1 ;;; calc-alg.el --- algebraic functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
4 ;; 2005 Free Software Foundation, Inc.
6 ;; Author: David Gillespie <daveg@synaptics.com>
7 ;; Maintainer: Jay Belanger <belanger@truman.edu>
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.
30 ;; This file is autoloaded from calc-ext.el.
37 (defun calc-alg-evaluate (arg)
40 (calc-with-default-simplification
41 (let ((math-simplify-only nil))
42 (calc-modify-simplify-mode arg)
43 (calc-enter-result 1 "dsmp" (calc-top 1))))))
45 (defun calc-modify-simplify-mode (arg)
46 (if (= (math-abs arg) 2)
47 (setq calc-simplify-mode 'alg)
48 (if (>= (math-abs arg) 3)
49 (setq calc-simplify-mode 'ext)))
51 (setq calc-simplify-mode (list calc-simplify-mode))))
53 (defun calc-simplify ()
56 (calc-with-default-simplification
57 (calc-enter-result 1 "simp" (math-simplify (calc-top-n 1))))))
59 (defun calc-simplify-extended ()
62 (calc-with-default-simplification
63 (calc-enter-result 1 "esmp" (math-simplify-extended (calc-top-n 1))))))
65 (defun calc-expand-formula (arg)
68 (calc-with-default-simplification
69 (let ((math-simplify-only nil))
70 (calc-modify-simplify-mode arg)
71 (calc-enter-result 1 "expf"
73 (let ((math-expand-formulas t))
75 (let ((top (calc-top-n 1)))
76 (or (math-expand-formula top)
79 (defun calc-factor (arg)
82 (calc-unary-op "fctr" (if (calc-is-hyperbolic)
83 'calcFunc-factors 'calcFunc-factor)
86 (defun calc-expand (n)
89 (calc-enter-result 1 "expa"
90 (append (list 'calcFunc-expand
92 (and n (list (prefix-numeric-value n)))))))
94 (defun calc-collect (&optional var)
95 (interactive "sCollect terms involving: ")
97 (if (or (equal var "") (equal var "$") (null var))
98 (calc-enter-result 2 "clct" (cons 'calcFunc-collect
100 (let ((var (math-read-expr var)))
101 (if (eq (car-safe var) 'error)
102 (error "Bad format in expression: %s" (nth 1 var)))
103 (calc-enter-result 1 "clct" (list 'calcFunc-collect
107 (defun calc-apart (arg)
110 (calc-unary-op "aprt" 'calcFunc-apart arg)))
112 (defun calc-normalize-rat (arg)
115 (calc-unary-op "nrat" 'calcFunc-nrat arg)))
117 (defun calc-poly-gcd (arg)
120 (calc-binary-op "pgcd" 'calcFunc-pgcd arg)))
123 (defun calc-poly-div (arg)
126 (let ((calc-poly-div-remainder nil))
127 (calc-binary-op "pdiv" 'calcFunc-pdiv arg)
128 (if (and calc-poly-div-remainder (null arg))
130 (calc-clear-command-flag 'clear-message)
131 (calc-record calc-poly-div-remainder "prem")
132 (if (not (Math-zerop calc-poly-div-remainder))
133 (message "(Remainder was %s)"
134 (math-format-flat-expr calc-poly-div-remainder 0))
135 (message "(No remainder)")))))))
137 (defun calc-poly-rem (arg)
140 (calc-binary-op "prem" 'calcFunc-prem arg)))
142 (defun calc-poly-div-rem (arg)
145 (if (calc-is-hyperbolic)
146 (calc-binary-op "pdvr" 'calcFunc-pdivide arg)
147 (calc-binary-op "pdvr" 'calcFunc-pdivrem arg))))
149 (defun calc-substitute (&optional oldname newname)
150 (interactive "sSubstitute old: ")
152 (let (old new (num 1) expr)
153 (if (or (equal oldname "") (equal oldname "$") (null oldname))
154 (setq new (calc-top-n 1)
159 (progn (calc-unread-command ?\C-a)
160 (setq newname (read-string (concat "Substitute old: "
164 (if (or (equal newname "") (equal newname "$") (null newname))
165 (setq new (calc-top-n 1)
168 (setq new (if (stringp newname) (math-read-expr newname) newname))
169 (if (eq (car-safe new) 'error)
170 (error "Bad format in expression: %s" (nth 1 new)))
171 (setq expr (calc-top-n 1)))
172 (setq old (if (stringp oldname) (math-read-expr oldname) oldname))
173 (if (eq (car-safe old) 'error)
174 (error "Bad format in expression: %s" (nth 1 old)))
175 (or (math-expr-contains expr old)
176 (error "No occurrences found")))
177 (calc-enter-result num "sbst" (math-expr-subst expr old new)))))
180 (defun calc-has-rules (name)
181 (setq name (calc-var-value name))
183 (memq (car name) '(vec calcFunc-assign calcFunc-condition))
186 ;; math-eval-rules-cache and math-eval-rules-cache-other are
187 ;; declared in calc.el, but are used here by math-recompile-eval-rules.
188 (defvar math-eval-rules-cache)
189 (defvar math-eval-rules-cache-other)
191 (defun math-recompile-eval-rules ()
192 (setq math-eval-rules-cache (and (calc-has-rules 'var-EvalRules)
193 (math-compile-rewrites
194 '(var EvalRules var-EvalRules)))
195 math-eval-rules-cache-other (assq nil math-eval-rules-cache)
196 math-eval-rules-cache-tag (calc-var-value 'var-EvalRules)))
199 ;;; Try to expand a formula according to its definition.
200 (defun math-expand-formula (expr)
203 (or (get (car expr) 'calc-user-defn)
204 (get (car expr) 'math-expandable))
205 (let ((res (let ((math-expand-formulas t))
206 (apply (car expr) (cdr expr)))))
207 (and (not (eq (car-safe res) (car expr)))
213 ;;; True if A comes before B in a canonical ordering of expressions. [P X X]
214 (defun math-beforep (a b) ; [Public]
215 (cond ((and (Math-realp a) (Math-realp b))
216 (let ((comp (math-compare a b)))
220 (> (length (memq (car-safe a)
221 '(bigneg nil bigpos frac float)))
222 (length (memq (car-safe b)
223 '(bigneg nil bigpos frac float))))))))
224 ((equal b '(neg (var inf var-inf))) nil)
225 ((equal a '(neg (var inf var-inf))) t)
226 ((equal a '(var inf var-inf)) nil)
227 ((equal b '(var inf var-inf)) t)
229 (if (and (eq (car-safe b) 'intv) (math-intv-constp b))
230 (if (or (math-beforep a (nth 2 b)) (Math-equal a (nth 2 b)))
235 (if (and (eq (car-safe a) 'intv) (math-intv-constp a))
236 (if (math-beforep (nth 2 a) b)
240 ((and (eq (car a) 'intv) (eq (car b) 'intv)
241 (math-intv-constp a) (math-intv-constp b))
242 (let ((comp (math-compare (nth 2 a) (nth 2 b))))
243 (cond ((eq comp -1) t)
245 ((and (memq (nth 1 a) '(2 3)) (memq (nth 1 b) '(0 1))) t)
246 ((and (memq (nth 1 a) '(0 1)) (memq (nth 1 b) '(2 3))) nil)
247 ((eq (setq comp (math-compare (nth 3 a) (nth 3 b))) -1) t)
249 ((and (memq (nth 1 a) '(0 2)) (memq (nth 1 b) '(1 3))) t)
251 ((not (eq (not (Math-objectp a)) (not (Math-objectp b))))
254 (if (eq (car b) 'var)
255 (string-lessp (symbol-name (nth 1 a)) (symbol-name (nth 1 b)))
256 (not (Math-numberp b))))
257 ((eq (car b) 'var) (Math-numberp a))
258 ((eq (car a) (car b))
259 (while (and (setq a (cdr a) b (cdr b)) a
260 (equal (car a) (car b))))
263 (math-beforep (car a) (car b)))))
264 (t (string-lessp (symbol-name (car a)) (symbol-name (car b))))))
267 (defsubst math-simplify-extended (a)
268 (let ((math-living-dangerously t))
271 (defalias 'calcFunc-esimplify 'math-simplify-extended)
273 ;; math-top-only is local to math-simplify, but is used by
274 ;; math-simplify-step, which is called by math-simplify.
275 (defvar math-top-only)
277 (defun math-simplify (top-expr)
278 (let ((math-simplifying t)
279 (math-top-only (consp calc-simplify-mode))
280 (simp-rules (append (and (calc-has-rules 'var-AlgSimpRules)
281 '((var AlgSimpRules var-AlgSimpRules)))
282 (and math-living-dangerously
283 (calc-has-rules 'var-ExtSimpRules)
284 '((var ExtSimpRules var-ExtSimpRules)))
285 (and math-simplifying-units
286 (calc-has-rules 'var-UnitSimpRules)
287 '((var UnitSimpRules var-UnitSimpRules)))
288 (and math-integrating
289 (calc-has-rules 'var-IntegSimpRules)
290 '((var IntegSimpRules var-IntegSimpRules)))))
293 (let ((r simp-rules))
294 (setq res (math-simplify-step (math-normalize top-expr))
295 calc-simplify-mode '(nil)
296 top-expr (math-normalize res))
298 (setq top-expr (math-rewrite top-expr (car r)
299 '(neg (var inf var-inf)))
301 (calc-with-default-simplification
302 (while (let ((r simp-rules))
303 (setq res (math-normalize top-expr))
305 (setq res (math-rewrite res (car r))
307 (not (equal top-expr (setq res (math-simplify-step res)))))
308 (setq top-expr res)))))
311 (defalias 'calcFunc-simplify 'math-simplify)
313 ;;; The following has a "bug" in that if any recursive simplifications
314 ;;; occur only the first handler will be tried; this doesn't really
315 ;;; matter, since math-simplify-step is iterated to a fixed point anyway.
316 (defun math-simplify-step (a)
319 (let ((aa (if (or math-top-only
320 (memq (car a) '(calcFunc-quote calcFunc-condition
323 (cons (car a) (mapcar 'math-simplify-step (cdr a))))))
324 (and (symbolp (car aa))
325 (let ((handler (get (car aa) 'math-simplify)))
328 (equal (setq aa (or (funcall (car handler) aa)
331 (setq handler (cdr handler))))))
335 (defmacro math-defsimplify (funcs &rest code)
339 (list 'put (list 'quote func) ''math-simplify
341 (list 'get (list 'quote func) ''math-simplify)
344 (append '(lambda (math-simplify-expr))
346 (if (symbolp funcs) (list funcs) funcs))))
347 (put 'math-defsimplify 'lisp-indent-hook 1)
349 ;; The function created by math-defsimplify uses the variable
350 ;; math-simplify-expr, and so is used by functions in math-defsimplify
351 (defvar math-simplify-expr)
353 (math-defsimplify (+ -)
354 (math-simplify-plus))
356 (defun math-simplify-plus ()
357 (cond ((and (memq (car-safe (nth 1 math-simplify-expr)) '(+ -))
358 (Math-numberp (nth 2 (nth 1 math-simplify-expr)))
359 (not (Math-numberp (nth 2 math-simplify-expr))))
360 (let ((x (nth 2 math-simplify-expr))
361 (op (car math-simplify-expr)))
362 (setcar (cdr (cdr math-simplify-expr)) (nth 2 (nth 1 math-simplify-expr)))
363 (setcar math-simplify-expr (car (nth 1 math-simplify-expr)))
364 (setcar (cdr (cdr (nth 1 math-simplify-expr))) x)
365 (setcar (nth 1 math-simplify-expr) op)))
366 ((and (eq (car math-simplify-expr) '+)
367 (Math-numberp (nth 1 math-simplify-expr))
368 (not (Math-numberp (nth 2 math-simplify-expr))))
369 (let ((x (nth 2 math-simplify-expr)))
370 (setcar (cdr (cdr math-simplify-expr)) (nth 1 math-simplify-expr))
371 (setcar (cdr math-simplify-expr) x))))
372 (let ((aa math-simplify-expr)
374 (while (memq (car-safe (setq aaa (nth 1 aa))) '(+ -))
375 (if (setq temp (math-combine-sum (nth 2 aaa) (nth 2 math-simplify-expr)
377 (eq (car math-simplify-expr) '-) t))
379 (setcar (cdr (cdr math-simplify-expr)) temp)
380 (setcar math-simplify-expr '+)
381 (setcar (cdr (cdr aaa)) 0)))
382 (setq aa (nth 1 aa)))
383 (if (setq temp (math-combine-sum aaa (nth 2 math-simplify-expr)
384 nil (eq (car math-simplify-expr) '-) t))
386 (setcar (cdr (cdr math-simplify-expr)) temp)
387 (setcar math-simplify-expr '+)
388 (setcar (cdr aa) 0)))
392 (math-simplify-times))
394 (defun math-simplify-times ()
395 (if (eq (car-safe (nth 2 math-simplify-expr)) '*)
396 (and (math-beforep (nth 1 (nth 2 math-simplify-expr)) (nth 1 math-simplify-expr))
397 (or (math-known-scalarp (nth 1 math-simplify-expr) t)
398 (math-known-scalarp (nth 1 (nth 2 math-simplify-expr)) t))
399 (let ((x (nth 1 math-simplify-expr)))
400 (setcar (cdr math-simplify-expr) (nth 1 (nth 2 math-simplify-expr)))
401 (setcar (cdr (nth 2 math-simplify-expr)) x)))
402 (and (math-beforep (nth 2 math-simplify-expr) (nth 1 math-simplify-expr))
403 (or (math-known-scalarp (nth 1 math-simplify-expr) t)
404 (math-known-scalarp (nth 2 math-simplify-expr) t))
405 (let ((x (nth 2 math-simplify-expr)))
406 (setcar (cdr (cdr math-simplify-expr)) (nth 1 math-simplify-expr))
407 (setcar (cdr math-simplify-expr) x))))
408 (let ((aa math-simplify-expr)
410 (safe t) (scalar (math-known-scalarp (nth 1 math-simplify-expr))))
411 (if (and (Math-ratp (nth 1 math-simplify-expr))
412 (setq temp (math-common-constant-factor (nth 2 math-simplify-expr))))
414 (setcar (cdr (cdr math-simplify-expr))
415 (math-cancel-common-factor (nth 2 math-simplify-expr) temp))
416 (setcar (cdr math-simplify-expr) (math-mul (nth 1 math-simplify-expr) temp))))
417 (while (and (eq (car-safe (setq aaa (nth 2 aa))) '*)
419 (if (setq temp (math-combine-prod (nth 1 math-simplify-expr)
420 (nth 1 aaa) nil nil t))
422 (setcar (cdr math-simplify-expr) temp)
423 (setcar (cdr aaa) 1)))
424 (setq safe (or scalar (math-known-scalarp (nth 1 aaa) t))
426 (if (and (setq temp (math-combine-prod aaa (nth 1 math-simplify-expr) nil nil t))
429 (setcar (cdr math-simplify-expr) temp)
430 (setcar (cdr (cdr aa)) 1)))
431 (if (and (eq (car-safe (nth 1 math-simplify-expr)) 'frac)
432 (memq (nth 1 (nth 1 math-simplify-expr)) '(1 -1)))
433 (math-div (math-mul (nth 2 math-simplify-expr)
434 (nth 1 (nth 1 math-simplify-expr)))
435 (nth 2 (nth 1 math-simplify-expr)))
436 math-simplify-expr)))
439 (math-simplify-divide))
441 (defun math-simplify-divide ()
442 (let ((np (cdr math-simplify-expr))
444 (nn (and (or (eq (car math-simplify-expr) '/)
445 (not (Math-realp (nth 2 math-simplify-expr))))
446 (math-common-constant-factor (nth 2 math-simplify-expr))))
450 (setq n (and (or (eq (car math-simplify-expr) '/)
451 (not (Math-realp (nth 1 math-simplify-expr))))
452 (math-common-constant-factor (nth 1 math-simplify-expr))))
453 (if (and (eq (car-safe nn) 'frac) (eq (nth 1 nn) 1) (not n))
455 (setcar (cdr math-simplify-expr)
456 (math-mul (nth 2 nn) (nth 1 math-simplify-expr)))
457 (setcar (cdr (cdr math-simplify-expr))
458 (math-cancel-common-factor (nth 2 math-simplify-expr) nn))
459 (if (and (math-negp nn)
460 (setq op (assq (car math-simplify-expr) calc-tweak-eqn-table)))
461 (setcar math-simplify-expr (nth 1 op))))
462 (if (and n (not (eq (setq n (math-frac-gcd n nn)) 1)))
464 (setcar (cdr math-simplify-expr)
465 (math-cancel-common-factor (nth 1 math-simplify-expr) n))
466 (setcar (cdr (cdr math-simplify-expr))
467 (math-cancel-common-factor (nth 2 math-simplify-expr) n))
468 (if (and (math-negp n)
469 (setq op (assq (car math-simplify-expr)
470 calc-tweak-eqn-table)))
471 (setcar math-simplify-expr (nth 1 op))))))))
472 (if (and (eq (car-safe (car np)) '/)
473 (math-known-scalarp (nth 2 math-simplify-expr) t))
475 (setq np (cdr (nth 1 math-simplify-expr)))
476 (while (eq (car-safe (setq n (car np))) '*)
477 (and (math-known-scalarp (nth 2 n) t)
478 (math-simplify-divisor (cdr n) (cdr (cdr math-simplify-expr)) nil t))
479 (setq np (cdr (cdr n))))
480 (math-simplify-divisor np (cdr (cdr math-simplify-expr)) nil t)
482 np (cdr (cdr (nth 1 math-simplify-expr))))))
483 (while (eq (car-safe (setq n (car np))) '*)
484 (and (math-known-scalarp (nth 2 n) t)
485 (math-simplify-divisor (cdr n) (cdr (cdr math-simplify-expr)) nover t))
486 (setq np (cdr (cdr n))))
487 (math-simplify-divisor np (cdr (cdr math-simplify-expr)) nover t)
490 ;; The variables math-simplify-divisor-nover and math-simplify-divisor-dover
491 ;; are local variables for math-simplify-divisor, but are used by
492 ;; math-simplify-one-divisor.
493 (defvar math-simplify-divisor-nover)
494 (defvar math-simplify-divisor-dover)
496 (defun math-simplify-divisor (np dp math-simplify-divisor-nover
497 math-simplify-divisor-dover)
498 (cond ((eq (car-safe (car dp)) '/)
499 (math-simplify-divisor np (cdr (car dp))
500 math-simplify-divisor-nover
501 math-simplify-divisor-dover)
502 (and (math-known-scalarp (nth 1 (car dp)) t)
503 (math-simplify-divisor np (cdr (cdr (car dp)))
504 math-simplify-divisor-nover
505 (not math-simplify-divisor-dover))))
506 ((or (or (eq (car math-simplify-expr) '/)
507 (let ((signs (math-possible-signs (car np))))
508 (or (memq signs '(1 4))
509 (and (memq (car math-simplify-expr) '(calcFunc-eq calcFunc-neq))
511 math-living-dangerously)))
512 (math-numberp (car np)))
515 (scalar (math-known-scalarp (car np))))
516 (while (and (eq (car-safe (setq d (car dp))) '*)
518 (math-simplify-one-divisor np (cdr d))
519 (setq safe (or scalar (math-known-scalarp (nth 1 d) t))
522 (math-simplify-one-divisor np dp))))))
524 (defun math-simplify-one-divisor (np dp)
525 (let ((temp (math-combine-prod (car np) (car dp) math-simplify-divisor-nover
526 math-simplify-divisor-dover t))
530 (and (not (memq (car math-simplify-expr) '(/ calcFunc-eq calcFunc-neq)))
531 (math-known-negp (car dp))
532 (setq op (assq (car math-simplify-expr) calc-tweak-eqn-table))
533 (setcar math-simplify-expr (nth 1 op)))
534 (setcar np (if math-simplify-divisor-nover (math-div 1 temp) temp))
536 (and math-simplify-divisor-dover (not math-simplify-divisor-nover)
537 (eq (car math-simplify-expr) '/)
538 (eq (car-safe (car dp)) 'calcFunc-sqrt)
539 (Math-integerp (nth 1 (car dp)))
541 (setcar np (math-mul (car np)
542 (list 'calcFunc-sqrt (nth 1 (car dp)))))
543 (setcar dp (nth 1 (car dp))))))))
545 (defun math-common-constant-factor (expr)
546 (if (Math-realp expr)
548 (and (not (memq expr '(0 1 -1)))
550 (if (math-ratp (setq expr (math-to-simple-fraction expr)))
551 (math-common-constant-factor expr)))
552 (if (memq (car expr) '(+ - cplx sdev))
553 (let ((f1 (math-common-constant-factor (nth 1 expr)))
554 (f2 (math-common-constant-factor (nth 2 expr))))
556 (not (eq (setq f1 (math-frac-gcd f1 f2)) 1))
558 (if (memq (car expr) '(* polar))
559 (math-common-constant-factor (nth 1 expr))
560 (if (eq (car expr) '/)
561 (or (math-common-constant-factor (nth 1 expr))
562 (and (Math-integerp (nth 2 expr))
563 (list 'frac 1 (math-abs (nth 2 expr))))))))))
565 (defun math-cancel-common-factor (expr val)
566 (if (memq (car-safe expr) '(+ - cplx sdev))
568 (setcar (cdr expr) (math-cancel-common-factor (nth 1 expr) val))
569 (setcar (cdr (cdr expr)) (math-cancel-common-factor (nth 2 expr) val))
571 (if (eq (car-safe expr) '*)
572 (math-mul (math-cancel-common-factor (nth 1 expr) val) (nth 2 expr))
573 (math-div expr val))))
575 (defun math-frac-gcd (a b)
580 (if (and (Math-integerp a)
583 (and (Math-integerp a) (setq a (list 'frac a 1)))
584 (and (Math-integerp b) (setq b (list 'frac b 1)))
585 (math-make-frac (math-gcd (nth 1 a) (nth 1 b))
586 (math-gcd (nth 2 a) (nth 2 b)))))))
591 (defun math-simplify-mod ()
592 (and (Math-realp (nth 2 math-simplify-expr))
593 (Math-posp (nth 2 math-simplify-expr))
594 (let ((lin (math-is-linear (nth 1 math-simplify-expr)))
597 (or (math-negp (car lin))
598 (not (Math-lessp (car lin) (nth 2 math-simplify-expr))))
601 (math-mul (nth 1 lin) (nth 2 lin))
602 (math-mod (car lin) (nth 2 math-simplify-expr)))
603 (nth 2 math-simplify-expr)))
605 (not (math-equal-int (nth 1 lin) 1))
606 (math-num-integerp (nth 1 lin))
607 (math-num-integerp (nth 2 math-simplify-expr))
608 (setq t1 (calcFunc-gcd (nth 1 lin) (nth 2 math-simplify-expr)))
609 (not (math-equal-int t1 1))
614 (math-mul (math-div (nth 1 lin) t1)
616 (let ((calc-prefer-frac t))
617 (math-div (car lin) t1)))
618 (math-div (nth 2 math-simplify-expr) t1))))
619 (and (math-equal-int (nth 2 math-simplify-expr) 1)
620 (math-known-integerp (if lin
621 (math-mul (nth 1 lin) (nth 2 lin))
622 (nth 1 math-simplify-expr)))
623 (if lin (math-mod (car lin) 1) 0))))))
625 (math-defsimplify (calcFunc-eq calcFunc-neq calcFunc-lt
626 calcFunc-gt calcFunc-leq calcFunc-geq)
627 (if (= (length math-simplify-expr) 3)
628 (math-simplify-ineq)))
630 (defun math-simplify-ineq ()
631 (let ((np (cdr math-simplify-expr))
633 (while (memq (car-safe (setq n (car np))) '(+ -))
634 (math-simplify-add-term (cdr (cdr n)) (cdr (cdr math-simplify-expr))
637 (math-simplify-add-term np (cdr (cdr math-simplify-expr)) nil
638 (eq np (cdr math-simplify-expr)))
639 (math-simplify-divide)
640 (let ((signs (math-possible-signs (cons '- (cdr math-simplify-expr)))))
641 (or (cond ((eq (car math-simplify-expr) 'calcFunc-eq)
642 (or (and (eq signs 2) 1)
643 (and (memq signs '(1 4 5)) 0)))
644 ((eq (car math-simplify-expr) 'calcFunc-neq)
645 (or (and (eq signs 2) 0)
646 (and (memq signs '(1 4 5)) 1)))
647 ((eq (car math-simplify-expr) 'calcFunc-lt)
648 (or (and (eq signs 1) 1)
649 (and (memq signs '(2 4 6)) 0)))
650 ((eq (car math-simplify-expr) 'calcFunc-gt)
651 (or (and (eq signs 4) 1)
652 (and (memq signs '(1 2 3)) 0)))
653 ((eq (car math-simplify-expr) 'calcFunc-leq)
654 (or (and (eq signs 4) 0)
655 (and (memq signs '(1 2 3)) 1)))
656 ((eq (car math-simplify-expr) 'calcFunc-geq)
657 (or (and (eq signs 1) 0)
658 (and (memq signs '(2 4 6)) 1))))
659 math-simplify-expr))))
661 (defun math-simplify-add-term (np dp minus lplain)
662 (or (math-vectorp (car np))
665 (while (memq (car-safe (setq n (car np) d (car dp))) '(+ -))
667 (if (setq temp (math-combine-sum n (nth 2 d)
668 minus (eq (car d) '+) t))
669 (if (or lplain (eq (math-looks-negp temp) minus))
671 (setcar np (setq n (if minus (math-neg temp) temp)))
672 (setcar (cdr (cdr d)) 0))
675 (setcar (cdr (cdr d)) (setq n (if (eq (car d) '+)
679 (if (setq temp (math-combine-sum n d minus t t))
682 (eq (math-looks-negp temp) minus)))
684 (setcar np (setq n (if minus (math-neg temp) temp)))
688 (setcar dp (setq n (math-neg temp)))))))))
690 (math-defsimplify calcFunc-sin
691 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
692 (nth 1 (nth 1 math-simplify-expr)))
693 (and (math-looks-negp (nth 1 math-simplify-expr))
694 (math-neg (list 'calcFunc-sin (math-neg (nth 1 math-simplify-expr)))))
695 (and (eq calc-angle-mode 'rad)
696 (let ((n (math-linear-in (nth 1 math-simplify-expr) '(var pi var-pi))))
698 (math-known-sin (car n) (nth 1 n) 120 0))))
699 (and (eq calc-angle-mode 'deg)
700 (let ((n (math-integer-plus (nth 1 math-simplify-expr))))
702 (math-known-sin (car n) (nth 1 n) '(frac 2 3) 0))))
703 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccos)
704 (list 'calcFunc-sqrt (math-sub 1 (math-sqr
705 (nth 1 (nth 1 math-simplify-expr))))))
706 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
707 (math-div (nth 1 (nth 1 math-simplify-expr))
709 (math-add 1 (math-sqr
710 (nth 1 (nth 1 math-simplify-expr)))))))
711 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr))))
712 (and m (integerp (car m))
713 (let ((n (car m)) (a (nth 1 m)))
715 (list '* (list 'calcFunc-sin (list '* (1- n) a))
716 (list 'calcFunc-cos a))
717 (list '* (list 'calcFunc-cos (list '* (1- n) a))
718 (list 'calcFunc-sin a))))))))
720 (math-defsimplify calcFunc-cos
721 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccos)
722 (nth 1 (nth 1 math-simplify-expr)))
723 (and (math-looks-negp (nth 1 math-simplify-expr))
724 (list 'calcFunc-cos (math-neg (nth 1 math-simplify-expr))))
725 (and (eq calc-angle-mode 'rad)
726 (let ((n (math-linear-in (nth 1 math-simplify-expr) '(var pi var-pi))))
728 (math-known-sin (car n) (nth 1 n) 120 300))))
729 (and (eq calc-angle-mode 'deg)
730 (let ((n (math-integer-plus (nth 1 math-simplify-expr))))
732 (math-known-sin (car n) (nth 1 n) '(frac 2 3) 300))))
733 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
735 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr))))))
736 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
740 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
741 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr))))
742 (and m (integerp (car m))
743 (let ((n (car m)) (a (nth 1 m)))
745 (list '* (list 'calcFunc-cos (list '* (1- n) a))
746 (list 'calcFunc-cos a))
747 (list '* (list 'calcFunc-sin (list '* (1- n) a))
748 (list 'calcFunc-sin a))))))))
750 (math-defsimplify calcFunc-sec
751 (or (and (math-looks-negp (nth 1 math-simplify-expr))
752 (list 'calcFunc-sec (math-neg (nth 1 math-simplify-expr))))
753 (and (eq calc-angle-mode 'rad)
754 (let ((n (math-linear-in (nth 1 math-simplify-expr) '(var pi var-pi))))
756 (math-div 1 (math-known-sin (car n) (nth 1 n) 120 300)))))
757 (and (eq calc-angle-mode 'deg)
758 (let ((n (math-integer-plus (nth 1 math-simplify-expr))))
760 (math-div 1 (math-known-sin (car n) (nth 1 n) '(frac 2 3) 300)))))
761 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
765 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
766 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccos)
769 (nth 1 (nth 1 math-simplify-expr))))
770 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
773 (math-sqr (nth 1 (nth 1 math-simplify-expr))))))))
775 (math-defsimplify calcFunc-csc
776 (or (and (math-looks-negp (nth 1 math-simplify-expr))
777 (math-neg (list 'calcFunc-csc (math-neg (nth 1 math-simplify-expr)))))
778 (and (eq calc-angle-mode 'rad)
779 (let ((n (math-linear-in (nth 1 math-simplify-expr) '(var pi var-pi))))
781 (math-div 1 (math-known-sin (car n) (nth 1 n) 120 0)))))
782 (and (eq calc-angle-mode 'deg)
783 (let ((n (math-integer-plus (nth 1 math-simplify-expr))))
785 (math-div 1 (math-known-sin (car n) (nth 1 n) '(frac 2 3) 0)))))
786 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
787 (math-div 1 (nth 1 (nth 1 math-simplify-expr))))
788 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccos)
791 (list 'calcFunc-sqrt (math-sub 1 (math-sqr
792 (nth 1 (nth 1 math-simplify-expr)))))))
793 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
794 (math-div (list 'calcFunc-sqrt
795 (math-add 1 (math-sqr
796 (nth 1 (nth 1 math-simplify-expr)))))
797 (nth 1 (nth 1 math-simplify-expr))))))
799 (defun math-should-expand-trig (x &optional hyperbolic)
800 (let ((m (math-is-multiple x)))
801 (and math-living-dangerously
802 m (or (and (integerp (car m)) (> (car m) 1))
803 (equal (car m) '(frac 1 2)))
805 (memq (car-safe (nth 1 m))
807 '(calcFunc-arcsinh calcFunc-arccosh calcFunc-arctanh)
808 '(calcFunc-arcsin calcFunc-arccos calcFunc-arctan)))
809 (and (eq (car-safe (nth 1 m)) 'calcFunc-ln)
810 (eq hyperbolic 'exp)))
813 (defun math-known-sin (plus n mul off)
814 (setq n (math-mul n mul))
815 (and (math-num-integerp n)
816 (setq n (math-mod (math-add (math-trunc n) off) 240))
818 (and (setq n (math-known-sin plus (- n 120) 1 0))
822 (if (math-zerop plus)
823 (and (or calc-symbolic-mode
827 (10 . (/ (calcFunc-sqrt
828 (- 2 (calcFunc-sqrt 3))) 2))
829 (12 . (/ (- (calcFunc-sqrt 5) 1) 4))
830 (15 . (/ (calcFunc-sqrt
831 (- 2 (calcFunc-sqrt 2))) 2))
833 (24 . (* (^ (/ 1 2) (/ 3 2))
835 (- 5 (calcFunc-sqrt 5)))))
836 (30 . (/ (calcFunc-sqrt 2) 2))
837 (36 . (/ (+ (calcFunc-sqrt 5) 1) 4))
838 (40 . (/ (calcFunc-sqrt 3) 2))
839 (45 . (/ (calcFunc-sqrt
840 (+ 2 (calcFunc-sqrt 2))) 2))
841 (48 . (* (^ (/ 1 2) (/ 3 2))
843 (+ 5 (calcFunc-sqrt 5)))))
844 (50 . (/ (calcFunc-sqrt
845 (+ 2 (calcFunc-sqrt 3))) 2))
847 (cond ((eq n 0) (math-normalize (list 'calcFunc-sin plus)))
848 ((eq n 60) (math-normalize (list 'calcFunc-cos plus)))
851 (math-defsimplify calcFunc-tan
852 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
853 (nth 1 (nth 1 math-simplify-expr)))
854 (and (math-looks-negp (nth 1 math-simplify-expr))
855 (math-neg (list 'calcFunc-tan (math-neg (nth 1 math-simplify-expr)))))
856 (and (eq calc-angle-mode 'rad)
857 (let ((n (math-linear-in (nth 1 math-simplify-expr) '(var pi var-pi))))
859 (math-known-tan (car n) (nth 1 n) 120))))
860 (and (eq calc-angle-mode 'deg)
861 (let ((n (math-integer-plus (nth 1 math-simplify-expr))))
863 (math-known-tan (car n) (nth 1 n) '(frac 2 3)))))
864 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
865 (math-div (nth 1 (nth 1 math-simplify-expr))
867 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
868 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccos)
869 (math-div (list 'calcFunc-sqrt
870 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))
871 (nth 1 (nth 1 math-simplify-expr))))
872 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr))))
874 (if (equal (car m) '(frac 1 2))
875 (math-div (math-sub 1 (list 'calcFunc-cos (nth 1 m)))
876 (list 'calcFunc-sin (nth 1 m)))
877 (math-div (list 'calcFunc-sin (nth 1 math-simplify-expr))
878 (list 'calcFunc-cos (nth 1 math-simplify-expr))))))))
880 (math-defsimplify calcFunc-cot
881 (or (and (math-looks-negp (nth 1 math-simplify-expr))
882 (math-neg (list 'calcFunc-cot (math-neg (nth 1 math-simplify-expr)))))
883 (and (eq calc-angle-mode 'rad)
884 (let ((n (math-linear-in (nth 1 math-simplify-expr) '(var pi var-pi))))
886 (math-div 1 (math-known-tan (car n) (nth 1 n) 120)))))
887 (and (eq calc-angle-mode 'deg)
888 (let ((n (math-integer-plus (nth 1 math-simplify-expr))))
890 (math-div 1 (math-known-tan (car n) (nth 1 n) '(frac 2 3))))))
891 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
892 (math-div (list 'calcFunc-sqrt
893 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))
894 (nth 1 (nth 1 math-simplify-expr))))
895 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccos)
896 (math-div (nth 1 (nth 1 math-simplify-expr))
898 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
899 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
900 (math-div 1 (nth 1 (nth 1 math-simplify-expr))))))
902 (defun math-known-tan (plus n mul)
903 (setq n (math-mul n mul))
904 (and (math-num-integerp n)
905 (setq n (math-mod (math-trunc n) 120))
907 (and (setq n (math-known-tan plus (- 120 n) 1))
909 (if (math-zerop plus)
910 (and (or calc-symbolic-mode
912 (cdr (assq n '( (0 . 0)
913 (10 . (- 2 (calcFunc-sqrt 3)))
915 (- 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
916 (15 . (- (calcFunc-sqrt 2) 1))
917 (20 . (/ (calcFunc-sqrt 3) 3))
919 (- 5 (* 2 (calcFunc-sqrt 5)))))
922 (+ 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
923 (40 . (calcFunc-sqrt 3))
924 (45 . (+ (calcFunc-sqrt 2) 1))
926 (+ 5 (* 2 (calcFunc-sqrt 5)))))
927 (50 . (+ 2 (calcFunc-sqrt 3)))
928 (60 . (var uinf var-uinf))))))
929 (cond ((eq n 0) (math-normalize (list 'calcFunc-tan plus)))
930 ((eq n 60) (math-normalize (list '/ -1
931 (list 'calcFunc-tan plus))))
934 (math-defsimplify calcFunc-sinh
935 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsinh)
936 (nth 1 (nth 1 math-simplify-expr)))
937 (and (math-looks-negp (nth 1 math-simplify-expr))
938 (math-neg (list 'calcFunc-sinh (math-neg (nth 1 math-simplify-expr)))))
939 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccosh)
940 math-living-dangerously
942 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1)))
943 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctanh)
944 math-living-dangerously
945 (math-div (nth 1 (nth 1 math-simplify-expr))
947 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
948 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr) t)))
949 (and m (integerp (car m))
950 (let ((n (car m)) (a (nth 1 m)))
953 (list '* (list 'calcFunc-sinh (list '* (1- n) a))
954 (list 'calcFunc-cosh a))
955 (list '* (list 'calcFunc-cosh (list '* (1- n) a))
956 (list 'calcFunc-sinh a)))))))))
958 (math-defsimplify calcFunc-cosh
959 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccosh)
960 (nth 1 (nth 1 math-simplify-expr)))
961 (and (math-looks-negp (nth 1 math-simplify-expr))
962 (list 'calcFunc-cosh (math-neg (nth 1 math-simplify-expr))))
963 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsinh)
964 math-living-dangerously
966 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1)))
967 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctanh)
968 math-living-dangerously
971 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
972 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr) t)))
973 (and m (integerp (car m))
974 (let ((n (car m)) (a (nth 1 m)))
977 (list '* (list 'calcFunc-cosh (list '* (1- n) a))
978 (list 'calcFunc-cosh a))
979 (list '* (list 'calcFunc-sinh (list '* (1- n) a))
980 (list 'calcFunc-sinh a)))))))))
982 (math-defsimplify calcFunc-tanh
983 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctanh)
984 (nth 1 (nth 1 math-simplify-expr)))
985 (and (math-looks-negp (nth 1 math-simplify-expr))
986 (math-neg (list 'calcFunc-tanh (math-neg (nth 1 math-simplify-expr)))))
987 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsinh)
988 math-living-dangerously
989 (math-div (nth 1 (nth 1 math-simplify-expr))
991 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1))))
992 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccosh)
993 math-living-dangerously
994 (math-div (list 'calcFunc-sqrt
995 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1))
996 (nth 1 (nth 1 math-simplify-expr))))
997 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr) t)))
999 (if (equal (car m) '(frac 1 2))
1000 (math-div (math-sub (list 'calcFunc-cosh (nth 1 m)) 1)
1001 (list 'calcFunc-sinh (nth 1 m)))
1002 (math-div (list 'calcFunc-sinh (nth 1 math-simplify-expr))
1003 (list 'calcFunc-cosh (nth 1 math-simplify-expr))))))))
1005 (math-defsimplify calcFunc-sech
1006 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1007 (list 'calcFunc-sech (math-neg (nth 1 math-simplify-expr))))
1008 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsinh)
1009 math-living-dangerously
1012 (list 'calcFunc-sqrt
1013 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1))))
1014 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccosh)
1015 math-living-dangerously
1016 (math-div 1 (nth 1 (nth 1 math-simplify-expr))) 1)
1017 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctanh)
1018 math-living-dangerously
1019 (list 'calcFunc-sqrt
1020 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr))))))))
1022 (math-defsimplify calcFunc-csch
1023 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1024 (math-neg (list 'calcFunc-csch (math-neg (nth 1 math-simplify-expr)))))
1025 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsinh)
1026 math-living-dangerously
1027 (math-div 1 (nth 1 (nth 1 math-simplify-expr))))
1028 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccosh)
1029 math-living-dangerously
1032 (list 'calcFunc-sqrt
1033 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1))))
1034 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctanh)
1035 math-living-dangerously
1036 (math-div (list 'calcFunc-sqrt
1037 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))
1038 (nth 1 (nth 1 math-simplify-expr))))))
1040 (math-defsimplify calcFunc-coth
1041 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1042 (math-neg (list 'calcFunc-coth (math-neg (nth 1 math-simplify-expr)))))
1043 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsinh)
1044 math-living-dangerously
1045 (math-div (list 'calcFunc-sqrt
1046 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1))
1047 (nth 1 (nth 1 math-simplify-expr))))
1048 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccosh)
1049 math-living-dangerously
1050 (math-div (nth 1 (nth 1 math-simplify-expr))
1051 (list 'calcFunc-sqrt
1052 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr))) 1))))
1053 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctanh)
1054 math-living-dangerously
1055 (math-div 1 (nth 1 (nth 1 math-simplify-expr))))))
1057 (math-defsimplify calcFunc-arcsin
1058 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1059 (math-neg (list 'calcFunc-arcsin (math-neg (nth 1 math-simplify-expr)))))
1060 (and (eq (nth 1 math-simplify-expr) 1)
1061 (math-quarter-circle t))
1062 (and (equal (nth 1 math-simplify-expr) '(frac 1 2))
1063 (math-div (math-half-circle t) 6))
1064 (and math-living-dangerously
1065 (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-sin)
1066 (nth 1 (nth 1 math-simplify-expr)))
1067 (and math-living-dangerously
1068 (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-cos)
1069 (math-sub (math-quarter-circle t)
1070 (nth 1 (nth 1 math-simplify-expr))))))
1072 (math-defsimplify calcFunc-arccos
1073 (or (and (eq (nth 1 math-simplify-expr) 0)
1074 (math-quarter-circle t))
1075 (and (eq (nth 1 math-simplify-expr) -1)
1076 (math-half-circle t))
1077 (and (equal (nth 1 math-simplify-expr) '(frac 1 2))
1078 (math-div (math-half-circle t) 3))
1079 (and (equal (nth 1 math-simplify-expr) '(frac -1 2))
1080 (math-div (math-mul (math-half-circle t) 2) 3))
1081 (and math-living-dangerously
1082 (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-cos)
1083 (nth 1 (nth 1 math-simplify-expr)))
1084 (and math-living-dangerously
1085 (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-sin)
1086 (math-sub (math-quarter-circle t)
1087 (nth 1 (nth 1 math-simplify-expr))))))
1089 (math-defsimplify calcFunc-arctan
1090 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1091 (math-neg (list 'calcFunc-arctan (math-neg (nth 1 math-simplify-expr)))))
1092 (and (eq (nth 1 math-simplify-expr) 1)
1093 (math-div (math-half-circle t) 4))
1094 (and math-living-dangerously
1095 (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-tan)
1096 (nth 1 (nth 1 math-simplify-expr)))))
1098 (math-defsimplify calcFunc-arcsinh
1099 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1100 (math-neg (list 'calcFunc-arcsinh (math-neg (nth 1 math-simplify-expr)))))
1101 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-sinh)
1102 (or math-living-dangerously
1103 (math-known-realp (nth 1 (nth 1 math-simplify-expr))))
1104 (nth 1 (nth 1 math-simplify-expr)))))
1106 (math-defsimplify calcFunc-arccosh
1107 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-cosh)
1108 (or math-living-dangerously
1109 (math-known-realp (nth 1 (nth 1 math-simplify-expr))))
1110 (nth 1 (nth 1 math-simplify-expr))))
1112 (math-defsimplify calcFunc-arctanh
1113 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1114 (math-neg (list 'calcFunc-arctanh (math-neg (nth 1 math-simplify-expr)))))
1115 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-tanh)
1116 (or math-living-dangerously
1117 (math-known-realp (nth 1 (nth 1 math-simplify-expr))))
1118 (nth 1 (nth 1 math-simplify-expr)))))
1120 (math-defsimplify calcFunc-sqrt
1121 (math-simplify-sqrt))
1123 (defun math-simplify-sqrt ()
1124 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'frac)
1125 (math-div (list 'calcFunc-sqrt
1126 (math-mul (nth 1 (nth 1 math-simplify-expr))
1127 (nth 2 (nth 1 math-simplify-expr))))
1128 (nth 2 (nth 1 math-simplify-expr))))
1129 (let ((fac (if (math-objectp (nth 1 math-simplify-expr))
1130 (math-squared-factor (nth 1 math-simplify-expr))
1131 (math-common-constant-factor (nth 1 math-simplify-expr)))))
1132 (and fac (not (eq fac 1))
1133 (math-mul (math-normalize (list 'calcFunc-sqrt fac))
1135 (list 'calcFunc-sqrt
1136 (math-cancel-common-factor
1137 (nth 1 math-simplify-expr) fac))))))
1138 (and math-living-dangerously
1139 (or (and (eq (car-safe (nth 1 math-simplify-expr)) '-)
1140 (math-equal-int (nth 1 (nth 1 math-simplify-expr)) 1)
1141 (eq (car-safe (nth 2 (nth 1 math-simplify-expr))) '^)
1142 (math-equal-int (nth 2 (nth 2 (nth 1 math-simplify-expr))) 2)
1143 (or (and (eq (car-safe (nth 1 (nth 2 (nth 1 math-simplify-expr))))
1146 (nth 1 (nth 1 (nth 2 (nth 1 math-simplify-expr))))))
1147 (and (eq (car-safe (nth 1 (nth 2 (nth 1 math-simplify-expr))))
1150 (nth 1 (nth 1 (nth 2
1151 (nth 1 math-simplify-expr))))))))
1152 (and (eq (car-safe (nth 1 math-simplify-expr)) '-)
1153 (math-equal-int (nth 2 (nth 1 math-simplify-expr)) 1)
1154 (eq (car-safe (nth 1 (nth 1 math-simplify-expr))) '^)
1155 (math-equal-int (nth 2 (nth 1 (nth 1 math-simplify-expr))) 2)
1156 (and (eq (car-safe (nth 1 (nth 1 (nth 1 math-simplify-expr))))
1158 (list 'calcFunc-sinh
1159 (nth 1 (nth 1 (nth 1 (nth 1 math-simplify-expr)))))))
1160 (and (eq (car-safe (nth 1 math-simplify-expr)) '+)
1161 (let ((a (nth 1 (nth 1 math-simplify-expr)))
1162 (b (nth 2 (nth 1 math-simplify-expr))))
1163 (and (or (and (math-equal-int a 1)
1164 (setq a b b (nth 1 (nth 1 math-simplify-expr))))
1165 (math-equal-int b 1))
1166 (eq (car-safe a) '^)
1167 (math-equal-int (nth 2 a) 2)
1168 (or (and (eq (car-safe (nth 1 a)) 'calcFunc-sinh)
1169 (list 'calcFunc-cosh (nth 1 (nth 1 a))))
1170 (and (eq (car-safe (nth 1 a)) 'calcFunc-csch)
1171 (list 'calcFunc-coth (nth 1 (nth 1 a))))
1172 (and (eq (car-safe (nth 1 a)) 'calcFunc-tan)
1173 (list '/ 1 (list 'calcFunc-cos
1174 (nth 1 (nth 1 a)))))
1175 (and (eq (car-safe (nth 1 a)) 'calcFunc-cot)
1176 (list '/ 1 (list 'calcFunc-sin
1177 (nth 1 (nth 1 a)))))))))
1178 (and (eq (car-safe (nth 1 math-simplify-expr)) '^)
1180 (nth 1 (nth 1 math-simplify-expr))
1181 (math-div (nth 2 (nth 1 math-simplify-expr)) 2)))
1182 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-sqrt)
1183 (list '^ (nth 1 (nth 1 math-simplify-expr)) (math-div 1 4)))
1184 (and (memq (car-safe (nth 1 math-simplify-expr)) '(* /))
1185 (list (car (nth 1 math-simplify-expr))
1186 (list 'calcFunc-sqrt (nth 1 (nth 1 math-simplify-expr)))
1187 (list 'calcFunc-sqrt (nth 2 (nth 1 math-simplify-expr)))))
1188 (and (memq (car-safe (nth 1 math-simplify-expr)) '(+ -))
1189 (not (math-any-floats (nth 1 math-simplify-expr)))
1190 (let ((f (calcFunc-factors (calcFunc-expand
1191 (nth 1 math-simplify-expr)))))
1192 (and (math-vectorp f)
1193 (or (> (length f) 2)
1194 (> (nth 2 (nth 1 f)) 1))
1195 (let ((out 1) (rest 1) (sums 1) fac pow)
1196 (while (setq f (cdr f))
1197 (setq fac (nth 1 (car f))
1198 pow (nth 2 (car f)))
1200 (setq out (math-mul out (math-pow
1204 (if (memq (car-safe fac) '(+ -))
1205 (setq sums (math-mul-thru sums fac))
1206 (setq rest (math-mul rest fac)))))
1207 (and (not (and (eq out 1) (memq rest '(1 -1))))
1210 (list 'calcFunc-sqrt
1211 (math-mul sums rest))))))))))))
1213 ;;; Rather than factoring x into primes, just check for the first ten primes.
1214 (defun math-squared-factor (x)
1215 (if (Math-integerp x)
1216 (let ((prsqr '(4 9 25 49 121 169 289 361 529 841))
1220 (if (eq (cdr (setq res (math-idivmod x (car prsqr)))) 0)
1222 fac (math-mul fac (car prsqr)))
1223 (setq prsqr (cdr prsqr))))
1226 (math-defsimplify calcFunc-exp
1227 (math-simplify-exp (nth 1 math-simplify-expr)))
1229 (defun math-simplify-exp (x)
1230 (or (and (eq (car-safe x) 'calcFunc-ln)
1232 (and math-living-dangerously
1233 (or (and (eq (car-safe x) 'calcFunc-arcsinh)
1235 (list 'calcFunc-sqrt
1236 (math-add (math-sqr (nth 1 x)) 1))))
1237 (and (eq (car-safe x) 'calcFunc-arccosh)
1239 (list 'calcFunc-sqrt
1240 (math-sub (math-sqr (nth 1 x)) 1))))
1241 (and (eq (car-safe x) 'calcFunc-arctanh)
1242 (math-div (list 'calcFunc-sqrt (math-add 1 (nth 1 x)))
1243 (list 'calcFunc-sqrt (math-sub 1 (nth 1 x)))))
1244 (let ((m (math-should-expand-trig x 'exp)))
1245 (and m (integerp (car m))
1246 (list '^ (list 'calcFunc-exp (nth 1 m)) (car m))))))
1247 (and calc-symbolic-mode
1248 (math-known-imagp x)
1249 (let* ((ip (calcFunc-im x))
1250 (n (math-linear-in ip '(var pi var-pi)))
1253 (setq s (math-known-sin (car n) (nth 1 n) 120 0))
1254 (setq c (math-known-sin (car n) (nth 1 n) 120 300))
1255 (list '+ c (list '* s '(var i var-i))))))))
1257 (math-defsimplify calcFunc-ln
1258 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-exp)
1259 (or math-living-dangerously
1260 (math-known-realp (nth 1 (nth 1 math-simplify-expr))))
1261 (nth 1 (nth 1 math-simplify-expr)))
1262 (and (eq (car-safe (nth 1 math-simplify-expr)) '^)
1263 (equal (nth 1 (nth 1 math-simplify-expr)) '(var e var-e))
1264 (or math-living-dangerously
1265 (math-known-realp (nth 2 (nth 1 math-simplify-expr))))
1266 (nth 2 (nth 1 math-simplify-expr)))
1267 (and calc-symbolic-mode
1268 (math-known-negp (nth 1 math-simplify-expr))
1269 (math-add (list 'calcFunc-ln (math-neg (nth 1 math-simplify-expr)))
1270 '(* (var pi var-pi) (var i var-i))))
1271 (and calc-symbolic-mode
1272 (math-known-imagp (nth 1 math-simplify-expr))
1273 (let* ((ip (calcFunc-im (nth 1 math-simplify-expr)))
1274 (ips (math-possible-signs ip)))
1275 (or (and (memq ips '(4 6))
1276 (math-add (list 'calcFunc-ln ip)
1277 '(/ (* (var pi var-pi) (var i var-i)) 2)))
1278 (and (memq ips '(1 3))
1279 (math-sub (list 'calcFunc-ln (math-neg ip))
1280 '(/ (* (var pi var-pi) (var i var-i)) 2))))))))
1283 (math-simplify-pow))
1285 (defun math-simplify-pow ()
1286 (or (and math-living-dangerously
1287 (or (and (eq (car-safe (nth 1 math-simplify-expr)) '^)
1289 (nth 1 (nth 1 math-simplify-expr))
1290 (math-mul (nth 2 math-simplify-expr)
1291 (nth 2 (nth 1 math-simplify-expr)))))
1292 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-sqrt)
1294 (nth 1 (nth 1 math-simplify-expr))
1295 (math-div (nth 2 math-simplify-expr) 2)))
1296 (and (memq (car-safe (nth 1 math-simplify-expr)) '(* /))
1297 (list (car (nth 1 math-simplify-expr))
1298 (list '^ (nth 1 (nth 1 math-simplify-expr))
1299 (nth 2 math-simplify-expr))
1300 (list '^ (nth 2 (nth 1 math-simplify-expr))
1301 (nth 2 math-simplify-expr))))))
1302 (and (math-equal-int (nth 1 math-simplify-expr) 10)
1303 (eq (car-safe (nth 2 math-simplify-expr)) 'calcFunc-log10)
1304 (nth 1 (nth 2 math-simplify-expr)))
1305 (and (equal (nth 1 math-simplify-expr) '(var e var-e))
1306 (math-simplify-exp (nth 2 math-simplify-expr)))
1307 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-exp)
1308 (not math-integrating)
1309 (list 'calcFunc-exp (math-mul (nth 1 (nth 1 math-simplify-expr))
1310 (nth 2 math-simplify-expr))))
1311 (and (equal (nth 1 math-simplify-expr) '(var i var-i))
1313 (math-num-integerp (nth 2 math-simplify-expr))
1314 (let ((x (math-mod (math-trunc (nth 2 math-simplify-expr)) 4)))
1316 ((eq x 1) (nth 1 math-simplify-expr))
1318 ((eq x 3) (math-neg (nth 1 math-simplify-expr))))))
1319 (and math-integrating
1320 (integerp (nth 2 math-simplify-expr))
1321 (>= (nth 2 math-simplify-expr) 2)
1322 (or (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-cos)
1323 (math-mul (math-pow (nth 1 math-simplify-expr)
1324 (- (nth 2 math-simplify-expr) 2))
1328 (nth 1 (nth 1 math-simplify-expr)))))))
1329 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-cosh)
1330 (math-mul (math-pow (nth 1 math-simplify-expr)
1331 (- (nth 2 math-simplify-expr) 2))
1334 (list 'calcFunc-sinh
1335 (nth 1 (nth 1 math-simplify-expr)))))))))
1336 (and (eq (car-safe (nth 2 math-simplify-expr)) 'frac)
1337 (Math-ratp (nth 1 math-simplify-expr))
1338 (Math-posp (nth 1 math-simplify-expr))
1339 (if (equal (nth 2 math-simplify-expr) '(frac 1 2))
1340 (list 'calcFunc-sqrt (nth 1 math-simplify-expr))
1341 (let ((flr (math-floor (nth 2 math-simplify-expr))))
1342 (and (not (Math-zerop flr))
1343 (list '* (list '^ (nth 1 math-simplify-expr) flr)
1344 (list '^ (nth 1 math-simplify-expr)
1345 (math-sub (nth 2 math-simplify-expr) flr)))))))
1346 (and (eq (math-quarter-integer (nth 2 math-simplify-expr)) 2)
1347 (let ((temp (math-simplify-sqrt)))
1349 (list '^ temp (math-mul (nth 2 math-simplify-expr) 2)))))))
1351 (math-defsimplify calcFunc-log10
1352 (and (eq (car-safe (nth 1 math-simplify-expr)) '^)
1353 (math-equal-int (nth 1 (nth 1 math-simplify-expr)) 10)
1354 (or math-living-dangerously
1355 (math-known-realp (nth 2 (nth 1 math-simplify-expr))))
1356 (nth 2 (nth 1 math-simplify-expr))))
1359 (math-defsimplify calcFunc-erf
1360 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1361 (math-neg (list 'calcFunc-erf (math-neg (nth 1 math-simplify-expr)))))
1362 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-conj)
1363 (list 'calcFunc-conj
1364 (list 'calcFunc-erf (nth 1 (nth 1 math-simplify-expr)))))))
1366 (math-defsimplify calcFunc-erfc
1367 (or (and (math-looks-negp (nth 1 math-simplify-expr))
1368 (math-sub 2 (list 'calcFunc-erfc (math-neg (nth 1 math-simplify-expr)))))
1369 (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-conj)
1370 (list 'calcFunc-conj
1371 (list 'calcFunc-erfc (nth 1 (nth 1 math-simplify-expr)))))))
1374 (defun math-linear-in (expr term &optional always)
1375 (if (math-expr-contains expr term)
1376 (let* ((calc-prefer-frac t)
1377 (p (math-is-polynomial expr term 1)))
1380 (and always (list expr 0))))
1382 (defun math-multiple-of (expr term)
1383 (let ((p (math-linear-in expr term)))
1385 (math-zerop (car p))
1388 ; not perfect, but it'll do
1389 (defun math-integer-plus (expr)
1390 (cond ((Math-integerp expr)
1392 ((and (memq (car expr) '(+ -))
1393 (Math-integerp (nth 1 expr)))
1394 (list (if (eq (car expr) '+) (nth 2 expr) (math-neg (nth 2 expr)))
1396 ((and (memq (car expr) '(+ -))
1397 (Math-integerp (nth 2 expr)))
1399 (if (eq (car expr) '+) (nth 2 expr) (math-neg (nth 2 expr)))))
1402 (defun math-is-linear (expr &optional always)
1405 (if (eq (car-safe expr) '+)
1406 (if (Math-objectp (nth 1 expr))
1407 (setq offset (nth 1 expr)
1409 (if (Math-objectp (nth 2 expr))
1410 (setq offset (nth 2 expr)
1411 expr (nth 1 expr))))
1412 (if (eq (car-safe expr) '-)
1413 (if (Math-objectp (nth 1 expr))
1414 (setq offset (nth 1 expr)
1415 expr (math-neg (nth 2 expr)))
1416 (if (Math-objectp (nth 2 expr))
1417 (setq offset (math-neg (nth 2 expr))
1418 expr (nth 1 expr))))))
1419 (setq coef (math-is-multiple expr always))
1421 (list offset (or (car coef) 1) (or (nth 1 coef) expr))
1425 (defun math-is-multiple (expr &optional always)
1426 (or (if (eq (car-safe expr) '*)
1427 (if (Math-objectp (nth 1 expr))
1428 (list (nth 1 expr) (nth 2 expr)))
1429 (if (eq (car-safe expr) '/)
1430 (if (and (Math-objectp (nth 1 expr))
1431 (not (math-equal-int (nth 1 expr) 1)))
1432 (list (nth 1 expr) (math-div 1 (nth 2 expr)))
1433 (if (Math-objectp (nth 2 expr))
1434 (list (math-div 1 (nth 2 expr)) (nth 1 expr))
1435 (let ((res (math-is-multiple (nth 1 expr))))
1438 (math-div (nth 2 (nth 1 expr)) (nth 2 expr)))
1439 (setq res (math-is-multiple (nth 2 expr)))
1441 (list (math-div 1 (car res))
1442 (math-div (nth 1 expr)
1443 (nth 2 (nth 2 expr)))))))))
1444 (if (eq (car-safe expr) 'neg)
1445 (list -1 (nth 1 expr)))))
1446 (if (Math-objvecp expr)
1452 (defun calcFunc-lin (expr &optional var)
1454 (let ((res (math-linear-in expr var t)))
1455 (or res (math-reject-arg expr "Linear term expected"))
1456 (list 'vec (car res) (nth 1 res) var))
1457 (let ((res (math-is-linear expr t)))
1458 (or res (math-reject-arg expr "Linear term expected"))
1461 (defun calcFunc-linnt (expr &optional var)
1463 (let ((res (math-linear-in expr var)))
1464 (or res (math-reject-arg expr "Linear term expected"))
1465 (list 'vec (car res) (nth 1 res) var))
1466 (let ((res (math-is-linear expr)))
1467 (or res (math-reject-arg expr "Linear term expected"))
1470 (defun calcFunc-islin (expr &optional var)
1471 (if (and (Math-objvecp expr) (not var))
1473 (calcFunc-lin expr var)
1476 (defun calcFunc-islinnt (expr &optional var)
1477 (if (Math-objvecp expr)
1479 (calcFunc-linnt expr var)
1485 ;;; Simple operations on expressions.
1487 ;;; Return number of occurrences of thing in expr, or nil if none.
1488 (defun math-expr-contains-count (expr thing)
1489 (cond ((equal expr thing) 1)
1490 ((Math-primp expr) nil)
1493 (while (setq expr (cdr expr))
1494 (setq num (+ num (or (math-expr-contains-count
1495 (car expr) thing) 0))))
1499 (defun math-expr-contains (expr thing)
1500 (cond ((equal expr thing) 1)
1501 ((Math-primp expr) nil)
1503 (while (and (setq expr (cdr expr))
1504 (not (math-expr-contains (car expr) thing))))
1507 ;;; Return non-nil if any variable of thing occurs in expr.
1508 (defun math-expr-depends (expr thing)
1509 (if (Math-primp thing)
1510 (and (eq (car-safe thing) 'var)
1511 (math-expr-contains expr thing))
1512 (while (and (setq thing (cdr thing))
1513 (not (math-expr-depends expr (car thing)))))
1516 ;;; Substitute all occurrences of old for new in expr (non-destructive).
1518 ;; The variables math-expr-subst-old and math-expr-subst-new are local
1519 ;; for math-expr-subst, but used by math-expr-subst-rec.
1520 (defvar math-expr-subst-old)
1521 (defvar math-expr-subst-new)
1523 (defun math-expr-subst (expr math-expr-subst-old math-expr-subst-new)
1524 (math-expr-subst-rec expr))
1526 (defalias 'calcFunc-subst 'math-expr-subst)
1528 (defun math-expr-subst-rec (expr)
1529 (cond ((equal expr math-expr-subst-old) math-expr-subst-new)
1530 ((Math-primp expr) expr)
1531 ((memq (car expr) '(calcFunc-deriv
1533 (if (= (length expr) 2)
1534 (if (equal (nth 1 expr) math-expr-subst-old)
1535 (append expr (list math-expr-subst-new))
1537 (list (car expr) (nth 1 expr)
1538 (math-expr-subst-rec (nth 2 expr)))))
1541 (mapcar 'math-expr-subst-rec (cdr expr))))))
1543 ;;; Various measures of the size of an expression.
1544 (defun math-expr-weight (expr)
1545 (if (Math-primp expr)
1548 (while (setq expr (cdr expr))
1549 (setq w (+ w (math-expr-weight (car expr)))))
1552 (defun math-expr-height (expr)
1553 (if (Math-primp expr)
1556 (while (setq expr (cdr expr))
1557 (setq h (max h (math-expr-height (car expr)))))
1563 ;;; Polynomial operations (to support the integrator and solve-for).
1565 (defun calcFunc-collect (expr base)
1566 (let ((p (math-is-polynomial expr base 50 t)))
1568 (math-normalize ; fix selection bug
1569 (math-build-polynomial-expr p base))
1572 ;;; If expr is of the form "a + bx + cx^2 + ...", return the list (a b c ...),
1573 ;;; else return nil if not in polynomial form. If "loose" (math-is-poly-loose),
1574 ;;; coefficients may contain x, e.g., sin(x) + cos(x) x^2 is a loose polynomial in x.
1576 ;; The variables math-is-poly-degree and math-is-poly-loose are local to
1577 ;; math-is-polynomial, but are used by math-is-poly-rec
1578 (defvar math-is-poly-degree)
1579 (defvar math-is-poly-loose)
1581 (defun math-is-polynomial (expr var &optional math-is-poly-degree math-is-poly-loose)
1582 (let* ((math-poly-base-variable (if math-is-poly-loose
1583 (if (eq math-is-poly-loose 'gen) var '(var XXX XXX))
1584 math-poly-base-variable))
1585 (poly (math-is-poly-rec expr math-poly-neg-powers)))
1586 (and (or (null math-is-poly-degree)
1587 (<= (length poly) (1+ math-is-poly-degree)))
1590 (defun math-is-poly-rec (expr negpow)
1592 (or (cond ((or (equal expr var)
1593 (eq (car-safe expr) '^))
1596 (or (equal expr var)
1597 (setq pow (nth 2 expr)
1599 (or (eq math-poly-mult-powers 1)
1600 (setq pow (let ((m (math-is-multiple pow 1)))
1601 (and (eq (car-safe (car m)) 'cplx)
1602 (Math-zerop (nth 1 (car m)))
1603 (setq m (list (nth 2 (car m))
1606 (and (if math-poly-mult-powers
1607 (equal math-poly-mult-powers
1609 (setq math-poly-mult-powers (nth 1 m)))
1610 (or (equal expr var)
1611 (eq math-poly-mult-powers 1))
1615 (setq pow (math-to-simple-fraction pow))
1616 (and (eq (car-safe pow) 'frac)
1617 math-poly-frac-powers
1619 (setq math-poly-frac-powers
1620 (calcFunc-lcm math-poly-frac-powers
1622 (or (memq math-poly-frac-powers '(1 nil))
1623 (setq pow (math-mul pow math-poly-frac-powers)))
1629 (let ((p1 (if (equal expr var)
1631 (math-is-poly-rec expr nil)))
1635 (or (null math-is-poly-degree)
1636 (<= (* (1- (length p1)) n) math-is-poly-degree))
1639 (setq accum (math-poly-mul accum p1)
1643 (math-is-poly-rec expr nil)
1644 (setq math-poly-neg-powers
1645 (cons (math-pow expr (- pow))
1646 math-poly-neg-powers))
1647 (list (list '^ expr pow))))))))
1648 ((Math-objectp expr)
1650 ((memq (car expr) '(+ -))
1651 (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
1653 (let ((p2 (math-is-poly-rec (nth 2 expr) negpow)))
1655 (math-poly-mix p1 1 p2
1656 (if (eq (car expr) '+) 1 -1)))))))
1657 ((eq (car expr) 'neg)
1658 (mapcar 'math-neg (math-is-poly-rec (nth 1 expr) negpow)))
1660 (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
1662 (let ((p2 (math-is-poly-rec (nth 2 expr) negpow)))
1664 (or (null math-is-poly-degree)
1665 (<= (- (+ (length p1) (length p2)) 2)
1666 math-is-poly-degree))
1667 (math-poly-mul p1 p2))))))
1669 (and (or (not (math-poly-depends (nth 2 expr) var))
1671 (math-is-poly-rec (nth 2 expr) nil)
1672 (setq math-poly-neg-powers
1673 (cons (nth 2 expr) math-poly-neg-powers))))
1674 (not (Math-zerop (nth 2 expr)))
1675 (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
1676 (mapcar (function (lambda (x) (math-div x (nth 2 expr))))
1678 ((and (eq (car expr) 'calcFunc-exp)
1679 (equal var '(var e var-e)))
1680 (math-is-poly-rec (list '^ var (nth 1 expr)) negpow))
1681 ((and (eq (car expr) 'calcFunc-sqrt)
1682 math-poly-frac-powers)
1683 (math-is-poly-rec (list '^ (nth 1 expr) '(frac 1 2)) negpow))
1685 (and (or (not (math-poly-depends expr var))
1687 (not (eq (car expr) 'vec))
1690 ;;; Check if expr is a polynomial in var; if so, return its degree.
1691 (defun math-polynomial-p (expr var)
1692 (cond ((equal expr var) 1)
1693 ((Math-primp expr) 0)
1694 ((memq (car expr) '(+ -))
1695 (let ((p1 (math-polynomial-p (nth 1 expr) var))
1697 (and p1 (setq p2 (math-polynomial-p (nth 2 expr) var))
1700 (let ((p1 (math-polynomial-p (nth 1 expr) var))
1702 (and p1 (setq p2 (math-polynomial-p (nth 2 expr) var))
1704 ((eq (car expr) 'neg)
1705 (math-polynomial-p (nth 1 expr) var))
1706 ((and (eq (car expr) '/)
1707 (not (math-poly-depends (nth 2 expr) var)))
1708 (math-polynomial-p (nth 1 expr) var))
1709 ((and (eq (car expr) '^)
1710 (natnump (nth 2 expr)))
1711 (let ((p1 (math-polynomial-p (nth 1 expr) var)))
1712 (and p1 (* p1 (nth 2 expr)))))
1713 ((math-poly-depends expr var) nil)
1716 (defun math-poly-depends (expr var)
1717 (if math-poly-base-variable
1718 (math-expr-contains expr math-poly-base-variable)
1719 (math-expr-depends expr var)))
1721 ;;; Find the variable (or sub-expression) which is the base of polynomial expr.
1722 ;; The variables math-poly-base-const-ok and math-poly-base-pred are
1723 ;; local to math-polynomial-base, but are used by math-polynomial-base-rec.
1724 (defvar math-poly-base-const-ok)
1725 (defvar math-poly-base-pred)
1727 ;; The variable math-poly-base-top-expr is local to math-polynomial-base,
1728 ;; but is used by math-polynomial-p1 in calc-poly.el, which is called
1729 ;; by math-polynomial-base.
1731 (defun math-polynomial-base (math-poly-base-top-expr &optional math-poly-base-pred)
1732 (or math-poly-base-pred
1733 (setq math-poly-base-pred (function (lambda (base) (math-polynomial-p
1734 math-poly-base-top-expr base)))))
1735 (or (let ((math-poly-base-const-ok nil))
1736 (math-polynomial-base-rec math-poly-base-top-expr))
1737 (let ((math-poly-base-const-ok t))
1738 (math-polynomial-base-rec math-poly-base-top-expr))))
1740 (defun math-polynomial-base-rec (mpb-expr)
1741 (and (not (Math-objvecp mpb-expr))
1742 (or (and (memq (car mpb-expr) '(+ - *))
1743 (or (math-polynomial-base-rec (nth 1 mpb-expr))
1744 (math-polynomial-base-rec (nth 2 mpb-expr))))
1745 (and (memq (car mpb-expr) '(/ neg))
1746 (math-polynomial-base-rec (nth 1 mpb-expr)))
1747 (and (eq (car mpb-expr) '^)
1748 (math-polynomial-base-rec (nth 1 mpb-expr)))
1749 (and (eq (car mpb-expr) 'calcFunc-exp)
1750 (math-polynomial-base-rec '(var e var-e)))
1751 (and (or math-poly-base-const-ok (math-expr-contains-vars mpb-expr))
1752 (funcall math-poly-base-pred mpb-expr)
1755 ;;; Return non-nil if expr refers to any variables.
1756 (defun math-expr-contains-vars (expr)
1757 (or (eq (car-safe expr) 'var)
1758 (and (not (Math-primp expr))
1760 (while (and (setq expr (cdr expr))
1761 (not (math-expr-contains-vars (car expr)))))
1764 ;;; Simplify a polynomial in list form by stripping off high-end zeros.
1765 ;;; This always leaves the constant part, i.e., nil->nil and nonnil->nonnil.
1766 (defun math-poly-simplify (p)
1768 (if (Math-zerop (nth (1- (length p)) p))
1769 (let ((pp (copy-sequence p)))
1770 (while (and (cdr pp)
1771 (Math-zerop (nth (1- (length pp)) pp)))
1772 (setcdr (nthcdr (- (length pp) 2) pp) nil))
1776 ;;; Compute ac*a + bc*b for polynomials in list form a, b and
1777 ;;; coefficients ac, bc. Result may be unsimplified.
1778 (defun math-poly-mix (a ac b bc)
1780 (cons (math-add (math-mul (or (car a) 0) ac)
1781 (math-mul (or (car b) 0) bc))
1782 (math-poly-mix (cdr a) ac (cdr b) bc))))
1784 (defun math-poly-zerop (a)
1786 (and (null (cdr a)) (Math-zerop (car a)))))
1788 ;;; Multiply two polynomials in list form.
1789 (defun math-poly-mul (a b)
1791 (math-poly-mix b (car a)
1792 (math-poly-mul (cdr a) (cons 0 b)) 1)))
1794 ;;; Build an expression from a polynomial list.
1795 (defun math-build-polynomial-expr (p var)
1797 (if (Math-numberp var)
1798 (math-with-extra-prec 1
1799 (let* ((rp (reverse p))
1801 (while (setq rp (cdr rp))
1802 (setq accum (math-add (car rp) (math-mul accum var))))
1804 (let* ((rp (reverse p))
1805 (n (1- (length rp)))
1806 (accum (math-mul (car rp) (math-pow var n)))
1808 (while (setq rp (cdr rp))
1810 (or (math-zerop (car rp))
1811 (setq accum (list (if (math-looks-negp (car rp)) '- '+)
1813 (math-mul (if (math-looks-negp (car rp))
1816 (math-pow var n))))))
1821 (defun math-to-simple-fraction (f)
1822 (or (and (eq (car-safe f) 'float)
1823 (or (and (>= (nth 2 f) 0)
1824 (math-scale-int (nth 1 f) (nth 2 f)))
1825 (and (integerp (nth 1 f))
1828 (math-make-frac (nth 1 f)
1829 (math-scale-int 1 (- (nth 2 f)))))))
1834 ;;; arch-tag: 52e7dcdf-9688-464d-a02b-4bbe789348d0
1835 ;;; calc-alg.el ends here