(displayed-month, displayed-year): Define for compiler.
[emacs.git] / lisp / calc / calc-alg.el
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1 ;;; calc-alg.el --- algebraic functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainers: D. Goel <deego@gnufans.org>
7 ;; Colin Walters <walters@debian.org>
9 ;; This file is part of GNU Emacs.
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.
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.
26 ;;; Commentary:
28 ;;; Code:
30 ;; This file is autoloaded from calc-ext.el.
31 (require 'calc-ext)
33 (require 'calc-macs)
35 (defun calc-Need-calc-alg () nil)
37 ;;; Algebra commands.
39 (defun calc-alg-evaluate (arg)
40 (interactive "p")
41 (calc-slow-wrapper
42 (calc-with-default-simplification
43 (let ((math-simplify-only nil))
44 (calc-modify-simplify-mode arg)
45 (calc-enter-result 1 "dsmp" (calc-top 1))))))
47 (defun calc-modify-simplify-mode (arg)
48 (if (= (math-abs arg) 2)
49 (setq calc-simplify-mode 'alg)
50 (if (>= (math-abs arg) 3)
51 (setq calc-simplify-mode 'ext)))
52 (if (< arg 0)
53 (setq calc-simplify-mode (list calc-simplify-mode))))
55 (defun calc-simplify ()
56 (interactive)
57 (calc-slow-wrapper
58 (calc-with-default-simplification
59 (calc-enter-result 1 "simp" (math-simplify (calc-top-n 1))))))
61 (defun calc-simplify-extended ()
62 (interactive)
63 (calc-slow-wrapper
64 (calc-with-default-simplification
65 (calc-enter-result 1 "esmp" (math-simplify-extended (calc-top-n 1))))))
67 (defun calc-expand-formula (arg)
68 (interactive "p")
69 (calc-slow-wrapper
70 (calc-with-default-simplification
71 (let ((math-simplify-only nil))
72 (calc-modify-simplify-mode arg)
73 (calc-enter-result 1 "expf"
74 (if (> arg 0)
75 (let ((math-expand-formulas t))
76 (calc-top-n 1))
77 (let ((top (calc-top-n 1)))
78 (or (math-expand-formula top)
79 top))))))))
81 (defun calc-factor (arg)
82 (interactive "P")
83 (calc-slow-wrapper
84 (calc-unary-op "fctr" (if (calc-is-hyperbolic)
85 'calcFunc-factors 'calcFunc-factor)
86 arg)))
88 (defun calc-expand (n)
89 (interactive "P")
90 (calc-slow-wrapper
91 (calc-enter-result 1 "expa"
92 (append (list 'calcFunc-expand
93 (calc-top-n 1))
94 (and n (list (prefix-numeric-value n)))))))
96 (defun calc-collect (&optional var)
97 (interactive "sCollect terms involving: ")
98 (calc-slow-wrapper
99 (if (or (equal var "") (equal var "$") (null var))
100 (calc-enter-result 2 "clct" (cons 'calcFunc-collect
101 (calc-top-list-n 2)))
102 (let ((var (math-read-expr var)))
103 (if (eq (car-safe var) 'error)
104 (error "Bad format in expression: %s" (nth 1 var)))
105 (calc-enter-result 1 "clct" (list 'calcFunc-collect
106 (calc-top-n 1)
107 var))))))
109 (defun calc-apart (arg)
110 (interactive "P")
111 (calc-slow-wrapper
112 (calc-unary-op "aprt" 'calcFunc-apart arg)))
114 (defun calc-normalize-rat (arg)
115 (interactive "P")
116 (calc-slow-wrapper
117 (calc-unary-op "nrat" 'calcFunc-nrat arg)))
119 (defun calc-poly-gcd (arg)
120 (interactive "P")
121 (calc-slow-wrapper
122 (calc-binary-op "pgcd" 'calcFunc-pgcd arg)))
124 (defun calc-poly-div (arg)
125 (interactive "P")
126 (calc-slow-wrapper
127 (setq calc-poly-div-remainder nil)
128 (calc-binary-op "pdiv" 'calcFunc-pdiv arg)
129 (if (and calc-poly-div-remainder (null arg))
130 (progn
131 (calc-clear-command-flag 'clear-message)
132 (calc-record calc-poly-div-remainder "prem")
133 (if (not (Math-zerop calc-poly-div-remainder))
134 (message "(Remainder was %s)"
135 (math-format-flat-expr calc-poly-div-remainder 0))
136 (message "(No remainder)"))))))
138 (defun calc-poly-rem (arg)
139 (interactive "P")
140 (calc-slow-wrapper
141 (calc-binary-op "prem" 'calcFunc-prem arg)))
143 (defun calc-poly-div-rem (arg)
144 (interactive "P")
145 (calc-slow-wrapper
146 (if (calc-is-hyperbolic)
147 (calc-binary-op "pdvr" 'calcFunc-pdivide arg)
148 (calc-binary-op "pdvr" 'calcFunc-pdivrem arg))))
150 (defun calc-substitute (&optional oldname newname)
151 (interactive "sSubstitute old: ")
152 (calc-slow-wrapper
153 (let (old new (num 1) expr)
154 (if (or (equal oldname "") (equal oldname "$") (null oldname))
155 (setq new (calc-top-n 1)
156 old (calc-top-n 2)
157 expr (calc-top-n 3)
158 num 3)
159 (or newname
160 (progn (calc-unread-command ?\C-a)
161 (setq newname (read-string (concat "Substitute old: "
162 oldname
163 ", new: ")
164 oldname))))
165 (if (or (equal newname "") (equal newname "$") (null newname))
166 (setq new (calc-top-n 1)
167 expr (calc-top-n 2)
168 num 2)
169 (setq new (if (stringp newname) (math-read-expr newname) newname))
170 (if (eq (car-safe new) 'error)
171 (error "Bad format in expression: %s" (nth 1 new)))
172 (setq expr (calc-top-n 1)))
173 (setq old (if (stringp oldname) (math-read-expr oldname) oldname))
174 (if (eq (car-safe old) 'error)
175 (error "Bad format in expression: %s" (nth 1 old)))
176 (or (math-expr-contains expr old)
177 (error "No occurrences found")))
178 (calc-enter-result num "sbst" (math-expr-subst expr old new)))))
181 (defun calc-has-rules (name)
182 (setq name (calc-var-value name))
183 (and (consp name)
184 (memq (car name) '(vec calcFunc-assign calcFunc-condition))
185 name))
187 (defun math-recompile-eval-rules ()
188 (setq math-eval-rules-cache (and (calc-has-rules 'var-EvalRules)
189 (math-compile-rewrites
190 '(var EvalRules var-EvalRules)))
191 math-eval-rules-cache-other (assq nil math-eval-rules-cache)
192 math-eval-rules-cache-tag (calc-var-value 'var-EvalRules)))
195 ;;; Try to expand a formula according to its definition.
196 (defun math-expand-formula (expr)
197 (and (consp expr)
198 (symbolp (car expr))
199 (or (get (car expr) 'calc-user-defn)
200 (get (car expr) 'math-expandable))
201 (let ((res (let ((math-expand-formulas t))
202 (apply (car expr) (cdr expr)))))
203 (and (not (eq (car-safe res) (car expr)))
204 res))))
209 ;;; True if A comes before B in a canonical ordering of expressions. [P X X]
210 (defun math-beforep (a b) ; [Public]
211 (cond ((and (Math-realp a) (Math-realp b))
212 (let ((comp (math-compare a b)))
213 (or (eq comp -1)
214 (and (eq comp 0)
215 (not (equal a b))
216 (> (length (memq (car-safe a)
217 '(bigneg nil bigpos frac float)))
218 (length (memq (car-safe b)
219 '(bigneg nil bigpos frac float))))))))
220 ((equal b '(neg (var inf var-inf))) nil)
221 ((equal a '(neg (var inf var-inf))) t)
222 ((equal a '(var inf var-inf)) nil)
223 ((equal b '(var inf var-inf)) t)
224 ((Math-realp a)
225 (if (and (eq (car-safe b) 'intv) (math-intv-constp b))
226 (if (or (math-beforep a (nth 2 b)) (Math-equal a (nth 2 b)))
228 nil)
230 ((Math-realp b)
231 (if (and (eq (car-safe a) 'intv) (math-intv-constp a))
232 (if (math-beforep (nth 2 a) b)
234 nil)
235 nil))
236 ((and (eq (car a) 'intv) (eq (car b) 'intv)
237 (math-intv-constp a) (math-intv-constp b))
238 (let ((comp (math-compare (nth 2 a) (nth 2 b))))
239 (cond ((eq comp -1) t)
240 ((eq comp 1) nil)
241 ((and (memq (nth 1 a) '(2 3)) (memq (nth 1 b) '(0 1))) t)
242 ((and (memq (nth 1 a) '(0 1)) (memq (nth 1 b) '(2 3))) nil)
243 ((eq (setq comp (math-compare (nth 3 a) (nth 3 b))) -1) t)
244 ((eq comp 1) nil)
245 ((and (memq (nth 1 a) '(0 2)) (memq (nth 1 b) '(1 3))) t)
246 (t nil))))
247 ((not (eq (not (Math-objectp a)) (not (Math-objectp b))))
248 (Math-objectp a))
249 ((eq (car a) 'var)
250 (if (eq (car b) 'var)
251 (string-lessp (symbol-name (nth 1 a)) (symbol-name (nth 1 b)))
252 (not (Math-numberp b))))
253 ((eq (car b) 'var) (Math-numberp a))
254 ((eq (car a) (car b))
255 (while (and (setq a (cdr a) b (cdr b)) a
256 (equal (car a) (car b))))
257 (and b
258 (or (null a)
259 (math-beforep (car a) (car b)))))
260 (t (string-lessp (symbol-name (car a)) (symbol-name (car b))))))
263 (defsubst math-simplify-extended (a)
264 (let ((math-living-dangerously t))
265 (math-simplify a)))
267 (defalias 'calcFunc-esimplify 'math-simplify-extended)
269 (defun math-simplify (top-expr)
270 (let ((math-simplifying t)
271 (top-only (consp calc-simplify-mode))
272 (simp-rules (append (and (calc-has-rules 'var-AlgSimpRules)
273 '((var AlgSimpRules var-AlgSimpRules)))
274 (and math-living-dangerously
275 (calc-has-rules 'var-ExtSimpRules)
276 '((var ExtSimpRules var-ExtSimpRules)))
277 (and math-simplifying-units
278 (calc-has-rules 'var-UnitSimpRules)
279 '((var UnitSimpRules var-UnitSimpRules)))
280 (and math-integrating
281 (calc-has-rules 'var-IntegSimpRules)
282 '((var IntegSimpRules var-IntegSimpRules)))))
283 res)
284 (if top-only
285 (let ((r simp-rules))
286 (setq res (math-simplify-step (math-normalize top-expr))
287 calc-simplify-mode '(nil)
288 top-expr (math-normalize res))
289 (while r
290 (setq top-expr (math-rewrite top-expr (car r)
291 '(neg (var inf var-inf)))
292 r (cdr r))))
293 (calc-with-default-simplification
294 (while (let ((r simp-rules))
295 (setq res (math-normalize top-expr))
296 (while r
297 (setq res (math-rewrite res (car r))
298 r (cdr r)))
299 (not (equal top-expr (setq res (math-simplify-step res)))))
300 (setq top-expr res)))))
301 top-expr)
303 (defalias 'calcFunc-simplify 'math-simplify)
305 ;;; The following has a "bug" in that if any recursive simplifications
306 ;;; occur only the first handler will be tried; this doesn't really
307 ;;; matter, since math-simplify-step is iterated to a fixed point anyway.
308 (defun math-simplify-step (a)
309 (if (Math-primp a)
311 (let ((aa (if (or top-only
312 (memq (car a) '(calcFunc-quote calcFunc-condition
313 calcFunc-evalto)))
315 (cons (car a) (mapcar 'math-simplify-step (cdr a))))))
316 (and (symbolp (car aa))
317 (let ((handler (get (car aa) 'math-simplify)))
318 (and handler
319 (while (and handler
320 (equal (setq aa (or (funcall (car handler) aa)
321 aa))
323 (setq handler (cdr handler))))))
324 aa)))
327 ;; Placeholder, to synchronize autoloading.
328 (defun math-need-std-simps ()
329 nil)
331 (math-defsimplify (+ -)
332 (math-simplify-plus))
334 (defun math-simplify-plus ()
335 (cond ((and (memq (car-safe (nth 1 expr)) '(+ -))
336 (Math-numberp (nth 2 (nth 1 expr)))
337 (not (Math-numberp (nth 2 expr))))
338 (let ((x (nth 2 expr))
339 (op (car expr)))
340 (setcar (cdr (cdr expr)) (nth 2 (nth 1 expr)))
341 (setcar expr (car (nth 1 expr)))
342 (setcar (cdr (cdr (nth 1 expr))) x)
343 (setcar (nth 1 expr) op)))
344 ((and (eq (car expr) '+)
345 (Math-numberp (nth 1 expr))
346 (not (Math-numberp (nth 2 expr))))
347 (let ((x (nth 2 expr)))
348 (setcar (cdr (cdr expr)) (nth 1 expr))
349 (setcar (cdr expr) x))))
350 (let ((aa expr)
351 aaa temp)
352 (while (memq (car-safe (setq aaa (nth 1 aa))) '(+ -))
353 (if (setq temp (math-combine-sum (nth 2 aaa) (nth 2 expr)
354 (eq (car aaa) '-) (eq (car expr) '-) t))
355 (progn
356 (setcar (cdr (cdr expr)) temp)
357 (setcar expr '+)
358 (setcar (cdr (cdr aaa)) 0)))
359 (setq aa (nth 1 aa)))
360 (if (setq temp (math-combine-sum aaa (nth 2 expr)
361 nil (eq (car expr) '-) t))
362 (progn
363 (setcar (cdr (cdr expr)) temp)
364 (setcar expr '+)
365 (setcar (cdr aa) 0)))
366 expr))
368 (math-defsimplify *
369 (math-simplify-times))
371 (defun math-simplify-times ()
372 (if (eq (car-safe (nth 2 expr)) '*)
373 (and (math-beforep (nth 1 (nth 2 expr)) (nth 1 expr))
374 (or (math-known-scalarp (nth 1 expr) t)
375 (math-known-scalarp (nth 1 (nth 2 expr)) t))
376 (let ((x (nth 1 expr)))
377 (setcar (cdr expr) (nth 1 (nth 2 expr)))
378 (setcar (cdr (nth 2 expr)) x)))
379 (and (math-beforep (nth 2 expr) (nth 1 expr))
380 (or (math-known-scalarp (nth 1 expr) t)
381 (math-known-scalarp (nth 2 expr) t))
382 (let ((x (nth 2 expr)))
383 (setcar (cdr (cdr expr)) (nth 1 expr))
384 (setcar (cdr expr) x))))
385 (let ((aa expr)
386 aaa temp
387 (safe t) (scalar (math-known-scalarp (nth 1 expr))))
388 (if (and (Math-ratp (nth 1 expr))
389 (setq temp (math-common-constant-factor (nth 2 expr))))
390 (progn
391 (setcar (cdr (cdr expr))
392 (math-cancel-common-factor (nth 2 expr) temp))
393 (setcar (cdr expr) (math-mul (nth 1 expr) temp))))
394 (while (and (eq (car-safe (setq aaa (nth 2 aa))) '*)
395 safe)
396 (if (setq temp (math-combine-prod (nth 1 expr) (nth 1 aaa) nil nil t))
397 (progn
398 (setcar (cdr expr) temp)
399 (setcar (cdr aaa) 1)))
400 (setq safe (or scalar (math-known-scalarp (nth 1 aaa) t))
401 aa (nth 2 aa)))
402 (if (and (setq temp (math-combine-prod aaa (nth 1 expr) nil nil t))
403 safe)
404 (progn
405 (setcar (cdr expr) temp)
406 (setcar (cdr (cdr aa)) 1)))
407 (if (and (eq (car-safe (nth 1 expr)) 'frac)
408 (memq (nth 1 (nth 1 expr)) '(1 -1)))
409 (math-div (math-mul (nth 2 expr) (nth 1 (nth 1 expr)))
410 (nth 2 (nth 1 expr)))
411 expr)))
413 (math-defsimplify /
414 (math-simplify-divide))
416 (defun math-simplify-divide ()
417 (let ((np (cdr expr))
418 (nover nil)
419 (nn (and (or (eq (car expr) '/) (not (Math-realp (nth 2 expr))))
420 (math-common-constant-factor (nth 2 expr))))
421 n op)
422 (if nn
423 (progn
424 (setq n (and (or (eq (car expr) '/) (not (Math-realp (nth 1 expr))))
425 (math-common-constant-factor (nth 1 expr))))
426 (if (and (eq (car-safe nn) 'frac) (eq (nth 1 nn) 1) (not n))
427 (progn
428 (setcar (cdr expr) (math-mul (nth 2 nn) (nth 1 expr)))
429 (setcar (cdr (cdr expr))
430 (math-cancel-common-factor (nth 2 expr) nn))
431 (if (and (math-negp nn)
432 (setq op (assq (car expr) calc-tweak-eqn-table)))
433 (setcar expr (nth 1 op))))
434 (if (and n (not (eq (setq n (math-frac-gcd n nn)) 1)))
435 (progn
436 (setcar (cdr expr)
437 (math-cancel-common-factor (nth 1 expr) n))
438 (setcar (cdr (cdr expr))
439 (math-cancel-common-factor (nth 2 expr) n))
440 (if (and (math-negp n)
441 (setq op (assq (car expr) calc-tweak-eqn-table)))
442 (setcar expr (nth 1 op))))))))
443 (if (and (eq (car-safe (car np)) '/)
444 (math-known-scalarp (nth 2 expr) t))
445 (progn
446 (setq np (cdr (nth 1 expr)))
447 (while (eq (car-safe (setq n (car np))) '*)
448 (and (math-known-scalarp (nth 2 n) t)
449 (math-simplify-divisor (cdr n) (cdr (cdr expr)) nil t))
450 (setq np (cdr (cdr n))))
451 (math-simplify-divisor np (cdr (cdr expr)) nil t)
452 (setq nover t
453 np (cdr (cdr (nth 1 expr))))))
454 (while (eq (car-safe (setq n (car np))) '*)
455 (and (math-known-scalarp (nth 2 n) t)
456 (math-simplify-divisor (cdr n) (cdr (cdr expr)) nover t))
457 (setq np (cdr (cdr n))))
458 (math-simplify-divisor np (cdr (cdr expr)) nover t)
459 expr))
461 (defun math-simplify-divisor (np dp nover dover)
462 (cond ((eq (car-safe (car dp)) '/)
463 (math-simplify-divisor np (cdr (car dp)) nover dover)
464 (and (math-known-scalarp (nth 1 (car dp)) t)
465 (math-simplify-divisor np (cdr (cdr (car dp)))
466 nover (not dover))))
467 ((or (or (eq (car expr) '/)
468 (let ((signs (math-possible-signs (car np))))
469 (or (memq signs '(1 4))
470 (and (memq (car expr) '(calcFunc-eq calcFunc-neq))
471 (eq signs 5))
472 math-living-dangerously)))
473 (math-numberp (car np)))
474 (let ((n (car np))
475 d dd temp op
476 (safe t) (scalar (math-known-scalarp n)))
477 (while (and (eq (car-safe (setq d (car dp))) '*)
478 safe)
479 (math-simplify-one-divisor np (cdr d))
480 (setq safe (or scalar (math-known-scalarp (nth 1 d) t))
481 dp (cdr (cdr d))))
482 (if safe
483 (math-simplify-one-divisor np dp))))))
485 (defun math-simplify-one-divisor (np dp)
486 (if (setq temp (math-combine-prod (car np) (car dp) nover dover t))
487 (progn
488 (and (not (memq (car expr) '(/ calcFunc-eq calcFunc-neq)))
489 (math-known-negp (car dp))
490 (setq op (assq (car expr) calc-tweak-eqn-table))
491 (setcar expr (nth 1 op)))
492 (setcar np (if nover (math-div 1 temp) temp))
493 (setcar dp 1))
494 (and dover (not nover) (eq (car expr) '/)
495 (eq (car-safe (car dp)) 'calcFunc-sqrt)
496 (Math-integerp (nth 1 (car dp)))
497 (progn
498 (setcar np (math-mul (car np)
499 (list 'calcFunc-sqrt (nth 1 (car dp)))))
500 (setcar dp (nth 1 (car dp)))))))
502 (defun math-common-constant-factor (expr)
503 (if (Math-realp expr)
504 (if (Math-ratp expr)
505 (and (not (memq expr '(0 1 -1)))
506 (math-abs expr))
507 (if (math-ratp (setq expr (math-to-simple-fraction expr)))
508 (math-common-constant-factor expr)))
509 (if (memq (car expr) '(+ - cplx sdev))
510 (let ((f1 (math-common-constant-factor (nth 1 expr)))
511 (f2 (math-common-constant-factor (nth 2 expr))))
512 (and f1 f2
513 (not (eq (setq f1 (math-frac-gcd f1 f2)) 1))
514 f1))
515 (if (memq (car expr) '(* polar))
516 (math-common-constant-factor (nth 1 expr))
517 (if (eq (car expr) '/)
518 (or (math-common-constant-factor (nth 1 expr))
519 (and (Math-integerp (nth 2 expr))
520 (list 'frac 1 (math-abs (nth 2 expr))))))))))
522 (defun math-cancel-common-factor (expr val)
523 (if (memq (car-safe expr) '(+ - cplx sdev))
524 (progn
525 (setcar (cdr expr) (math-cancel-common-factor (nth 1 expr) val))
526 (setcar (cdr (cdr expr)) (math-cancel-common-factor (nth 2 expr) val))
527 expr)
528 (if (eq (car-safe expr) '*)
529 (math-mul (math-cancel-common-factor (nth 1 expr) val) (nth 2 expr))
530 (math-div expr val))))
532 (defun math-frac-gcd (a b)
533 (if (Math-zerop a)
535 (if (Math-zerop b)
537 (if (and (Math-integerp a)
538 (Math-integerp b))
539 (math-gcd a b)
540 (and (Math-integerp a) (setq a (list 'frac a 1)))
541 (and (Math-integerp b) (setq b (list 'frac b 1)))
542 (math-make-frac (math-gcd (nth 1 a) (nth 1 b))
543 (math-gcd (nth 2 a) (nth 2 b)))))))
545 (math-defsimplify %
546 (math-simplify-mod))
548 (defun math-simplify-mod ()
549 (and (Math-realp (nth 2 expr))
550 (Math-posp (nth 2 expr))
551 (let ((lin (math-is-linear (nth 1 expr)))
552 t1 t2 t3)
553 (or (and lin
554 (or (math-negp (car lin))
555 (not (Math-lessp (car lin) (nth 2 expr))))
556 (list '%
557 (list '+
558 (math-mul (nth 1 lin) (nth 2 lin))
559 (math-mod (car lin) (nth 2 expr)))
560 (nth 2 expr)))
561 (and lin
562 (not (math-equal-int (nth 1 lin) 1))
563 (math-num-integerp (nth 1 lin))
564 (math-num-integerp (nth 2 expr))
565 (setq t1 (calcFunc-gcd (nth 1 lin) (nth 2 expr)))
566 (not (math-equal-int t1 1))
567 (list '*
569 (list '%
570 (list '+
571 (math-mul (math-div (nth 1 lin) t1)
572 (nth 2 lin))
573 (let ((calc-prefer-frac t))
574 (math-div (car lin) t1)))
575 (math-div (nth 2 expr) t1))))
576 (and (math-equal-int (nth 2 expr) 1)
577 (math-known-integerp (if lin
578 (math-mul (nth 1 lin) (nth 2 lin))
579 (nth 1 expr)))
580 (if lin (math-mod (car lin) 1) 0))))))
582 (math-defsimplify (calcFunc-eq calcFunc-neq calcFunc-lt
583 calcFunc-gt calcFunc-leq calcFunc-geq)
584 (if (= (length expr) 3)
585 (math-simplify-ineq)))
587 (defun math-simplify-ineq ()
588 (let ((np (cdr expr))
590 (while (memq (car-safe (setq n (car np))) '(+ -))
591 (math-simplify-add-term (cdr (cdr n)) (cdr (cdr expr))
592 (eq (car n) '-) nil)
593 (setq np (cdr n)))
594 (math-simplify-add-term np (cdr (cdr expr)) nil (eq np (cdr expr)))
595 (math-simplify-divide)
596 (let ((signs (math-possible-signs (cons '- (cdr expr)))))
597 (or (cond ((eq (car expr) 'calcFunc-eq)
598 (or (and (eq signs 2) 1)
599 (and (memq signs '(1 4 5)) 0)))
600 ((eq (car expr) 'calcFunc-neq)
601 (or (and (eq signs 2) 0)
602 (and (memq signs '(1 4 5)) 1)))
603 ((eq (car expr) 'calcFunc-lt)
604 (or (and (eq signs 1) 1)
605 (and (memq signs '(2 4 6)) 0)))
606 ((eq (car expr) 'calcFunc-gt)
607 (or (and (eq signs 4) 1)
608 (and (memq signs '(1 2 3)) 0)))
609 ((eq (car expr) 'calcFunc-leq)
610 (or (and (eq signs 4) 0)
611 (and (memq signs '(1 2 3)) 1)))
612 ((eq (car expr) 'calcFunc-geq)
613 (or (and (eq signs 1) 0)
614 (and (memq signs '(2 4 6)) 1))))
615 expr))))
617 (defun math-simplify-add-term (np dp minus lplain)
618 (or (math-vectorp (car np))
619 (let ((rplain t)
620 n d dd temp)
621 (while (memq (car-safe (setq n (car np) d (car dp))) '(+ -))
622 (setq rplain nil)
623 (if (setq temp (math-combine-sum n (nth 2 d)
624 minus (eq (car d) '+) t))
625 (if (or lplain (eq (math-looks-negp temp) minus))
626 (progn
627 (setcar np (setq n (if minus (math-neg temp) temp)))
628 (setcar (cdr (cdr d)) 0))
629 (progn
630 (setcar np 0)
631 (setcar (cdr (cdr d)) (setq n (if (eq (car d) '+)
632 (math-neg temp)
633 temp))))))
634 (setq dp (cdr d)))
635 (if (setq temp (math-combine-sum n d minus t t))
636 (if (or lplain
637 (and (not rplain)
638 (eq (math-looks-negp temp) minus)))
639 (progn
640 (setcar np (setq n (if minus (math-neg temp) temp)))
641 (setcar dp 0))
642 (progn
643 (setcar np 0)
644 (setcar dp (setq n (math-neg temp)))))))))
646 (math-defsimplify calcFunc-sin
647 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
648 (nth 1 (nth 1 expr)))
649 (and (math-looks-negp (nth 1 expr))
650 (math-neg (list 'calcFunc-sin (math-neg (nth 1 expr)))))
651 (and (eq calc-angle-mode 'rad)
652 (let ((n (math-linear-in (nth 1 expr) '(var pi var-pi))))
653 (and n
654 (math-known-sin (car n) (nth 1 n) 120 0))))
655 (and (eq calc-angle-mode 'deg)
656 (let ((n (math-integer-plus (nth 1 expr))))
657 (and n
658 (math-known-sin (car n) (nth 1 n) '(frac 2 3) 0))))
659 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
660 (list 'calcFunc-sqrt (math-sub 1 (math-sqr (nth 1 (nth 1 expr))))))
661 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
662 (math-div (nth 1 (nth 1 expr))
663 (list 'calcFunc-sqrt
664 (math-add 1 (math-sqr (nth 1 (nth 1 expr)))))))
665 (let ((m (math-should-expand-trig (nth 1 expr))))
666 (and m (integerp (car m))
667 (let ((n (car m)) (a (nth 1 m)))
668 (list '+
669 (list '* (list 'calcFunc-sin (list '* (1- n) a))
670 (list 'calcFunc-cos a))
671 (list '* (list 'calcFunc-cos (list '* (1- n) a))
672 (list 'calcFunc-sin a))))))))
674 (math-defsimplify calcFunc-cos
675 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
676 (nth 1 (nth 1 expr)))
677 (and (math-looks-negp (nth 1 expr))
678 (list 'calcFunc-cos (math-neg (nth 1 expr))))
679 (and (eq calc-angle-mode 'rad)
680 (let ((n (math-linear-in (nth 1 expr) '(var pi var-pi))))
681 (and n
682 (math-known-sin (car n) (nth 1 n) 120 300))))
683 (and (eq calc-angle-mode 'deg)
684 (let ((n (math-integer-plus (nth 1 expr))))
685 (and n
686 (math-known-sin (car n) (nth 1 n) '(frac 2 3) 300))))
687 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
688 (list 'calcFunc-sqrt (math-sub 1 (math-sqr (nth 1 (nth 1 expr))))))
689 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
690 (math-div 1
691 (list 'calcFunc-sqrt
692 (math-add 1 (math-sqr (nth 1 (nth 1 expr)))))))
693 (let ((m (math-should-expand-trig (nth 1 expr))))
694 (and m (integerp (car m))
695 (let ((n (car m)) (a (nth 1 m)))
696 (list '-
697 (list '* (list 'calcFunc-cos (list '* (1- n) a))
698 (list 'calcFunc-cos a))
699 (list '* (list 'calcFunc-sin (list '* (1- n) a))
700 (list 'calcFunc-sin a))))))))
702 (defun math-should-expand-trig (x &optional hyperbolic)
703 (let ((m (math-is-multiple x)))
704 (and math-living-dangerously
705 m (or (and (integerp (car m)) (> (car m) 1))
706 (equal (car m) '(frac 1 2)))
707 (or math-integrating
708 (memq (car-safe (nth 1 m))
709 (if hyperbolic
710 '(calcFunc-arcsinh calcFunc-arccosh calcFunc-arctanh)
711 '(calcFunc-arcsin calcFunc-arccos calcFunc-arctan)))
712 (and (eq (car-safe (nth 1 m)) 'calcFunc-ln)
713 (eq hyperbolic 'exp)))
714 m)))
716 (defun math-known-sin (plus n mul off)
717 (setq n (math-mul n mul))
718 (and (math-num-integerp n)
719 (setq n (math-mod (math-add (math-trunc n) off) 240))
720 (if (>= n 120)
721 (and (setq n (math-known-sin plus (- n 120) 1 0))
722 (math-neg n))
723 (if (> n 60)
724 (setq n (- 120 n)))
725 (if (math-zerop plus)
726 (and (or calc-symbolic-mode
727 (memq n '(0 20 60)))
728 (cdr (assq n
729 '( (0 . 0)
730 (10 . (/ (calcFunc-sqrt
731 (- 2 (calcFunc-sqrt 3))) 2))
732 (12 . (/ (- (calcFunc-sqrt 5) 1) 4))
733 (15 . (/ (calcFunc-sqrt
734 (- 2 (calcFunc-sqrt 2))) 2))
735 (20 . (/ 1 2))
736 (24 . (* (^ (/ 1 2) (/ 3 2))
737 (calcFunc-sqrt
738 (- 5 (calcFunc-sqrt 5)))))
739 (30 . (/ (calcFunc-sqrt 2) 2))
740 (36 . (/ (+ (calcFunc-sqrt 5) 1) 4))
741 (40 . (/ (calcFunc-sqrt 3) 2))
742 (45 . (/ (calcFunc-sqrt
743 (+ 2 (calcFunc-sqrt 2))) 2))
744 (48 . (* (^ (/ 1 2) (/ 3 2))
745 (calcFunc-sqrt
746 (+ 5 (calcFunc-sqrt 5)))))
747 (50 . (/ (calcFunc-sqrt
748 (+ 2 (calcFunc-sqrt 3))) 2))
749 (60 . 1)))))
750 (cond ((eq n 0) (math-normalize (list 'calcFunc-sin plus)))
751 ((eq n 60) (math-normalize (list 'calcFunc-cos plus)))
752 (t nil))))))
754 (math-defsimplify calcFunc-tan
755 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
756 (nth 1 (nth 1 expr)))
757 (and (math-looks-negp (nth 1 expr))
758 (math-neg (list 'calcFunc-tan (math-neg (nth 1 expr)))))
759 (and (eq calc-angle-mode 'rad)
760 (let ((n (math-linear-in (nth 1 expr) '(var pi var-pi))))
761 (and n
762 (math-known-tan (car n) (nth 1 n) 120))))
763 (and (eq calc-angle-mode 'deg)
764 (let ((n (math-integer-plus (nth 1 expr))))
765 (and n
766 (math-known-tan (car n) (nth 1 n) '(frac 2 3)))))
767 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
768 (math-div (nth 1 (nth 1 expr))
769 (list 'calcFunc-sqrt
770 (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))))
771 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
772 (math-div (list 'calcFunc-sqrt
773 (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))
774 (nth 1 (nth 1 expr))))
775 (let ((m (math-should-expand-trig (nth 1 expr))))
776 (and m
777 (if (equal (car m) '(frac 1 2))
778 (math-div (math-sub 1 (list 'calcFunc-cos (nth 1 m)))
779 (list 'calcFunc-sin (nth 1 m)))
780 (math-div (list 'calcFunc-sin (nth 1 expr))
781 (list 'calcFunc-cos (nth 1 expr))))))))
783 (defun math-known-tan (plus n mul)
784 (setq n (math-mul n mul))
785 (and (math-num-integerp n)
786 (setq n (math-mod (math-trunc n) 120))
787 (if (> n 60)
788 (and (setq n (math-known-tan plus (- 120 n) 1))
789 (math-neg n))
790 (if (math-zerop plus)
791 (and (or calc-symbolic-mode
792 (memq n '(0 30 60)))
793 (cdr (assq n '( (0 . 0)
794 (10 . (- 2 (calcFunc-sqrt 3)))
795 (12 . (calcFunc-sqrt
796 (- 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
797 (15 . (- (calcFunc-sqrt 2) 1))
798 (20 . (/ (calcFunc-sqrt 3) 3))
799 (24 . (calcFunc-sqrt
800 (- 5 (* 2 (calcFunc-sqrt 5)))))
801 (30 . 1)
802 (36 . (calcFunc-sqrt
803 (+ 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
804 (40 . (calcFunc-sqrt 3))
805 (45 . (+ (calcFunc-sqrt 2) 1))
806 (48 . (calcFunc-sqrt
807 (+ 5 (* 2 (calcFunc-sqrt 5)))))
808 (50 . (+ 2 (calcFunc-sqrt 3)))
809 (60 . (var uinf var-uinf))))))
810 (cond ((eq n 0) (math-normalize (list 'calcFunc-tan plus)))
811 ((eq n 60) (math-normalize (list '/ -1
812 (list 'calcFunc-tan plus))))
813 (t nil))))))
815 (math-defsimplify calcFunc-sinh
816 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsinh)
817 (nth 1 (nth 1 expr)))
818 (and (math-looks-negp (nth 1 expr))
819 (math-neg (list 'calcFunc-sinh (math-neg (nth 1 expr)))))
820 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccosh)
821 math-living-dangerously
822 (list 'calcFunc-sqrt (math-sub (math-sqr (nth 1 (nth 1 expr))) 1)))
823 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctanh)
824 math-living-dangerously
825 (math-div (nth 1 (nth 1 expr))
826 (list 'calcFunc-sqrt
827 (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))))
828 (let ((m (math-should-expand-trig (nth 1 expr) t)))
829 (and m (integerp (car m))
830 (let ((n (car m)) (a (nth 1 m)))
831 (if (> n 1)
832 (list '+
833 (list '* (list 'calcFunc-sinh (list '* (1- n) a))
834 (list 'calcFunc-cosh a))
835 (list '* (list 'calcFunc-cosh (list '* (1- n) a))
836 (list 'calcFunc-sinh a)))))))))
838 (math-defsimplify calcFunc-cosh
839 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccosh)
840 (nth 1 (nth 1 expr)))
841 (and (math-looks-negp (nth 1 expr))
842 (list 'calcFunc-cosh (math-neg (nth 1 expr))))
843 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsinh)
844 math-living-dangerously
845 (list 'calcFunc-sqrt (math-add (math-sqr (nth 1 (nth 1 expr))) 1)))
846 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctanh)
847 math-living-dangerously
848 (math-div 1
849 (list 'calcFunc-sqrt
850 (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))))
851 (let ((m (math-should-expand-trig (nth 1 expr) t)))
852 (and m (integerp (car m))
853 (let ((n (car m)) (a (nth 1 m)))
854 (if (> n 1)
855 (list '+
856 (list '* (list 'calcFunc-cosh (list '* (1- n) a))
857 (list 'calcFunc-cosh a))
858 (list '* (list 'calcFunc-sinh (list '* (1- n) a))
859 (list 'calcFunc-sinh a)))))))))
861 (math-defsimplify calcFunc-tanh
862 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctanh)
863 (nth 1 (nth 1 expr)))
864 (and (math-looks-negp (nth 1 expr))
865 (math-neg (list 'calcFunc-tanh (math-neg (nth 1 expr)))))
866 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsinh)
867 math-living-dangerously
868 (math-div (nth 1 (nth 1 expr))
869 (list 'calcFunc-sqrt
870 (math-add (math-sqr (nth 1 (nth 1 expr))) 1))))
871 (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccosh)
872 math-living-dangerously
873 (math-div (list 'calcFunc-sqrt
874 (math-sub (math-sqr (nth 1 (nth 1 expr))) 1))
875 (nth 1 (nth 1 expr))))
876 (let ((m (math-should-expand-trig (nth 1 expr) t)))
877 (and m
878 (if (equal (car m) '(frac 1 2))
879 (math-div (math-sub (list 'calcFunc-cosh (nth 1 m)) 1)
880 (list 'calcFunc-sinh (nth 1 m)))
881 (math-div (list 'calcFunc-sinh (nth 1 expr))
882 (list 'calcFunc-cosh (nth 1 expr))))))))
884 (math-defsimplify calcFunc-arcsin
885 (or (and (math-looks-negp (nth 1 expr))
886 (math-neg (list 'calcFunc-arcsin (math-neg (nth 1 expr)))))
887 (and (eq (nth 1 expr) 1)
888 (math-quarter-circle t))
889 (and (equal (nth 1 expr) '(frac 1 2))
890 (math-div (math-half-circle t) 6))
891 (and math-living-dangerously
892 (eq (car-safe (nth 1 expr)) 'calcFunc-sin)
893 (nth 1 (nth 1 expr)))
894 (and math-living-dangerously
895 (eq (car-safe (nth 1 expr)) 'calcFunc-cos)
896 (math-sub (math-quarter-circle t)
897 (nth 1 (nth 1 expr))))))
899 (math-defsimplify calcFunc-arccos
900 (or (and (eq (nth 1 expr) 0)
901 (math-quarter-circle t))
902 (and (eq (nth 1 expr) -1)
903 (math-half-circle t))
904 (and (equal (nth 1 expr) '(frac 1 2))
905 (math-div (math-half-circle t) 3))
906 (and (equal (nth 1 expr) '(frac -1 2))
907 (math-div (math-mul (math-half-circle t) 2) 3))
908 (and math-living-dangerously
909 (eq (car-safe (nth 1 expr)) 'calcFunc-cos)
910 (nth 1 (nth 1 expr)))
911 (and math-living-dangerously
912 (eq (car-safe (nth 1 expr)) 'calcFunc-sin)
913 (math-sub (math-quarter-circle t)
914 (nth 1 (nth 1 expr))))))
916 (math-defsimplify calcFunc-arctan
917 (or (and (math-looks-negp (nth 1 expr))
918 (math-neg (list 'calcFunc-arctan (math-neg (nth 1 expr)))))
919 (and (eq (nth 1 expr) 1)
920 (math-div (math-half-circle t) 4))
921 (and math-living-dangerously
922 (eq (car-safe (nth 1 expr)) 'calcFunc-tan)
923 (nth 1 (nth 1 expr)))))
925 (math-defsimplify calcFunc-arcsinh
926 (or (and (math-looks-negp (nth 1 expr))
927 (math-neg (list 'calcFunc-arcsinh (math-neg (nth 1 expr)))))
928 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
929 (or math-living-dangerously
930 (math-known-realp (nth 1 (nth 1 expr))))
931 (nth 1 (nth 1 expr)))))
933 (math-defsimplify calcFunc-arccosh
934 (and (eq (car-safe (nth 1 expr)) 'calcFunc-cosh)
935 (or math-living-dangerously
936 (math-known-realp (nth 1 (nth 1 expr))))
937 (nth 1 (nth 1 expr))))
939 (math-defsimplify calcFunc-arctanh
940 (or (and (math-looks-negp (nth 1 expr))
941 (math-neg (list 'calcFunc-arctanh (math-neg (nth 1 expr)))))
942 (and (eq (car-safe (nth 1 expr)) 'calcFunc-tanh)
943 (or math-living-dangerously
944 (math-known-realp (nth 1 (nth 1 expr))))
945 (nth 1 (nth 1 expr)))))
947 (math-defsimplify calcFunc-sqrt
948 (math-simplify-sqrt))
950 (defun math-simplify-sqrt ()
951 (or (and (eq (car-safe (nth 1 expr)) 'frac)
952 (math-div (list 'calcFunc-sqrt (math-mul (nth 1 (nth 1 expr))
953 (nth 2 (nth 1 expr))))
954 (nth 2 (nth 1 expr))))
955 (let ((fac (if (math-objectp (nth 1 expr))
956 (math-squared-factor (nth 1 expr))
957 (math-common-constant-factor (nth 1 expr)))))
958 (and fac (not (eq fac 1))
959 (math-mul (math-normalize (list 'calcFunc-sqrt fac))
960 (math-normalize
961 (list 'calcFunc-sqrt
962 (math-cancel-common-factor (nth 1 expr) fac))))))
963 (and math-living-dangerously
964 (or (and (eq (car-safe (nth 1 expr)) '-)
965 (math-equal-int (nth 1 (nth 1 expr)) 1)
966 (eq (car-safe (nth 2 (nth 1 expr))) '^)
967 (math-equal-int (nth 2 (nth 2 (nth 1 expr))) 2)
968 (or (and (eq (car-safe (nth 1 (nth 2 (nth 1 expr))))
969 'calcFunc-sin)
970 (list 'calcFunc-cos
971 (nth 1 (nth 1 (nth 2 (nth 1 expr))))))
972 (and (eq (car-safe (nth 1 (nth 2 (nth 1 expr))))
973 'calcFunc-cos)
974 (list 'calcFunc-sin
975 (nth 1 (nth 1 (nth 2 (nth 1 expr))))))))
976 (and (eq (car-safe (nth 1 expr)) '-)
977 (math-equal-int (nth 2 (nth 1 expr)) 1)
978 (eq (car-safe (nth 1 (nth 1 expr))) '^)
979 (math-equal-int (nth 2 (nth 1 (nth 1 expr))) 2)
980 (and (eq (car-safe (nth 1 (nth 1 (nth 1 expr))))
981 'calcFunc-cosh)
982 (list 'calcFunc-sinh
983 (nth 1 (nth 1 (nth 1 (nth 1 expr)))))))
984 (and (eq (car-safe (nth 1 expr)) '+)
985 (let ((a (nth 1 (nth 1 expr)))
986 (b (nth 2 (nth 1 expr))))
987 (and (or (and (math-equal-int a 1)
988 (setq a b b (nth 1 (nth 1 expr))))
989 (math-equal-int b 1))
990 (eq (car-safe a) '^)
991 (math-equal-int (nth 2 a) 2)
992 (or (and (eq (car-safe (nth 1 a)) 'calcFunc-sinh)
993 (list 'calcFunc-cosh (nth 1 (nth 1 a))))
994 (and (eq (car-safe (nth 1 a)) 'calcFunc-tan)
995 (list '/ 1 (list 'calcFunc-cos
996 (nth 1 (nth 1 a)))))))))
997 (and (eq (car-safe (nth 1 expr)) '^)
998 (list '^
999 (nth 1 (nth 1 expr))
1000 (math-div (nth 2 (nth 1 expr)) 2)))
1001 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sqrt)
1002 (list '^ (nth 1 (nth 1 expr)) (math-div 1 4)))
1003 (and (memq (car-safe (nth 1 expr)) '(* /))
1004 (list (car (nth 1 expr))
1005 (list 'calcFunc-sqrt (nth 1 (nth 1 expr)))
1006 (list 'calcFunc-sqrt (nth 2 (nth 1 expr)))))
1007 (and (memq (car-safe (nth 1 expr)) '(+ -))
1008 (not (math-any-floats (nth 1 expr)))
1009 (let ((f (calcFunc-factors (calcFunc-expand
1010 (nth 1 expr)))))
1011 (and (math-vectorp f)
1012 (or (> (length f) 2)
1013 (> (nth 2 (nth 1 f)) 1))
1014 (let ((out 1) (rest 1) (sums 1) fac pow)
1015 (while (setq f (cdr f))
1016 (setq fac (nth 1 (car f))
1017 pow (nth 2 (car f)))
1018 (if (> pow 1)
1019 (setq out (math-mul out (math-pow
1020 fac (/ pow 2)))
1021 pow (% pow 2)))
1022 (if (> pow 0)
1023 (if (memq (car-safe fac) '(+ -))
1024 (setq sums (math-mul-thru sums fac))
1025 (setq rest (math-mul rest fac)))))
1026 (and (not (and (eq out 1) (memq rest '(1 -1))))
1027 (math-mul
1029 (list 'calcFunc-sqrt
1030 (math-mul sums rest))))))))))))
1032 ;;; Rather than factoring x into primes, just check for the first ten primes.
1033 (defun math-squared-factor (x)
1034 (if (Math-integerp x)
1035 (let ((prsqr '(4 9 25 49 121 169 289 361 529 841))
1036 (fac 1)
1037 res)
1038 (while prsqr
1039 (if (eq (cdr (setq res (math-idivmod x (car prsqr)))) 0)
1040 (setq x (car res)
1041 fac (math-mul fac (car prsqr)))
1042 (setq prsqr (cdr prsqr))))
1043 fac)))
1045 (math-defsimplify calcFunc-exp
1046 (math-simplify-exp (nth 1 expr)))
1048 (defun math-simplify-exp (x)
1049 (or (and (eq (car-safe x) 'calcFunc-ln)
1050 (nth 1 x))
1051 (and math-living-dangerously
1052 (or (and (eq (car-safe x) 'calcFunc-arcsinh)
1053 (math-add (nth 1 x)
1054 (list 'calcFunc-sqrt
1055 (math-add (math-sqr (nth 1 x)) 1))))
1056 (and (eq (car-safe x) 'calcFunc-arccosh)
1057 (math-add (nth 1 x)
1058 (list 'calcFunc-sqrt
1059 (math-sub (math-sqr (nth 1 x)) 1))))
1060 (and (eq (car-safe x) 'calcFunc-arctanh)
1061 (math-div (list 'calcFunc-sqrt (math-add 1 (nth 1 x)))
1062 (list 'calcFunc-sqrt (math-sub 1 (nth 1 x)))))
1063 (let ((m (math-should-expand-trig x 'exp)))
1064 (and m (integerp (car m))
1065 (list '^ (list 'calcFunc-exp (nth 1 m)) (car m))))))
1066 (and calc-symbolic-mode
1067 (math-known-imagp x)
1068 (let* ((ip (calcFunc-im x))
1069 (n (math-linear-in ip '(var pi var-pi)))
1070 s c)
1071 (and n
1072 (setq s (math-known-sin (car n) (nth 1 n) 120 0))
1073 (setq c (math-known-sin (car n) (nth 1 n) 120 300))
1074 (list '+ c (list '* s '(var i var-i))))))))
1076 (math-defsimplify calcFunc-ln
1077 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-exp)
1078 (or math-living-dangerously
1079 (math-known-realp (nth 1 (nth 1 expr))))
1080 (nth 1 (nth 1 expr)))
1081 (and (eq (car-safe (nth 1 expr)) '^)
1082 (equal (nth 1 (nth 1 expr)) '(var e var-e))
1083 (or math-living-dangerously
1084 (math-known-realp (nth 2 (nth 1 expr))))
1085 (nth 2 (nth 1 expr)))
1086 (and calc-symbolic-mode
1087 (math-known-negp (nth 1 expr))
1088 (math-add (list 'calcFunc-ln (math-neg (nth 1 expr)))
1089 '(* (var pi var-pi) (var i var-i))))
1090 (and calc-symbolic-mode
1091 (math-known-imagp (nth 1 expr))
1092 (let* ((ip (calcFunc-im (nth 1 expr)))
1093 (ips (math-possible-signs ip)))
1094 (or (and (memq ips '(4 6))
1095 (math-add (list 'calcFunc-ln ip)
1096 '(/ (* (var pi var-pi) (var i var-i)) 2)))
1097 (and (memq ips '(1 3))
1098 (math-sub (list 'calcFunc-ln (math-neg ip))
1099 '(/ (* (var pi var-pi) (var i var-i)) 2))))))))
1101 (math-defsimplify ^
1102 (math-simplify-pow))
1104 (defun math-simplify-pow ()
1105 (or (and math-living-dangerously
1106 (or (and (eq (car-safe (nth 1 expr)) '^)
1107 (list '^
1108 (nth 1 (nth 1 expr))
1109 (math-mul (nth 2 expr) (nth 2 (nth 1 expr)))))
1110 (and (eq (car-safe (nth 1 expr)) 'calcFunc-sqrt)
1111 (list '^
1112 (nth 1 (nth 1 expr))
1113 (math-div (nth 2 expr) 2)))
1114 (and (memq (car-safe (nth 1 expr)) '(* /))
1115 (list (car (nth 1 expr))
1116 (list '^ (nth 1 (nth 1 expr)) (nth 2 expr))
1117 (list '^ (nth 2 (nth 1 expr)) (nth 2 expr))))))
1118 (and (math-equal-int (nth 1 expr) 10)
1119 (eq (car-safe (nth 2 expr)) 'calcFunc-log10)
1120 (nth 1 (nth 2 expr)))
1121 (and (equal (nth 1 expr) '(var e var-e))
1122 (math-simplify-exp (nth 2 expr)))
1123 (and (eq (car-safe (nth 1 expr)) 'calcFunc-exp)
1124 (not math-integrating)
1125 (list 'calcFunc-exp (math-mul (nth 1 (nth 1 expr)) (nth 2 expr))))
1126 (and (equal (nth 1 expr) '(var i var-i))
1127 (math-imaginary-i)
1128 (math-num-integerp (nth 2 expr))
1129 (let ((x (math-mod (math-trunc (nth 2 expr)) 4)))
1130 (cond ((eq x 0) 1)
1131 ((eq x 1) (nth 1 expr))
1132 ((eq x 2) -1)
1133 ((eq x 3) (math-neg (nth 1 expr))))))
1134 (and math-integrating
1135 (integerp (nth 2 expr))
1136 (>= (nth 2 expr) 2)
1137 (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-cos)
1138 (math-mul (math-pow (nth 1 expr) (- (nth 2 expr) 2))
1139 (math-sub 1
1140 (math-sqr
1141 (list 'calcFunc-sin
1142 (nth 1 (nth 1 expr)))))))
1143 (and (eq (car-safe (nth 1 expr)) 'calcFunc-cosh)
1144 (math-mul (math-pow (nth 1 expr) (- (nth 2 expr) 2))
1145 (math-add 1
1146 (math-sqr
1147 (list 'calcFunc-sinh
1148 (nth 1 (nth 1 expr)))))))))
1149 (and (eq (car-safe (nth 2 expr)) 'frac)
1150 (Math-ratp (nth 1 expr))
1151 (Math-posp (nth 1 expr))
1152 (if (equal (nth 2 expr) '(frac 1 2))
1153 (list 'calcFunc-sqrt (nth 1 expr))
1154 (let ((flr (math-floor (nth 2 expr))))
1155 (and (not (Math-zerop flr))
1156 (list '* (list '^ (nth 1 expr) flr)
1157 (list '^ (nth 1 expr)
1158 (math-sub (nth 2 expr) flr)))))))
1159 (and (eq (math-quarter-integer (nth 2 expr)) 2)
1160 (let ((temp (math-simplify-sqrt)))
1161 (and temp
1162 (list '^ temp (math-mul (nth 2 expr) 2)))))))
1164 (math-defsimplify calcFunc-log10
1165 (and (eq (car-safe (nth 1 expr)) '^)
1166 (math-equal-int (nth 1 (nth 1 expr)) 10)
1167 (or math-living-dangerously
1168 (math-known-realp (nth 2 (nth 1 expr))))
1169 (nth 2 (nth 1 expr))))
1172 (math-defsimplify calcFunc-erf
1173 (or (and (math-looks-negp (nth 1 expr))
1174 (math-neg (list 'calcFunc-erf (math-neg (nth 1 expr)))))
1175 (and (eq (car-safe (nth 1 expr)) 'calcFunc-conj)
1176 (list 'calcFunc-conj (list 'calcFunc-erf (nth 1 (nth 1 expr)))))))
1178 (math-defsimplify calcFunc-erfc
1179 (or (and (math-looks-negp (nth 1 expr))
1180 (math-sub 2 (list 'calcFunc-erfc (math-neg (nth 1 expr)))))
1181 (and (eq (car-safe (nth 1 expr)) 'calcFunc-conj)
1182 (list 'calcFunc-conj (list 'calcFunc-erfc (nth 1 (nth 1 expr)))))))
1185 (defun math-linear-in (expr term &optional always)
1186 (if (math-expr-contains expr term)
1187 (let* ((calc-prefer-frac t)
1188 (p (math-is-polynomial expr term 1)))
1189 (and (cdr p)
1191 (and always (list expr 0))))
1193 (defun math-multiple-of (expr term)
1194 (let ((p (math-linear-in expr term)))
1195 (and p
1196 (math-zerop (car p))
1197 (nth 1 p))))
1199 ; not perfect, but it'll do
1200 (defun math-integer-plus (expr)
1201 (cond ((Math-integerp expr)
1202 (list 0 expr))
1203 ((and (memq (car expr) '(+ -))
1204 (Math-integerp (nth 1 expr)))
1205 (list (if (eq (car expr) '+) (nth 2 expr) (math-neg (nth 2 expr)))
1206 (nth 1 expr)))
1207 ((and (memq (car expr) '(+ -))
1208 (Math-integerp (nth 2 expr)))
1209 (list (nth 1 expr)
1210 (if (eq (car expr) '+) (nth 2 expr) (math-neg (nth 2 expr)))))
1211 (t nil)))
1213 (defun math-is-linear (expr &optional always)
1214 (let ((offset nil)
1215 (coef nil))
1216 (if (eq (car-safe expr) '+)
1217 (if (Math-objectp (nth 1 expr))
1218 (setq offset (nth 1 expr)
1219 expr (nth 2 expr))
1220 (if (Math-objectp (nth 2 expr))
1221 (setq offset (nth 2 expr)
1222 expr (nth 1 expr))))
1223 (if (eq (car-safe expr) '-)
1224 (if (Math-objectp (nth 1 expr))
1225 (setq offset (nth 1 expr)
1226 expr (math-neg (nth 2 expr)))
1227 (if (Math-objectp (nth 2 expr))
1228 (setq offset (math-neg (nth 2 expr))
1229 expr (nth 1 expr))))))
1230 (setq coef (math-is-multiple expr always))
1231 (if offset
1232 (list offset (or (car coef) 1) (or (nth 1 coef) expr))
1233 (if coef
1234 (cons 0 coef)))))
1236 (defun math-is-multiple (expr &optional always)
1237 (or (if (eq (car-safe expr) '*)
1238 (if (Math-objectp (nth 1 expr))
1239 (list (nth 1 expr) (nth 2 expr)))
1240 (if (eq (car-safe expr) '/)
1241 (if (and (Math-objectp (nth 1 expr))
1242 (not (math-equal-int (nth 1 expr) 1)))
1243 (list (nth 1 expr) (math-div 1 (nth 2 expr)))
1244 (if (Math-objectp (nth 2 expr))
1245 (list (math-div 1 (nth 2 expr)) (nth 1 expr))
1246 (let ((res (math-is-multiple (nth 1 expr))))
1247 (if res
1248 (list (car res)
1249 (math-div (nth 2 (nth 1 expr)) (nth 2 expr)))
1250 (setq res (math-is-multiple (nth 2 expr)))
1251 (if res
1252 (list (math-div 1 (car res))
1253 (math-div (nth 1 expr)
1254 (nth 2 (nth 2 expr)))))))))
1255 (if (eq (car-safe expr) 'neg)
1256 (list -1 (nth 1 expr)))))
1257 (if (Math-objvecp expr)
1258 (and (eq always 1)
1259 (list expr 1))
1260 (and always
1261 (list 1 expr)))))
1263 (defun calcFunc-lin (expr &optional var)
1264 (if var
1265 (let ((res (math-linear-in expr var t)))
1266 (or res (math-reject-arg expr "Linear term expected"))
1267 (list 'vec (car res) (nth 1 res) var))
1268 (let ((res (math-is-linear expr t)))
1269 (or res (math-reject-arg expr "Linear term expected"))
1270 (cons 'vec res))))
1272 (defun calcFunc-linnt (expr &optional var)
1273 (if var
1274 (let ((res (math-linear-in expr var)))
1275 (or res (math-reject-arg expr "Linear term expected"))
1276 (list 'vec (car res) (nth 1 res) var))
1277 (let ((res (math-is-linear expr)))
1278 (or res (math-reject-arg expr "Linear term expected"))
1279 (cons 'vec res))))
1281 (defun calcFunc-islin (expr &optional var)
1282 (if (and (Math-objvecp expr) (not var))
1284 (calcFunc-lin expr var)
1287 (defun calcFunc-islinnt (expr &optional var)
1288 (if (Math-objvecp expr)
1290 (calcFunc-linnt expr var)
1296 ;;; Simple operations on expressions.
1298 ;;; Return number of occurrences of thing in expr, or nil if none.
1299 (defun math-expr-contains-count (expr thing)
1300 (cond ((equal expr thing) 1)
1301 ((Math-primp expr) nil)
1303 (let ((num 0))
1304 (while (setq expr (cdr expr))
1305 (setq num (+ num (or (math-expr-contains-count
1306 (car expr) thing) 0))))
1307 (and (> num 0)
1308 num)))))
1310 (defun math-expr-contains (expr thing)
1311 (cond ((equal expr thing) 1)
1312 ((Math-primp expr) nil)
1314 (while (and (setq expr (cdr expr))
1315 (not (math-expr-contains (car expr) thing))))
1316 expr)))
1318 ;;; Return non-nil if any variable of thing occurs in expr.
1319 (defun math-expr-depends (expr thing)
1320 (if (Math-primp thing)
1321 (and (eq (car-safe thing) 'var)
1322 (math-expr-contains expr thing))
1323 (while (and (setq thing (cdr thing))
1324 (not (math-expr-depends expr (car thing)))))
1325 thing))
1327 ;;; Substitute all occurrences of old for new in expr (non-destructive).
1328 (defun math-expr-subst (expr old new)
1329 (math-expr-subst-rec expr))
1331 (defalias 'calcFunc-subst 'math-expr-subst)
1333 (defun math-expr-subst-rec (expr)
1334 (cond ((equal expr old) new)
1335 ((Math-primp expr) expr)
1336 ((memq (car expr) '(calcFunc-deriv
1337 calcFunc-tderiv))
1338 (if (= (length expr) 2)
1339 (if (equal (nth 1 expr) old)
1340 (append expr (list new))
1341 expr)
1342 (list (car expr) (nth 1 expr)
1343 (math-expr-subst-rec (nth 2 expr)))))
1345 (cons (car expr)
1346 (mapcar 'math-expr-subst-rec (cdr expr))))))
1348 ;;; Various measures of the size of an expression.
1349 (defun math-expr-weight (expr)
1350 (if (Math-primp expr)
1352 (let ((w 1))
1353 (while (setq expr (cdr expr))
1354 (setq w (+ w (math-expr-weight (car expr)))))
1355 w)))
1357 (defun math-expr-height (expr)
1358 (if (Math-primp expr)
1360 (let ((h 0))
1361 (while (setq expr (cdr expr))
1362 (setq h (max h (math-expr-height (car expr)))))
1363 (1+ h))))
1368 ;;; Polynomial operations (to support the integrator and solve-for).
1370 (defun calcFunc-collect (expr base)
1371 (let ((p (math-is-polynomial expr base 50 t)))
1372 (if (cdr p)
1373 (math-normalize ; fix selection bug
1374 (math-build-polynomial-expr p base))
1375 expr)))
1377 ;;; If expr is of the form "a + bx + cx^2 + ...", return the list (a b c ...),
1378 ;;; else return nil if not in polynomial form. If "loose", coefficients
1379 ;;; may contain x, e.g., sin(x) + cos(x) x^2 is a loose polynomial in x.
1380 (defun math-is-polynomial (expr var &optional degree loose)
1381 (let* ((math-poly-base-variable (if loose
1382 (if (eq loose 'gen) var '(var XXX XXX))
1383 math-poly-base-variable))
1384 (poly (math-is-poly-rec expr math-poly-neg-powers)))
1385 (and (or (null degree)
1386 (<= (length poly) (1+ degree)))
1387 poly)))
1389 (defun math-is-poly-rec (expr negpow)
1390 (math-poly-simplify
1391 (or (cond ((or (equal expr var)
1392 (eq (car-safe expr) '^))
1393 (let ((pow 1)
1394 (expr expr))
1395 (or (equal expr var)
1396 (setq pow (nth 2 expr)
1397 expr (nth 1 expr)))
1398 (or (eq math-poly-mult-powers 1)
1399 (setq pow (let ((m (math-is-multiple pow 1)))
1400 (and (eq (car-safe (car m)) 'cplx)
1401 (Math-zerop (nth 1 (car m)))
1402 (setq m (list (nth 2 (car m))
1403 (math-mul (nth 1 m)
1404 '(var i var-i)))))
1405 (and (if math-poly-mult-powers
1406 (equal math-poly-mult-powers
1407 (nth 1 m))
1408 (setq math-poly-mult-powers (nth 1 m)))
1409 (or (equal expr var)
1410 (eq math-poly-mult-powers 1))
1411 (car m)))))
1412 (if (consp pow)
1413 (progn
1414 (setq pow (math-to-simple-fraction pow))
1415 (and (eq (car-safe pow) 'frac)
1416 math-poly-frac-powers
1417 (equal expr var)
1418 (setq math-poly-frac-powers
1419 (calcFunc-lcm math-poly-frac-powers
1420 (nth 2 pow))))))
1421 (or (memq math-poly-frac-powers '(1 nil))
1422 (setq pow (math-mul pow math-poly-frac-powers)))
1423 (if (integerp pow)
1424 (if (and (= pow 1)
1425 (equal expr var))
1426 (list 0 1)
1427 (if (natnump pow)
1428 (let ((p1 (if (equal expr var)
1429 (list 0 1)
1430 (math-is-poly-rec expr nil)))
1431 (n pow)
1432 (accum (list 1)))
1433 (and p1
1434 (or (null degree)
1435 (<= (* (1- (length p1)) n) degree))
1436 (progn
1437 (while (>= n 1)
1438 (setq accum (math-poly-mul accum p1)
1439 n (1- n)))
1440 accum)))
1441 (and negpow
1442 (math-is-poly-rec expr nil)
1443 (setq math-poly-neg-powers
1444 (cons (math-pow expr (- pow))
1445 math-poly-neg-powers))
1446 (list (list '^ expr pow))))))))
1447 ((Math-objectp expr)
1448 (list expr))
1449 ((memq (car expr) '(+ -))
1450 (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
1451 (and p1
1452 (let ((p2 (math-is-poly-rec (nth 2 expr) negpow)))
1453 (and p2
1454 (math-poly-mix p1 1 p2
1455 (if (eq (car expr) '+) 1 -1)))))))
1456 ((eq (car expr) 'neg)
1457 (mapcar 'math-neg (math-is-poly-rec (nth 1 expr) negpow)))
1458 ((eq (car expr) '*)
1459 (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
1460 (and p1
1461 (let ((p2 (math-is-poly-rec (nth 2 expr) negpow)))
1462 (and p2
1463 (or (null degree)
1464 (<= (- (+ (length p1) (length p2)) 2) degree))
1465 (math-poly-mul p1 p2))))))
1466 ((eq (car expr) '/)
1467 (and (or (not (math-poly-depends (nth 2 expr) var))
1468 (and negpow
1469 (math-is-poly-rec (nth 2 expr) nil)
1470 (setq math-poly-neg-powers
1471 (cons (nth 2 expr) math-poly-neg-powers))))
1472 (not (Math-zerop (nth 2 expr)))
1473 (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
1474 (mapcar (function (lambda (x) (math-div x (nth 2 expr))))
1475 p1))))
1476 ((and (eq (car expr) 'calcFunc-exp)
1477 (equal var '(var e var-e)))
1478 (math-is-poly-rec (list '^ var (nth 1 expr)) negpow))
1479 ((and (eq (car expr) 'calcFunc-sqrt)
1480 math-poly-frac-powers)
1481 (math-is-poly-rec (list '^ (nth 1 expr) '(frac 1 2)) negpow))
1482 (t nil))
1483 (and (or (not (math-poly-depends expr var))
1484 loose)
1485 (not (eq (car expr) 'vec))
1486 (list expr)))))
1488 ;;; Check if expr is a polynomial in var; if so, return its degree.
1489 (defun math-polynomial-p (expr var)
1490 (cond ((equal expr var) 1)
1491 ((Math-primp expr) 0)
1492 ((memq (car expr) '(+ -))
1493 (let ((p1 (math-polynomial-p (nth 1 expr) var))
1495 (and p1 (setq p2 (math-polynomial-p (nth 2 expr) var))
1496 (max p1 p2))))
1497 ((eq (car expr) '*)
1498 (let ((p1 (math-polynomial-p (nth 1 expr) var))
1500 (and p1 (setq p2 (math-polynomial-p (nth 2 expr) var))
1501 (+ p1 p2))))
1502 ((eq (car expr) 'neg)
1503 (math-polynomial-p (nth 1 expr) var))
1504 ((and (eq (car expr) '/)
1505 (not (math-poly-depends (nth 2 expr) var)))
1506 (math-polynomial-p (nth 1 expr) var))
1507 ((and (eq (car expr) '^)
1508 (natnump (nth 2 expr)))
1509 (let ((p1 (math-polynomial-p (nth 1 expr) var)))
1510 (and p1 (* p1 (nth 2 expr)))))
1511 ((math-poly-depends expr var) nil)
1512 (t 0)))
1514 (defun math-poly-depends (expr var)
1515 (if math-poly-base-variable
1516 (math-expr-contains expr math-poly-base-variable)
1517 (math-expr-depends expr var)))
1519 ;;; Find the variable (or sub-expression) which is the base of polynomial expr.
1520 (defun math-polynomial-base (mpb-top-expr &optional mpb-pred)
1521 (or mpb-pred
1522 (setq mpb-pred (function (lambda (base) (math-polynomial-p
1523 mpb-top-expr base)))))
1524 (or (let ((const-ok nil))
1525 (math-polynomial-base-rec mpb-top-expr))
1526 (let ((const-ok t))
1527 (math-polynomial-base-rec mpb-top-expr))))
1529 (defun math-polynomial-base-rec (mpb-expr)
1530 (and (not (Math-objvecp mpb-expr))
1531 (or (and (memq (car mpb-expr) '(+ - *))
1532 (or (math-polynomial-base-rec (nth 1 mpb-expr))
1533 (math-polynomial-base-rec (nth 2 mpb-expr))))
1534 (and (memq (car mpb-expr) '(/ neg))
1535 (math-polynomial-base-rec (nth 1 mpb-expr)))
1536 (and (eq (car mpb-expr) '^)
1537 (math-polynomial-base-rec (nth 1 mpb-expr)))
1538 (and (eq (car mpb-expr) 'calcFunc-exp)
1539 (math-polynomial-base-rec '(var e var-e)))
1540 (and (or const-ok (math-expr-contains-vars mpb-expr))
1541 (funcall mpb-pred mpb-expr)
1542 mpb-expr))))
1544 ;;; Return non-nil if expr refers to any variables.
1545 (defun math-expr-contains-vars (expr)
1546 (or (eq (car-safe expr) 'var)
1547 (and (not (Math-primp expr))
1548 (progn
1549 (while (and (setq expr (cdr expr))
1550 (not (math-expr-contains-vars (car expr)))))
1551 expr))))
1553 ;;; Simplify a polynomial in list form by stripping off high-end zeros.
1554 ;;; This always leaves the constant part, i.e., nil->nil and nonnil->nonnil.
1555 (defun math-poly-simplify (p)
1556 (and p
1557 (if (Math-zerop (nth (1- (length p)) p))
1558 (let ((pp (copy-sequence p)))
1559 (while (and (cdr pp)
1560 (Math-zerop (nth (1- (length pp)) pp)))
1561 (setcdr (nthcdr (- (length pp) 2) pp) nil))
1563 p)))
1565 ;;; Compute ac*a + bc*b for polynomials in list form a, b and
1566 ;;; coefficients ac, bc. Result may be unsimplified.
1567 (defun math-poly-mix (a ac b bc)
1568 (and (or a b)
1569 (cons (math-add (math-mul (or (car a) 0) ac)
1570 (math-mul (or (car b) 0) bc))
1571 (math-poly-mix (cdr a) ac (cdr b) bc))))
1573 (defun math-poly-zerop (a)
1574 (or (null a)
1575 (and (null (cdr a)) (Math-zerop (car a)))))
1577 ;;; Multiply two polynomials in list form.
1578 (defun math-poly-mul (a b)
1579 (and a b
1580 (math-poly-mix b (car a)
1581 (math-poly-mul (cdr a) (cons 0 b)) 1)))
1583 ;;; Build an expression from a polynomial list.
1584 (defun math-build-polynomial-expr (p var)
1585 (if p
1586 (if (Math-numberp var)
1587 (math-with-extra-prec 1
1588 (let* ((rp (reverse p))
1589 (accum (car rp)))
1590 (while (setq rp (cdr rp))
1591 (setq accum (math-add (car rp) (math-mul accum var))))
1592 accum))
1593 (let* ((rp (reverse p))
1594 (n (1- (length rp)))
1595 (accum (math-mul (car rp) (math-pow var n)))
1596 term)
1597 (while (setq rp (cdr rp))
1598 (setq n (1- n))
1599 (or (math-zerop (car rp))
1600 (setq accum (list (if (math-looks-negp (car rp)) '- '+)
1601 accum
1602 (math-mul (if (math-looks-negp (car rp))
1603 (math-neg (car rp))
1604 (car rp))
1605 (math-pow var n))))))
1606 accum))
1610 (defun math-to-simple-fraction (f)
1611 (or (and (eq (car-safe f) 'float)
1612 (or (and (>= (nth 2 f) 0)
1613 (math-scale-int (nth 1 f) (nth 2 f)))
1614 (and (integerp (nth 1 f))
1615 (> (nth 1 f) -1000)
1616 (< (nth 1 f) 1000)
1617 (math-make-frac (nth 1 f)
1618 (math-scale-int 1 (- (nth 2 f)))))))
1621 ;;; calc-alg.el ends here