or degrees to radians. | |
1658 | (defun calcFunc-rad (a) ; [R R] [Public] | |
1659 | (cond ((or (Math-numberp a) | |
1660 | (eq (car a) 'intv)) | |
1661 | (math-with-extra-prec 2 | |
1662 | (math-mul a (math-pi-over-180)))) | |
1663 | ((eq (car a) 'hms) | |
1664 | (math-from-hms a 'rad)) | |
1665 | ((eq (car a) 'sdev) | |
1666 | (math-make-sdev (calcFunc-rad (nth 1 a)) | |
1667 | (calcFunc-rad (nth 2 a)))) | |
1668 | (math-expand-formulas | |
1669 | (math-div (math-mul a '(var pi var-pi)) 180)) | |
1670 | ((math-infinitep a) a) | |
491c3062 | 1671 | (t (list 'calcFunc-rad a)))) |
136211a9 EZ |
1672 | (put 'calcFunc-rad 'math-expandable t) |
1673 | ||
1674 | ;;; Convert A from HMS or radians to degrees. | |
1675 | (defun calcFunc-deg (a) ; [R R] [Public] | |
1676 | (cond ((or (Math-numberp a) | |
1677 | (eq (car a) 'intv)) | |
1678 | (math-with-extra-prec 2 | |
1679 | (math-div a (math-pi-over-180)))) | |
1680 | ((eq (car a) 'hms) | |
1681 | (math-from-hms a 'deg)) | |
1682 | ((eq (car a) 'sdev) | |
1683 | (math-make-sdev (calcFunc-deg (nth 1 a)) | |
1684 | (calcFunc-deg (nth 2 a)))) | |
1685 | (math-expand-formulas | |
1686 | (math-div (math-mul 180 a) '(var pi var-pi))) | |
1687 | ((math-infinitep a) a) | |
491c3062 | 1688 | (t (list 'calcFunc-deg a)))) |
136211a9 EZ |
1689 | (put 'calcFunc-deg 'math-expandable t) |
1690 | ||
ab5796a9 | 1691 | ;;; arch-tag: c7367e8e-d0b8-4f70-8577-2fb3f31dbb4c |
491c3062 | 1692 | ;;; calc-math.el ends here |