1 ;;; calc-alg.el --- algebraic functions for Calc
3 ;; Copyright (C) 1990-1993, 2001-2014 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
8 ;; This file is part of GNU Emacs.
10 ;; GNU Emacs is free software: you can redistribute it and/or modify
11 ;; it under the terms of the GNU General Public License as published by
12 ;; the Free Software Foundation, either version 3 of the License, or
13 ;; (at your option) any later version.
15 ;; GNU Emacs is distributed in the hope that it will be useful,
16 ;; but WITHOUT ANY WARRANTY; without even the implied warranty of
17 ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
18 ;; GNU General Public License for more details.
20 ;; You should have received a copy of the GNU General Public License
21 ;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>.
27 ;; This file is autoloaded from calc-ext.el.
34 (defun calc-alg-evaluate (arg)
37 (calc-with-default-simplification
38 (let ((math-simplify-only nil
))
39 (calc-modify-simplify-mode arg
)
40 (calc-enter-result 1 "dsmp" (calc-top 1))))))
42 (defun calc-modify-simplify-mode (arg)
43 (if (= (math-abs arg
) 2)
44 (setq calc-simplify-mode
'alg
)
45 (if (>= (math-abs arg
) 3)
46 (setq calc-simplify-mode
'ext
)))
48 (setq calc-simplify-mode
(list calc-simplify-mode
))))
50 (defun calc-simplify ()
53 (let ((top (calc-top-n 1)))
56 (let ((calc-simplify-mode nil
))
57 (math-normalize (math-trig-rewrite top
)))))
58 (if (calc-is-hyperbolic)
60 (let ((calc-simplify-mode nil
))
61 (math-normalize (math-hyperbolic-trig-rewrite top
)))))
62 (calc-with-default-simplification
63 (calc-enter-result 1 "simp" (math-simplify top
))))))
65 (defun calc-simplify-extended ()
68 (calc-with-default-simplification
69 (calc-enter-result 1 "esmp" (math-simplify-extended (calc-top-n 1))))))
71 (defun calc-expand-formula (arg)
74 (calc-with-default-simplification
75 (let ((math-simplify-only nil
))
76 (calc-modify-simplify-mode arg
)
77 (calc-enter-result 1 "expf"
79 (let ((math-expand-formulas t
))
81 (let ((top (calc-top-n 1)))
82 (or (math-expand-formula top
)
85 (defun calc-factor (arg)
88 (calc-unary-op "fctr" (if (calc-is-hyperbolic)
89 'calcFunc-factors
'calcFunc-factor
)
92 (defun calc-expand (n)
95 (calc-enter-result 1 "expa"
96 (append (list 'calcFunc-expand
98 (and n
(list (prefix-numeric-value n
)))))))
100 ;;; Write out powers (a*b*...)^n as a*b*...*a*b*...
101 (defun calcFunc-powerexpand (expr)
102 (math-normalize (math-map-tree 'math-powerexpand expr
)))
104 (defun math-powerexpand (expr)
105 (if (eq (car-safe expr
) '^
)
106 (let ((n (nth 2 expr
)))
107 (cond ((and (integerp n
)
113 (setq prod
(math-mul prod a
))
119 (a (math-pow (nth 1 expr
) -
1))
120 (prod (math-pow (nth 1 expr
) -
1)))
122 (setq prod
(math-mul a prod
))
129 (defun calc-powerexpand ()
132 (calc-enter-result 1 "pexp"
133 (calcFunc-powerexpand (calc-top-n 1)))))
135 (defun calc-collect (&optional var
)
136 (interactive "sCollect terms involving: ")
138 (if (or (equal var
"") (equal var
"$") (null var
))
139 (calc-enter-result 2 "clct" (cons 'calcFunc-collect
140 (calc-top-list-n 2)))
141 (let ((var (math-read-expr var
)))
142 (if (eq (car-safe var
) 'error
)
143 (error "Bad format in expression: %s" (nth 1 var
)))
144 (calc-enter-result 1 "clct" (list 'calcFunc-collect
148 (defun calc-apart (arg)
151 (calc-unary-op "aprt" 'calcFunc-apart arg
)))
153 (defun calc-normalize-rat (arg)
156 (calc-unary-op "nrat" 'calcFunc-nrat arg
)))
158 (defun calc-poly-gcd (arg)
161 (calc-binary-op "pgcd" 'calcFunc-pgcd arg
)))
164 (defun calc-poly-div (arg)
167 (let ((calc-poly-div-remainder nil
))
168 (calc-binary-op "pdiv" 'calcFunc-pdiv arg
)
169 (if (and calc-poly-div-remainder
(null arg
))
171 (calc-clear-command-flag 'clear-message
)
172 (calc-record calc-poly-div-remainder
"prem")
173 (if (not (Math-zerop calc-poly-div-remainder
))
174 (message "(Remainder was %s)"
175 (math-format-flat-expr calc-poly-div-remainder
0))
176 (message "(No remainder)")))))))
178 (defun calc-poly-rem (arg)
181 (calc-binary-op "prem" 'calcFunc-prem arg
)))
183 (defun calc-poly-div-rem (arg)
186 (if (calc-is-hyperbolic)
187 (calc-binary-op "pdvr" 'calcFunc-pdivide arg
)
188 (calc-binary-op "pdvr" 'calcFunc-pdivrem arg
))))
190 (defun calc-substitute (&optional oldname newname
)
191 (interactive "sSubstitute old: ")
193 (let (old new
(num 1) expr
)
194 (if (or (equal oldname
"") (equal oldname
"$") (null oldname
))
195 (setq new
(calc-top-n 1)
200 (progn (calc-unread-command ?\C-a
)
201 (setq newname
(read-string (concat "Substitute old: "
205 (if (or (equal newname
"") (equal newname
"$") (null newname
))
206 (setq new
(calc-top-n 1)
209 (setq new
(if (stringp newname
) (math-read-expr newname
) newname
))
210 (if (eq (car-safe new
) 'error
)
211 (error "Bad format in expression: %s" (nth 1 new
)))
212 (setq expr
(calc-top-n 1)))
213 (setq old
(if (stringp oldname
) (math-read-expr oldname
) oldname
))
214 (if (eq (car-safe old
) 'error
)
215 (error "Bad format in expression: %s" (nth 1 old
)))
216 (or (math-expr-contains expr old
)
217 (error "No occurrences found")))
218 (calc-enter-result num
"sbst" (math-expr-subst expr old new
)))))
221 (defun calc-has-rules (name)
222 (setq name
(calc-var-value name
))
224 (memq (car name
) '(vec calcFunc-assign calcFunc-condition
))
227 ;; math-eval-rules-cache and math-eval-rules-cache-other are
228 ;; declared in calc.el, but are used here by math-recompile-eval-rules.
229 (defvar math-eval-rules-cache
)
230 (defvar math-eval-rules-cache-other
)
232 (defun math-recompile-eval-rules ()
233 (setq math-eval-rules-cache
(and (calc-has-rules 'var-EvalRules
)
234 (math-compile-rewrites
235 '(var EvalRules var-EvalRules
)))
236 math-eval-rules-cache-other
(assq nil math-eval-rules-cache
)
237 math-eval-rules-cache-tag
(calc-var-value 'var-EvalRules
)))
240 ;;; Try to expand a formula according to its definition.
241 (defun math-expand-formula (expr)
244 (or (get (car expr
) 'calc-user-defn
)
245 (get (car expr
) 'math-expandable
))
246 (let ((res (let ((math-expand-formulas t
))
247 (apply (car expr
) (cdr expr
)))))
248 (and (not (eq (car-safe res
) (car expr
)))
254 ;;; True if A comes before B in a canonical ordering of expressions. [P X X]
255 (defun math-beforep (a b
) ; [Public]
256 (cond ((and (Math-realp a
) (Math-realp b
))
257 (let ((comp (math-compare a b
)))
261 (> (length (memq (car-safe a
)
262 '(bigneg nil bigpos frac float
)))
263 (length (memq (car-safe b
)
264 '(bigneg nil bigpos frac float
))))))))
265 ((equal b
'(neg (var inf var-inf
))) nil
)
266 ((equal a
'(neg (var inf var-inf
))) t
)
267 ((equal a
'(var inf var-inf
)) nil
)
268 ((equal b
'(var inf var-inf
)) t
)
270 (if (and (eq (car-safe b
) 'intv
) (math-intv-constp b
))
271 (if (or (math-beforep a
(nth 2 b
)) (Math-equal a
(nth 2 b
)))
276 (if (and (eq (car-safe a
) 'intv
) (math-intv-constp a
))
277 (if (math-beforep (nth 2 a
) b
)
281 ((and (eq (car a
) 'intv
) (eq (car b
) 'intv
)
282 (math-intv-constp a
) (math-intv-constp b
))
283 (let ((comp (math-compare (nth 2 a
) (nth 2 b
))))
284 (cond ((eq comp -
1) t
)
286 ((and (memq (nth 1 a
) '(2 3)) (memq (nth 1 b
) '(0 1))) t
)
287 ((and (memq (nth 1 a
) '(0 1)) (memq (nth 1 b
) '(2 3))) nil
)
288 ((eq (setq comp
(math-compare (nth 3 a
) (nth 3 b
))) -
1) t
)
290 ((and (memq (nth 1 a
) '(0 2)) (memq (nth 1 b
) '(1 3))) t
)
292 ((not (eq (not (Math-objectp a
)) (not (Math-objectp b
))))
295 (if (eq (car b
) 'var
)
296 (string-lessp (nth 1 a
) (nth 1 b
))
297 (not (Math-numberp b
))))
298 ((eq (car b
) 'var
) (Math-numberp a
))
299 ((eq (car a
) (car b
))
300 (while (and (setq a
(cdr a
) b
(cdr b
)) a
301 (equal (car a
) (car b
))))
304 (math-beforep (car a
) (car b
)))))
305 (t (string-lessp (car a
) (car b
)))))
308 (defsubst math-simplify-extended
(a)
309 (let ((math-living-dangerously t
))
312 (defalias 'calcFunc-esimplify
'math-simplify-extended
)
314 ;;; Rewrite the trig functions in a form easier to simplify.
315 (defun math-trig-rewrite (fn)
316 "Rewrite trigonometric functions in terms of sines and cosines."
320 ((eq (car-safe fn
) 'calcFunc-sec
)
321 (list '/ 1 (cons 'calcFunc-cos
(math-trig-rewrite (cdr fn
)))))
322 ((eq (car-safe fn
) 'calcFunc-csc
)
323 (list '/ 1 (cons 'calcFunc-sin
(math-trig-rewrite (cdr fn
)))))
324 ((eq (car-safe fn
) 'calcFunc-tan
)
325 (let ((newfn (math-trig-rewrite (cdr fn
))))
326 (list '/ (cons 'calcFunc-sin newfn
)
327 (cons 'calcFunc-cos newfn
))))
328 ((eq (car-safe fn
) 'calcFunc-cot
)
329 (let ((newfn (math-trig-rewrite (cdr fn
))))
330 (list '/ (cons 'calcFunc-cos newfn
)
331 (cons 'calcFunc-sin newfn
))))
333 (mapcar 'math-trig-rewrite fn
))))
335 (defun math-hyperbolic-trig-rewrite (fn)
336 "Rewrite hyperbolic functions in terms of sinhs and coshs."
340 ((eq (car-safe fn
) 'calcFunc-sech
)
341 (list '/ 1 (cons 'calcFunc-cosh
(math-hyperbolic-trig-rewrite (cdr fn
)))))
342 ((eq (car-safe fn
) 'calcFunc-csch
)
343 (list '/ 1 (cons 'calcFunc-sinh
(math-hyperbolic-trig-rewrite (cdr fn
)))))
344 ((eq (car-safe fn
) 'calcFunc-tanh
)
345 (let ((newfn (math-hyperbolic-trig-rewrite (cdr fn
))))
346 (list '/ (cons 'calcFunc-sinh newfn
)
347 (cons 'calcFunc-cosh newfn
))))
348 ((eq (car-safe fn
) 'calcFunc-coth
)
349 (let ((newfn (math-hyperbolic-trig-rewrite (cdr fn
))))
350 (list '/ (cons 'calcFunc-cosh newfn
)
351 (cons 'calcFunc-sinh newfn
))))
353 (mapcar 'math-hyperbolic-trig-rewrite fn
))))
355 ;; math-top-only is local to math-simplify, but is used by
356 ;; math-simplify-step, which is called by math-simplify.
357 (defvar math-top-only
)
359 ;; math-normalize-error is declared in calc.el.
360 (defvar math-normalize-error
)
361 (defun math-simplify (top-expr)
362 (let ((math-simplifying t
)
363 (math-top-only (consp calc-simplify-mode
))
364 (simp-rules (append (and (calc-has-rules 'var-AlgSimpRules
)
365 '((var AlgSimpRules var-AlgSimpRules
)))
366 (and math-living-dangerously
367 (calc-has-rules 'var-ExtSimpRules
)
368 '((var ExtSimpRules var-ExtSimpRules
)))
369 (and math-simplifying-units
370 (calc-has-rules 'var-UnitSimpRules
)
371 '((var UnitSimpRules var-UnitSimpRules
)))
372 (and math-integrating
373 (calc-has-rules 'var-IntegSimpRules
)
374 '((var IntegSimpRules var-IntegSimpRules
)))))
377 (let ((r simp-rules
))
378 (setq res
(math-simplify-step (math-normalize top-expr
))
379 calc-simplify-mode
'(nil)
380 top-expr
(math-normalize res
))
382 (setq top-expr
(math-rewrite top-expr
(car r
)
383 '(neg (var inf var-inf
)))
385 (calc-with-default-simplification
386 (while (let ((r simp-rules
))
387 (setq res
(math-normalize top-expr
))
388 (if (not math-normalize-error
)
391 (setq res
(math-rewrite res
(car r
))
393 (not (equal top-expr
(setq res
(math-simplify-step res
)))))))
394 (setq top-expr res
)))))
397 (defalias 'calcFunc-simplify
'math-simplify
)
399 ;;; The following has a "bug" in that if any recursive simplifications
400 ;;; occur only the first handler will be tried; this doesn't really
401 ;;; matter, since math-simplify-step is iterated to a fixed point anyway.
402 (defun math-simplify-step (a)
405 (let ((aa (if (or math-top-only
406 (memq (car a
) '(calcFunc-quote calcFunc-condition
409 (cons (car a
) (mapcar 'math-simplify-step
(cdr a
))))))
410 (and (symbolp (car aa
))
411 (let ((handler (get (car aa
) 'math-simplify
)))
414 (equal (setq aa
(or (funcall (car handler
) aa
)
417 (setq handler
(cdr handler
))))))
421 (defmacro math-defsimplify
(funcs &rest code
)
423 (mapcar #'(lambda (func)
424 `(put ',func
'math-simplify
426 (get ',func
'math-simplify
)
428 #'(lambda (math-simplify-expr) ,@code
)))))
429 (if (symbolp funcs
) (list funcs
) funcs
))))
430 (put 'math-defsimplify
'lisp-indent-hook
1)
432 ;; The function created by math-defsimplify uses the variable
433 ;; math-simplify-expr, and so is used by functions in math-defsimplify
434 (defvar math-simplify-expr
)
436 (math-defsimplify (+ -
)
437 (math-simplify-plus))
439 (defun math-simplify-plus ()
440 (cond ((and (memq (car-safe (nth 1 math-simplify-expr
)) '(+ -
))
441 (Math-numberp (nth 2 (nth 1 math-simplify-expr
)))
442 (not (Math-numberp (nth 2 math-simplify-expr
))))
443 (let ((x (nth 2 math-simplify-expr
))
444 (op (car math-simplify-expr
)))
445 (setcar (cdr (cdr math-simplify-expr
)) (nth 2 (nth 1 math-simplify-expr
)))
446 (setcar math-simplify-expr
(car (nth 1 math-simplify-expr
)))
447 (setcar (cdr (cdr (nth 1 math-simplify-expr
))) x
)
448 (setcar (nth 1 math-simplify-expr
) op
)))
449 ((and (eq (car math-simplify-expr
) '+)
450 (Math-numberp (nth 1 math-simplify-expr
))
451 (not (Math-numberp (nth 2 math-simplify-expr
))))
452 (let ((x (nth 2 math-simplify-expr
)))
453 (setcar (cdr (cdr math-simplify-expr
)) (nth 1 math-simplify-expr
))
454 (setcar (cdr math-simplify-expr
) x
))))
455 (let ((aa math-simplify-expr
)
457 (while (memq (car-safe (setq aaa
(nth 1 aa
))) '(+ -
))
458 (if (setq temp
(math-combine-sum (nth 2 aaa
) (nth 2 math-simplify-expr
)
460 (eq (car math-simplify-expr
) '-
) t
))
462 (setcar (cdr (cdr math-simplify-expr
)) temp
)
463 (setcar math-simplify-expr
'+)
464 (setcar (cdr (cdr aaa
)) 0)))
465 (setq aa
(nth 1 aa
)))
466 (if (setq temp
(math-combine-sum aaa
(nth 2 math-simplify-expr
)
467 nil
(eq (car math-simplify-expr
) '-
) t
))
469 (setcar (cdr (cdr math-simplify-expr
)) temp
)
470 (setcar math-simplify-expr
'+)
471 (setcar (cdr aa
) 0)))
475 (math-simplify-times))
477 (defun math-simplify-times ()
478 (if (eq (car-safe (nth 2 math-simplify-expr
)) '*)
479 (and (math-beforep (nth 1 (nth 2 math-simplify-expr
)) (nth 1 math-simplify-expr
))
480 (or (math-known-scalarp (nth 1 math-simplify-expr
) t
)
481 (math-known-scalarp (nth 1 (nth 2 math-simplify-expr
)) t
))
482 (let ((x (nth 1 math-simplify-expr
)))
483 (setcar (cdr math-simplify-expr
) (nth 1 (nth 2 math-simplify-expr
)))
484 (setcar (cdr (nth 2 math-simplify-expr
)) x
)))
485 (and (math-beforep (nth 2 math-simplify-expr
) (nth 1 math-simplify-expr
))
486 (or (math-known-scalarp (nth 1 math-simplify-expr
) t
)
487 (math-known-scalarp (nth 2 math-simplify-expr
) t
))
488 (let ((x (nth 2 math-simplify-expr
)))
489 (setcar (cdr (cdr math-simplify-expr
)) (nth 1 math-simplify-expr
))
490 (setcar (cdr math-simplify-expr
) x
))))
491 (let ((aa math-simplify-expr
)
493 (safe t
) (scalar (math-known-scalarp (nth 1 math-simplify-expr
))))
494 (if (and (Math-ratp (nth 1 math-simplify-expr
))
495 (setq temp
(math-common-constant-factor (nth 2 math-simplify-expr
))))
497 (setcar (cdr (cdr math-simplify-expr
))
498 (math-cancel-common-factor (nth 2 math-simplify-expr
) temp
))
499 (setcar (cdr math-simplify-expr
) (math-mul (nth 1 math-simplify-expr
) temp
))))
500 (while (and (eq (car-safe (setq aaa
(nth 2 aa
))) '*)
502 (if (setq temp
(math-combine-prod (nth 1 math-simplify-expr
)
503 (nth 1 aaa
) nil nil t
))
505 (setcar (cdr math-simplify-expr
) temp
)
506 (setcar (cdr aaa
) 1)))
507 (setq safe
(or scalar
(math-known-scalarp (nth 1 aaa
) t
))
509 (if (and (setq temp
(math-combine-prod aaa
(nth 1 math-simplify-expr
) nil nil t
))
512 (setcar (cdr math-simplify-expr
) temp
)
513 (setcar (cdr (cdr aa
)) 1)))
514 (if (and (eq (car-safe (nth 1 math-simplify-expr
)) 'frac
)
515 (memq (nth 1 (nth 1 math-simplify-expr
)) '(1 -
1)))
516 (math-div (math-mul (nth 2 math-simplify-expr
)
517 (nth 1 (nth 1 math-simplify-expr
)))
518 (nth 2 (nth 1 math-simplify-expr
)))
519 math-simplify-expr
)))
522 (math-simplify-divide))
524 (defun math-simplify-divide ()
525 (let ((np (cdr math-simplify-expr
))
527 (nn (and (or (eq (car math-simplify-expr
) '/)
528 (not (Math-realp (nth 2 math-simplify-expr
))))
529 (math-common-constant-factor (nth 2 math-simplify-expr
))))
533 (setq n
(and (or (eq (car math-simplify-expr
) '/)
534 (not (Math-realp (nth 1 math-simplify-expr
))))
535 (math-common-constant-factor (nth 1 math-simplify-expr
))))
536 (if (and (eq (car-safe nn
) 'frac
) (eq (nth 1 nn
) 1) (not n
))
537 (unless (and (eq (car-safe math-simplify-expr
) 'calcFunc-eq
)
538 (eq (car-safe (nth 1 math-simplify-expr
)) 'var
)
539 (not (math-expr-contains (nth 2 math-simplify-expr
)
540 (nth 1 math-simplify-expr
))))
541 (setcar (cdr math-simplify-expr
)
542 (math-mul (nth 2 nn
) (nth 1 math-simplify-expr
)))
543 (setcar (cdr (cdr math-simplify-expr
))
544 (math-cancel-common-factor (nth 2 math-simplify-expr
) nn
))
545 (if (and (math-negp nn
)
546 (setq op
(assq (car math-simplify-expr
) calc-tweak-eqn-table
)))
547 (setcar math-simplify-expr
(nth 1 op
))))
548 (if (and n
(not (eq (setq n
(math-frac-gcd n nn
)) 1)))
550 (setcar (cdr math-simplify-expr
)
551 (math-cancel-common-factor (nth 1 math-simplify-expr
) n
))
552 (setcar (cdr (cdr math-simplify-expr
))
553 (math-cancel-common-factor (nth 2 math-simplify-expr
) n
))
554 (if (and (math-negp n
)
555 (setq op
(assq (car math-simplify-expr
)
556 calc-tweak-eqn-table
)))
557 (setcar math-simplify-expr
(nth 1 op
))))))))
558 (if (and (eq (car-safe (car np
)) '/)
559 (math-known-scalarp (nth 2 math-simplify-expr
) t
))
561 (setq np
(cdr (nth 1 math-simplify-expr
)))
562 (while (eq (car-safe (setq n
(car np
))) '*)
563 (and (math-known-scalarp (nth 2 n
) t
)
564 (math-simplify-divisor (cdr n
) (cdr (cdr math-simplify-expr
)) nil t
))
565 (setq np
(cdr (cdr n
))))
566 (math-simplify-divisor np
(cdr (cdr math-simplify-expr
)) nil t
)
568 np
(cdr (cdr (nth 1 math-simplify-expr
))))))
569 (while (eq (car-safe (setq n
(car np
))) '*)
570 (and (math-known-scalarp (nth 2 n
) t
)
571 (math-simplify-divisor (cdr n
) (cdr (cdr math-simplify-expr
)) nover t
))
572 (setq np
(cdr (cdr n
))))
573 (math-simplify-divisor np
(cdr (cdr math-simplify-expr
)) nover t
)
576 ;; The variables math-simplify-divisor-nover and math-simplify-divisor-dover
577 ;; are local variables for math-simplify-divisor, but are used by
578 ;; math-simplify-one-divisor.
579 (defvar math-simplify-divisor-nover
)
580 (defvar math-simplify-divisor-dover
)
582 (defun math-simplify-divisor (np dp math-simplify-divisor-nover
583 math-simplify-divisor-dover
)
584 (cond ((eq (car-safe (car dp
)) '/)
585 (math-simplify-divisor np
(cdr (car dp
))
586 math-simplify-divisor-nover
587 math-simplify-divisor-dover
)
588 (and (math-known-scalarp (nth 1 (car dp
)) t
)
589 (math-simplify-divisor np
(cdr (cdr (car dp
)))
590 math-simplify-divisor-nover
591 (not math-simplify-divisor-dover
))))
592 ((or (or (eq (car math-simplify-expr
) '/)
593 (let ((signs (math-possible-signs (car np
))))
594 (or (memq signs
'(1 4))
595 (and (memq (car math-simplify-expr
) '(calcFunc-eq calcFunc-neq
))
597 math-living-dangerously
)))
598 (math-numberp (car np
)))
601 (scalar (math-known-scalarp (car np
))))
602 (while (and (eq (car-safe (setq d
(car dp
))) '*)
604 (math-simplify-one-divisor np
(cdr d
))
605 (setq safe
(or scalar
(math-known-scalarp (nth 1 d
) t
))
608 (math-simplify-one-divisor np dp
))))))
610 (defun math-simplify-one-divisor (np dp
)
611 (let ((temp (math-combine-prod (car np
) (car dp
) math-simplify-divisor-nover
612 math-simplify-divisor-dover t
))
616 (and (not (memq (car math-simplify-expr
) '(/ calcFunc-eq calcFunc-neq
)))
617 (math-known-negp (car dp
))
618 (setq op
(assq (car math-simplify-expr
) calc-tweak-eqn-table
))
619 (setcar math-simplify-expr
(nth 1 op
)))
620 (setcar np
(if math-simplify-divisor-nover
(math-div 1 temp
) temp
))
622 (and math-simplify-divisor-dover
(not math-simplify-divisor-nover
)
623 (eq (car math-simplify-expr
) '/)
624 (eq (car-safe (car dp
)) 'calcFunc-sqrt
)
625 (Math-integerp (nth 1 (car dp
)))
627 (setcar np
(math-mul (car np
)
628 (list 'calcFunc-sqrt
(nth 1 (car dp
)))))
629 (setcar dp
(nth 1 (car dp
))))))))
631 (defun math-common-constant-factor (expr)
632 (if (Math-realp expr
)
634 (and (not (memq expr
'(0 1 -
1)))
636 (if (math-ratp (setq expr
(math-to-simple-fraction expr
)))
637 (math-common-constant-factor expr
)))
638 (if (memq (car expr
) '(+ - cplx sdev
))
639 (let ((f1 (math-common-constant-factor (nth 1 expr
)))
640 (f2 (math-common-constant-factor (nth 2 expr
))))
642 (not (eq (setq f1
(math-frac-gcd f1 f2
)) 1))
644 (if (memq (car expr
) '(* polar
))
645 (math-common-constant-factor (nth 1 expr
))
646 (if (eq (car expr
) '/)
647 (or (math-common-constant-factor (nth 1 expr
))
648 (and (Math-integerp (nth 2 expr
))
649 (list 'frac
1 (math-abs (nth 2 expr
))))))))))
651 (defun math-cancel-common-factor (expr val
)
652 (if (memq (car-safe expr
) '(+ - cplx sdev
))
654 (setcar (cdr expr
) (math-cancel-common-factor (nth 1 expr
) val
))
655 (setcar (cdr (cdr expr
)) (math-cancel-common-factor (nth 2 expr
) val
))
657 (if (eq (car-safe expr
) '*)
658 (math-mul (math-cancel-common-factor (nth 1 expr
) val
) (nth 2 expr
))
659 (math-div expr val
))))
661 (defun math-frac-gcd (a b
)
666 (if (and (Math-integerp a
)
669 (and (Math-integerp a
) (setq a
(list 'frac a
1)))
670 (and (Math-integerp b
) (setq b
(list 'frac b
1)))
671 (math-make-frac (math-gcd (nth 1 a
) (nth 1 b
))
672 (math-gcd (nth 2 a
) (nth 2 b
)))))))
677 (defun math-simplify-mod ()
678 (and (Math-realp (nth 2 math-simplify-expr
))
679 (Math-posp (nth 2 math-simplify-expr
))
680 (let ((lin (math-is-linear (nth 1 math-simplify-expr
)))
683 (or (math-negp (car lin
))
684 (not (Math-lessp (car lin
) (nth 2 math-simplify-expr
))))
687 (math-mul (nth 1 lin
) (nth 2 lin
))
688 (math-mod (car lin
) (nth 2 math-simplify-expr
)))
689 (nth 2 math-simplify-expr
)))
691 (not (math-equal-int (nth 1 lin
) 1))
692 (math-num-integerp (nth 1 lin
))
693 (math-num-integerp (nth 2 math-simplify-expr
))
694 (setq t1
(calcFunc-gcd (nth 1 lin
) (nth 2 math-simplify-expr
)))
695 (not (math-equal-int t1
1))
700 (math-mul (math-div (nth 1 lin
) t1
)
702 (let ((calc-prefer-frac t
))
703 (math-div (car lin
) t1
)))
704 (math-div (nth 2 math-simplify-expr
) t1
))))
705 (and (math-equal-int (nth 2 math-simplify-expr
) 1)
706 (math-known-integerp (if lin
707 (math-mul (nth 1 lin
) (nth 2 lin
))
708 (nth 1 math-simplify-expr
)))
709 (if lin
(math-mod (car lin
) 1) 0))))))
711 (math-defsimplify (calcFunc-eq calcFunc-neq calcFunc-lt
712 calcFunc-gt calcFunc-leq calcFunc-geq
)
713 (if (= (length math-simplify-expr
) 3)
714 (math-simplify-ineq)))
716 (defun math-simplify-ineq ()
717 (let ((np (cdr math-simplify-expr
))
719 (while (memq (car-safe (setq n
(car np
))) '(+ -
))
720 (math-simplify-add-term (cdr (cdr n
)) (cdr (cdr math-simplify-expr
))
723 (math-simplify-add-term np
(cdr (cdr math-simplify-expr
)) nil
724 (eq np
(cdr math-simplify-expr
)))
725 (math-simplify-divide)
726 (let ((signs (math-possible-signs (cons '-
(cdr math-simplify-expr
)))))
727 (or (cond ((eq (car math-simplify-expr
) 'calcFunc-eq
)
728 (or (and (eq signs
2) 1)
729 (and (memq signs
'(1 4 5)) 0)))
730 ((eq (car math-simplify-expr
) 'calcFunc-neq
)
731 (or (and (eq signs
2) 0)
732 (and (memq signs
'(1 4 5)) 1)))
733 ((eq (car math-simplify-expr
) 'calcFunc-lt
)
734 (or (and (eq signs
1) 1)
735 (and (memq signs
'(2 4 6)) 0)))
736 ((eq (car math-simplify-expr
) 'calcFunc-gt
)
737 (or (and (eq signs
4) 1)
738 (and (memq signs
'(1 2 3)) 0)))
739 ((eq (car math-simplify-expr
) 'calcFunc-leq
)
740 (or (and (eq signs
4) 0)
741 (and (memq signs
'(1 2 3)) 1)))
742 ((eq (car math-simplify-expr
) 'calcFunc-geq
)
743 (or (and (eq signs
1) 0)
744 (and (memq signs
'(2 4 6)) 1))))
745 math-simplify-expr
))))
747 (defun math-simplify-add-term (np dp minus lplain
)
748 (or (math-vectorp (car np
))
751 (while (memq (car-safe (setq n
(car np
) d
(car dp
))) '(+ -
))
753 (if (setq temp
(math-combine-sum n
(nth 2 d
)
754 minus
(eq (car d
) '+) t
))
755 (if (or lplain
(eq (math-looks-negp temp
) minus
))
757 (setcar np
(setq n
(if minus
(math-neg temp
) temp
)))
758 (setcar (cdr (cdr d
)) 0))
761 (setcar (cdr (cdr d
)) (setq n
(if (eq (car d
) '+)
765 (if (setq temp
(math-combine-sum n d minus t t
))
768 (eq (math-looks-negp temp
) minus
)))
770 (setcar np
(setq n
(if minus
(math-neg temp
) temp
)))
774 (setcar dp
(setq n
(math-neg temp
)))))))))
776 (math-defsimplify calcFunc-sin
777 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
778 (nth 1 (nth 1 math-simplify-expr
)))
779 (and (math-looks-negp (nth 1 math-simplify-expr
))
780 (math-neg (list 'calcFunc-sin
(math-neg (nth 1 math-simplify-expr
)))))
781 (and (eq calc-angle-mode
'rad
)
782 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
784 (math-known-sin (car n
) (nth 1 n
) 120 0))))
785 (and (eq calc-angle-mode
'deg
)
786 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
788 (math-known-sin (car n
) (nth 1 n
) '(frac 2 3) 0))))
789 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
790 (list 'calcFunc-sqrt
(math-sub 1 (math-sqr
791 (nth 1 (nth 1 math-simplify-expr
))))))
792 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
793 (math-div (nth 1 (nth 1 math-simplify-expr
))
795 (math-add 1 (math-sqr
796 (nth 1 (nth 1 math-simplify-expr
)))))))
797 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
))))
798 (and m
(integerp (car m
))
799 (let ((n (car m
)) (a (nth 1 m
)))
801 (list '* (list 'calcFunc-sin
(list '* (1- n
) a
))
802 (list 'calcFunc-cos a
))
803 (list '* (list 'calcFunc-cos
(list '* (1- n
) a
))
804 (list 'calcFunc-sin a
))))))))
806 (math-defsimplify calcFunc-cos
807 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
808 (nth 1 (nth 1 math-simplify-expr
)))
809 (and (math-looks-negp (nth 1 math-simplify-expr
))
810 (list 'calcFunc-cos
(math-neg (nth 1 math-simplify-expr
))))
811 (and (eq calc-angle-mode
'rad
)
812 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
814 (math-known-sin (car n
) (nth 1 n
) 120 300))))
815 (and (eq calc-angle-mode
'deg
)
816 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
818 (math-known-sin (car n
) (nth 1 n
) '(frac 2 3) 300))))
819 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
821 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
))))))
822 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
826 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
827 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
))))
828 (and m
(integerp (car m
))
829 (let ((n (car m
)) (a (nth 1 m
)))
831 (list '* (list 'calcFunc-cos
(list '* (1- n
) a
))
832 (list 'calcFunc-cos a
))
833 (list '* (list 'calcFunc-sin
(list '* (1- n
) a
))
834 (list 'calcFunc-sin a
))))))))
836 (math-defsimplify calcFunc-sec
837 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
838 (list 'calcFunc-sec
(math-neg (nth 1 math-simplify-expr
))))
839 (and (eq calc-angle-mode
'rad
)
840 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
842 (math-div 1 (math-known-sin (car n
) (nth 1 n
) 120 300)))))
843 (and (eq calc-angle-mode
'deg
)
844 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
846 (math-div 1 (math-known-sin (car n
) (nth 1 n
) '(frac 2 3) 300)))))
847 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
851 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
852 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
855 (nth 1 (nth 1 math-simplify-expr
))))
856 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
859 (math-sqr (nth 1 (nth 1 math-simplify-expr
))))))))
861 (math-defsimplify calcFunc-csc
862 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
863 (math-neg (list 'calcFunc-csc
(math-neg (nth 1 math-simplify-expr
)))))
864 (and (eq calc-angle-mode
'rad
)
865 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
867 (math-div 1 (math-known-sin (car n
) (nth 1 n
) 120 0)))))
868 (and (eq calc-angle-mode
'deg
)
869 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
871 (math-div 1 (math-known-sin (car n
) (nth 1 n
) '(frac 2 3) 0)))))
872 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
873 (math-div 1 (nth 1 (nth 1 math-simplify-expr
))))
874 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
877 (list 'calcFunc-sqrt
(math-sub 1 (math-sqr
878 (nth 1 (nth 1 math-simplify-expr
)))))))
879 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
880 (math-div (list 'calcFunc-sqrt
881 (math-add 1 (math-sqr
882 (nth 1 (nth 1 math-simplify-expr
)))))
883 (nth 1 (nth 1 math-simplify-expr
))))))
885 (defun math-should-expand-trig (x &optional hyperbolic
)
886 (let ((m (math-is-multiple x
)))
887 (and math-living-dangerously
888 m
(or (and (integerp (car m
)) (> (car m
) 1))
889 (equal (car m
) '(frac 1 2)))
891 (memq (car-safe (nth 1 m
))
893 '(calcFunc-arcsinh calcFunc-arccosh calcFunc-arctanh
)
894 '(calcFunc-arcsin calcFunc-arccos calcFunc-arctan
)))
895 (and (eq (car-safe (nth 1 m
)) 'calcFunc-ln
)
896 (eq hyperbolic
'exp
)))
899 (defun math-known-sin (plus n mul off
)
900 (setq n
(math-mul n mul
))
901 (and (math-num-integerp n
)
902 (setq n
(math-mod (math-add (math-trunc n
) off
) 240))
904 (and (setq n
(math-known-sin plus
(- n
120) 1 0))
908 (if (math-zerop plus
)
909 (and (or calc-symbolic-mode
913 (10 .
(/ (calcFunc-sqrt
914 (- 2 (calcFunc-sqrt 3))) 2))
915 (12 .
(/ (- (calcFunc-sqrt 5) 1) 4))
916 (15 .
(/ (calcFunc-sqrt
917 (- 2 (calcFunc-sqrt 2))) 2))
919 (24 .
(* (^
(/ 1 2) (/ 3 2))
921 (- 5 (calcFunc-sqrt 5)))))
922 (30 .
(/ (calcFunc-sqrt 2) 2))
923 (36 .
(/ (+ (calcFunc-sqrt 5) 1) 4))
924 (40 .
(/ (calcFunc-sqrt 3) 2))
925 (45 .
(/ (calcFunc-sqrt
926 (+ 2 (calcFunc-sqrt 2))) 2))
927 (48 .
(* (^
(/ 1 2) (/ 3 2))
929 (+ 5 (calcFunc-sqrt 5)))))
930 (50 .
(/ (calcFunc-sqrt
931 (+ 2 (calcFunc-sqrt 3))) 2))
933 (cond ((eq n
0) (math-normalize (list 'calcFunc-sin plus
)))
934 ((eq n
60) (math-normalize (list 'calcFunc-cos plus
)))
937 (math-defsimplify calcFunc-tan
938 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
939 (nth 1 (nth 1 math-simplify-expr
)))
940 (and (math-looks-negp (nth 1 math-simplify-expr
))
941 (math-neg (list 'calcFunc-tan
(math-neg (nth 1 math-simplify-expr
)))))
942 (and (eq calc-angle-mode
'rad
)
943 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
945 (math-known-tan (car n
) (nth 1 n
) 120))))
946 (and (eq calc-angle-mode
'deg
)
947 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
949 (math-known-tan (car n
) (nth 1 n
) '(frac 2 3)))))
950 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
951 (math-div (nth 1 (nth 1 math-simplify-expr
))
953 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
954 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
955 (math-div (list 'calcFunc-sqrt
956 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))
957 (nth 1 (nth 1 math-simplify-expr
))))
958 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
))))
960 (if (equal (car m
) '(frac 1 2))
961 (math-div (math-sub 1 (list 'calcFunc-cos
(nth 1 m
)))
962 (list 'calcFunc-sin
(nth 1 m
)))
963 (math-div (list 'calcFunc-sin
(nth 1 math-simplify-expr
))
964 (list 'calcFunc-cos
(nth 1 math-simplify-expr
))))))))
966 (math-defsimplify calcFunc-cot
967 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
968 (math-neg (list 'calcFunc-cot
(math-neg (nth 1 math-simplify-expr
)))))
969 (and (eq calc-angle-mode
'rad
)
970 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
972 (math-div 1 (math-known-tan (car n
) (nth 1 n
) 120)))))
973 (and (eq calc-angle-mode
'deg
)
974 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
976 (math-div 1 (math-known-tan (car n
) (nth 1 n
) '(frac 2 3))))))
977 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
978 (math-div (list 'calcFunc-sqrt
979 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))
980 (nth 1 (nth 1 math-simplify-expr
))))
981 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
982 (math-div (nth 1 (nth 1 math-simplify-expr
))
984 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
985 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
986 (math-div 1 (nth 1 (nth 1 math-simplify-expr
))))))
988 (defun math-known-tan (plus n mul
)
989 (setq n
(math-mul n mul
))
990 (and (math-num-integerp n
)
991 (setq n
(math-mod (math-trunc n
) 120))
993 (and (setq n
(math-known-tan plus
(- 120 n
) 1))
995 (if (math-zerop plus
)
996 (and (or calc-symbolic-mode
998 (cdr (assq n
'( (0 .
0)
999 (10 .
(- 2 (calcFunc-sqrt 3)))
1000 (12 .
(calcFunc-sqrt
1001 (- 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
1002 (15 .
(- (calcFunc-sqrt 2) 1))
1003 (20 .
(/ (calcFunc-sqrt 3) 3))
1004 (24 .
(calcFunc-sqrt
1005 (- 5 (* 2 (calcFunc-sqrt 5)))))
1007 (36 .
(calcFunc-sqrt
1008 (+ 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
1009 (40 .
(calcFunc-sqrt 3))
1010 (45 .
(+ (calcFunc-sqrt 2) 1))
1011 (48 .
(calcFunc-sqrt
1012 (+ 5 (* 2 (calcFunc-sqrt 5)))))
1013 (50 .
(+ 2 (calcFunc-sqrt 3)))
1014 (60 .
(var uinf var-uinf
))))))
1015 (cond ((eq n
0) (math-normalize (list 'calcFunc-tan plus
)))
1016 ((eq n
60) (math-normalize (list '/ -
1
1017 (list 'calcFunc-tan plus
))))
1020 (math-defsimplify calcFunc-sinh
1021 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
1022 (nth 1 (nth 1 math-simplify-expr
)))
1023 (and (math-looks-negp (nth 1 math-simplify-expr
))
1024 (math-neg (list 'calcFunc-sinh
(math-neg (nth 1 math-simplify-expr
)))))
1025 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
1026 math-living-dangerously
1027 (list 'calcFunc-sqrt
1028 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1)))
1029 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
1030 math-living-dangerously
1031 (math-div (nth 1 (nth 1 math-simplify-expr
))
1032 (list 'calcFunc-sqrt
1033 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
1034 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
) t
)))
1035 (and m
(integerp (car m
))
1036 (let ((n (car m
)) (a (nth 1 m
)))
1039 (list '* (list 'calcFunc-sinh
(list '* (1- n
) a
))
1040 (list 'calcFunc-cosh a
))
1041 (list '* (list 'calcFunc-cosh
(list '* (1- n
) a
))
1042 (list 'calcFunc-sinh a
)))))))))
1044 (math-defsimplify calcFunc-cosh
1045 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
1046 (nth 1 (nth 1 math-simplify-expr
)))
1047 (and (math-looks-negp (nth 1 math-simplify-expr
))
1048 (list 'calcFunc-cosh
(math-neg (nth 1 math-simplify-expr
))))
1049 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
1050 math-living-dangerously
1051 (list 'calcFunc-sqrt
1052 (math-add (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
1056 (list 'calcFunc-sqrt
1057 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
1058 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
) t
)))
1059 (and m
(integerp (car m
))
1060 (let ((n (car m
)) (a (nth 1 m
)))
1063 (list '* (list 'calcFunc-cosh
(list '* (1- n
) a
))
1064 (list 'calcFunc-cosh a
))
1065 (list '* (list 'calcFunc-sinh
(list '* (1- n
) a
))
1066 (list 'calcFunc-sinh a
)))))))))
1068 (math-defsimplify calcFunc-tanh
1069 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
1070 (nth 1 (nth 1 math-simplify-expr
)))
1071 (and (math-looks-negp (nth 1 math-simplify-expr
))
1072 (math-neg (list 'calcFunc-tanh
(math-neg (nth 1 math-simplify-expr
)))))
1073 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
1074 math-living-dangerously
1075 (math-div (nth 1 (nth 1 math-simplify-expr
))
1076 (list 'calcFunc-sqrt
1077 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))))
1078 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
1079 math-living-dangerously
1080 (math-div (list 'calcFunc-sqrt
1081 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))
1082 (nth 1 (nth 1 math-simplify-expr
))))
1083 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
) t
)))
1085 (if (equal (car m
) '(frac 1 2))
1086 (math-div (math-sub (list 'calcFunc-cosh
(nth 1 m
)) 1)
1087 (list 'calcFunc-sinh
(nth 1 m
)))
1088 (math-div (list 'calcFunc-sinh
(nth 1 math-simplify-expr
))
1089 (list 'calcFunc-cosh
(nth 1 math-simplify-expr
))))))))
1091 (math-defsimplify calcFunc-sech
1092 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1093 (list 'calcFunc-sech
(math-neg (nth 1 math-simplify-expr
))))
1094 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
1095 math-living-dangerously
1098 (list 'calcFunc-sqrt
1099 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))))
1100 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
1101 math-living-dangerously
1102 (math-div 1 (nth 1 (nth 1 math-simplify-expr
))) 1)
1103 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
1104 math-living-dangerously
1105 (list 'calcFunc-sqrt
1106 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
))))))))
1108 (math-defsimplify calcFunc-csch
1109 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1110 (math-neg (list 'calcFunc-csch
(math-neg (nth 1 math-simplify-expr
)))))
1111 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
1112 math-living-dangerously
1113 (math-div 1 (nth 1 (nth 1 math-simplify-expr
))))
1114 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
1115 math-living-dangerously
1118 (list 'calcFunc-sqrt
1119 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))))
1120 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
1121 math-living-dangerously
1122 (math-div (list 'calcFunc-sqrt
1123 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))
1124 (nth 1 (nth 1 math-simplify-expr
))))))
1126 (math-defsimplify calcFunc-coth
1127 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1128 (math-neg (list 'calcFunc-coth
(math-neg (nth 1 math-simplify-expr
)))))
1129 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
1130 math-living-dangerously
1131 (math-div (list 'calcFunc-sqrt
1132 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))
1133 (nth 1 (nth 1 math-simplify-expr
))))
1134 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
1135 math-living-dangerously
1136 (math-div (nth 1 (nth 1 math-simplify-expr
))
1137 (list 'calcFunc-sqrt
1138 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))))
1139 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
1140 math-living-dangerously
1141 (math-div 1 (nth 1 (nth 1 math-simplify-expr
))))))
1143 (math-defsimplify calcFunc-arcsin
1144 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1145 (math-neg (list 'calcFunc-arcsin
(math-neg (nth 1 math-simplify-expr
)))))
1146 (and (eq (nth 1 math-simplify-expr
) 1)
1147 (math-quarter-circle t
))
1148 (and (equal (nth 1 math-simplify-expr
) '(frac 1 2))
1149 (math-div (math-half-circle t
) 6))
1150 (and math-living-dangerously
1151 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sin
)
1152 (nth 1 (nth 1 math-simplify-expr
)))
1153 (and math-living-dangerously
1154 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cos
)
1155 (math-sub (math-quarter-circle t
)
1156 (nth 1 (nth 1 math-simplify-expr
))))))
1158 (math-defsimplify calcFunc-arccos
1159 (or (and (eq (nth 1 math-simplify-expr
) 0)
1160 (math-quarter-circle t
))
1161 (and (eq (nth 1 math-simplify-expr
) -
1)
1162 (math-half-circle t
))
1163 (and (equal (nth 1 math-simplify-expr
) '(frac 1 2))
1164 (math-div (math-half-circle t
) 3))
1165 (and (equal (nth 1 math-simplify-expr
) '(frac -
1 2))
1166 (math-div (math-mul (math-half-circle t
) 2) 3))
1167 (and math-living-dangerously
1168 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cos
)
1169 (nth 1 (nth 1 math-simplify-expr
)))
1170 (and math-living-dangerously
1171 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sin
)
1172 (math-sub (math-quarter-circle t
)
1173 (nth 1 (nth 1 math-simplify-expr
))))))
1175 (math-defsimplify calcFunc-arctan
1176 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1177 (math-neg (list 'calcFunc-arctan
(math-neg (nth 1 math-simplify-expr
)))))
1178 (and (eq (nth 1 math-simplify-expr
) 1)
1179 (math-div (math-half-circle t
) 4))
1180 (and math-living-dangerously
1181 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-tan
)
1182 (nth 1 (nth 1 math-simplify-expr
)))))
1184 (math-defsimplify calcFunc-arcsinh
1185 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1186 (math-neg (list 'calcFunc-arcsinh
(math-neg (nth 1 math-simplify-expr
)))))
1187 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sinh
)
1188 (or math-living-dangerously
1189 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
1190 (nth 1 (nth 1 math-simplify-expr
)))))
1192 (math-defsimplify calcFunc-arccosh
1193 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cosh
)
1194 (or math-living-dangerously
1195 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
1196 (nth 1 (nth 1 math-simplify-expr
))))
1198 (math-defsimplify calcFunc-arctanh
1199 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1200 (math-neg (list 'calcFunc-arctanh
(math-neg (nth 1 math-simplify-expr
)))))
1201 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-tanh
)
1202 (or math-living-dangerously
1203 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
1204 (nth 1 (nth 1 math-simplify-expr
)))))
1206 (math-defsimplify calcFunc-sqrt
1207 (math-simplify-sqrt))
1209 (defun math-simplify-sqrt ()
1210 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'frac
)
1211 (math-div (list 'calcFunc-sqrt
1212 (math-mul (nth 1 (nth 1 math-simplify-expr
))
1213 (nth 2 (nth 1 math-simplify-expr
))))
1214 (nth 2 (nth 1 math-simplify-expr
))))
1215 (let ((fac (if (math-objectp (nth 1 math-simplify-expr
))
1216 (math-squared-factor (nth 1 math-simplify-expr
))
1217 (math-common-constant-factor (nth 1 math-simplify-expr
)))))
1218 (and fac
(not (eq fac
1))
1219 (math-mul (math-normalize (list 'calcFunc-sqrt fac
))
1221 (list 'calcFunc-sqrt
1222 (math-cancel-common-factor
1223 (nth 1 math-simplify-expr
) fac
))))))
1224 (and math-living-dangerously
1225 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) '-
)
1226 (math-equal-int (nth 1 (nth 1 math-simplify-expr
)) 1)
1227 (eq (car-safe (nth 2 (nth 1 math-simplify-expr
))) '^
)
1228 (math-equal-int (nth 2 (nth 2 (nth 1 math-simplify-expr
))) 2)
1229 (or (and (eq (car-safe (nth 1 (nth 2 (nth 1 math-simplify-expr
))))
1232 (nth 1 (nth 1 (nth 2 (nth 1 math-simplify-expr
))))))
1233 (and (eq (car-safe (nth 1 (nth 2 (nth 1 math-simplify-expr
))))
1236 (nth 1 (nth 1 (nth 2
1237 (nth 1 math-simplify-expr
))))))))
1238 (and (eq (car-safe (nth 1 math-simplify-expr
)) '-
)
1239 (math-equal-int (nth 2 (nth 1 math-simplify-expr
)) 1)
1240 (eq (car-safe (nth 1 (nth 1 math-simplify-expr
))) '^
)
1241 (math-equal-int (nth 2 (nth 1 (nth 1 math-simplify-expr
))) 2)
1242 (and (eq (car-safe (nth 1 (nth 1 (nth 1 math-simplify-expr
))))
1244 (list 'calcFunc-sinh
1245 (nth 1 (nth 1 (nth 1 (nth 1 math-simplify-expr
)))))))
1246 (and (eq (car-safe (nth 1 math-simplify-expr
)) '+)
1247 (let ((a (nth 1 (nth 1 math-simplify-expr
)))
1248 (b (nth 2 (nth 1 math-simplify-expr
))))
1249 (and (or (and (math-equal-int a
1)
1250 (setq a b b
(nth 1 (nth 1 math-simplify-expr
))))
1251 (math-equal-int b
1))
1252 (eq (car-safe a
) '^
)
1253 (math-equal-int (nth 2 a
) 2)
1254 (or (and (eq (car-safe (nth 1 a
)) 'calcFunc-sinh
)
1255 (list 'calcFunc-cosh
(nth 1 (nth 1 a
))))
1256 (and (eq (car-safe (nth 1 a
)) 'calcFunc-csch
)
1257 (list 'calcFunc-coth
(nth 1 (nth 1 a
))))
1258 (and (eq (car-safe (nth 1 a
)) 'calcFunc-tan
)
1259 (list '/ 1 (list 'calcFunc-cos
1260 (nth 1 (nth 1 a
)))))
1261 (and (eq (car-safe (nth 1 a
)) 'calcFunc-cot
)
1262 (list '/ 1 (list 'calcFunc-sin
1263 (nth 1 (nth 1 a
)))))))))
1264 (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1266 (nth 1 (nth 1 math-simplify-expr
))
1267 (math-div (nth 2 (nth 1 math-simplify-expr
)) 2)))
1268 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sqrt
)
1269 (list '^
(nth 1 (nth 1 math-simplify-expr
)) (math-div 1 4)))
1270 (and (memq (car-safe (nth 1 math-simplify-expr
)) '(* /))
1271 (list (car (nth 1 math-simplify-expr
))
1272 (list 'calcFunc-sqrt
(nth 1 (nth 1 math-simplify-expr
)))
1273 (list 'calcFunc-sqrt
(nth 2 (nth 1 math-simplify-expr
)))))
1274 (and (memq (car-safe (nth 1 math-simplify-expr
)) '(+ -
))
1275 (not (math-any-floats (nth 1 math-simplify-expr
)))
1276 (let ((f (calcFunc-factors (calcFunc-expand
1277 (nth 1 math-simplify-expr
)))))
1278 (and (math-vectorp f
)
1279 (or (> (length f
) 2)
1280 (> (nth 2 (nth 1 f
)) 1))
1281 (let ((out 1) (rest 1) (sums 1) fac pow
)
1282 (while (setq f
(cdr f
))
1283 (setq fac
(nth 1 (car f
))
1284 pow
(nth 2 (car f
)))
1286 (setq out
(math-mul out
(math-pow
1290 (if (memq (car-safe fac
) '(+ -
))
1291 (setq sums
(math-mul-thru sums fac
))
1292 (setq rest
(math-mul rest fac
)))))
1293 (and (not (and (eq out
1) (memq rest
'(1 -
1))))
1296 (list 'calcFunc-sqrt
1297 (math-mul sums rest
))))))))))))
1299 ;;; Rather than factoring x into primes, just check for the first ten primes.
1300 (defun math-squared-factor (x)
1301 (if (Math-integerp x
)
1302 (let ((prsqr '(4 9 25 49 121 169 289 361 529 841))
1306 (if (eq (cdr (setq res
(math-idivmod x
(car prsqr
)))) 0)
1308 fac
(math-mul fac
(car prsqr
)))
1309 (setq prsqr
(cdr prsqr
))))
1312 (math-defsimplify calcFunc-exp
1313 (math-simplify-exp (nth 1 math-simplify-expr
)))
1315 (defun math-simplify-exp (x)
1316 (or (and (eq (car-safe x
) 'calcFunc-ln
)
1318 (and math-living-dangerously
1319 (or (and (eq (car-safe x
) 'calcFunc-arcsinh
)
1321 (list 'calcFunc-sqrt
1322 (math-add (math-sqr (nth 1 x
)) 1))))
1323 (and (eq (car-safe x
) 'calcFunc-arccosh
)
1325 (list 'calcFunc-sqrt
1326 (math-sub (math-sqr (nth 1 x
)) 1))))
1327 (and (eq (car-safe x
) 'calcFunc-arctanh
)
1328 (math-div (list 'calcFunc-sqrt
(math-add 1 (nth 1 x
)))
1329 (list 'calcFunc-sqrt
(math-sub 1 (nth 1 x
)))))
1330 (let ((m (math-should-expand-trig x
'exp
)))
1331 (and m
(integerp (car m
))
1332 (list '^
(list 'calcFunc-exp
(nth 1 m
)) (car m
))))))
1333 (and calc-symbolic-mode
1334 (math-known-imagp x
)
1335 (let* ((ip (calcFunc-im x
))
1336 (n (math-linear-in ip
'(var pi var-pi
)))
1339 (setq s
(math-known-sin (car n
) (nth 1 n
) 120 0))
1340 (setq c
(math-known-sin (car n
) (nth 1 n
) 120 300))
1341 (list '+ c
(list '* s
'(var i var-i
))))))))
1343 (math-defsimplify calcFunc-ln
1344 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-exp
)
1345 (or math-living-dangerously
1346 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
1347 (nth 1 (nth 1 math-simplify-expr
)))
1348 (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1349 (equal (nth 1 (nth 1 math-simplify-expr
)) '(var e var-e
))
1350 (or math-living-dangerously
1351 (math-known-realp (nth 2 (nth 1 math-simplify-expr
))))
1352 (nth 2 (nth 1 math-simplify-expr
)))
1353 (and calc-symbolic-mode
1354 (math-known-negp (nth 1 math-simplify-expr
))
1355 (math-add (list 'calcFunc-ln
(math-neg (nth 1 math-simplify-expr
)))
1356 '(* (var pi var-pi
) (var i var-i
))))
1357 (and calc-symbolic-mode
1358 (math-known-imagp (nth 1 math-simplify-expr
))
1359 (let* ((ip (calcFunc-im (nth 1 math-simplify-expr
)))
1360 (ips (math-possible-signs ip
)))
1361 (or (and (memq ips
'(4 6))
1362 (math-add (list 'calcFunc-ln ip
)
1363 '(/ (* (var pi var-pi
) (var i var-i
)) 2)))
1364 (and (memq ips
'(1 3))
1365 (math-sub (list 'calcFunc-ln
(math-neg ip
))
1366 '(/ (* (var pi var-pi
) (var i var-i
)) 2))))))))
1369 (math-simplify-pow))
1371 (defun math-simplify-pow ()
1372 (or (and math-living-dangerously
1373 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1375 (nth 1 (nth 1 math-simplify-expr
))
1376 (math-mul (nth 2 math-simplify-expr
)
1377 (nth 2 (nth 1 math-simplify-expr
)))))
1378 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sqrt
)
1380 (nth 1 (nth 1 math-simplify-expr
))
1381 (math-div (nth 2 math-simplify-expr
) 2)))
1382 (and (memq (car-safe (nth 1 math-simplify-expr
)) '(* /))
1383 (list (car (nth 1 math-simplify-expr
))
1384 (list '^
(nth 1 (nth 1 math-simplify-expr
))
1385 (nth 2 math-simplify-expr
))
1386 (list '^
(nth 2 (nth 1 math-simplify-expr
))
1387 (nth 2 math-simplify-expr
))))))
1388 (and (math-equal-int (nth 1 math-simplify-expr
) 10)
1389 (eq (car-safe (nth 2 math-simplify-expr
)) 'calcFunc-log10
)
1390 (nth 1 (nth 2 math-simplify-expr
)))
1391 (and (equal (nth 1 math-simplify-expr
) '(var e var-e
))
1392 (math-simplify-exp (nth 2 math-simplify-expr
)))
1393 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-exp
)
1394 (not math-integrating
)
1395 (list 'calcFunc-exp
(math-mul (nth 1 (nth 1 math-simplify-expr
))
1396 (nth 2 math-simplify-expr
))))
1397 (and (equal (nth 1 math-simplify-expr
) '(var i var-i
))
1399 (math-num-integerp (nth 2 math-simplify-expr
))
1400 (let ((x (math-mod (math-trunc (nth 2 math-simplify-expr
)) 4)))
1402 ((eq x
1) (nth 1 math-simplify-expr
))
1404 ((eq x
3) (math-neg (nth 1 math-simplify-expr
))))))
1405 (and math-integrating
1406 (integerp (nth 2 math-simplify-expr
))
1407 (>= (nth 2 math-simplify-expr
) 2)
1408 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cos
)
1409 (math-mul (math-pow (nth 1 math-simplify-expr
)
1410 (- (nth 2 math-simplify-expr
) 2))
1414 (nth 1 (nth 1 math-simplify-expr
)))))))
1415 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cosh
)
1416 (math-mul (math-pow (nth 1 math-simplify-expr
)
1417 (- (nth 2 math-simplify-expr
) 2))
1420 (list 'calcFunc-sinh
1421 (nth 1 (nth 1 math-simplify-expr
)))))))))
1422 (and (eq (car-safe (nth 2 math-simplify-expr
)) 'frac
)
1423 (Math-ratp (nth 1 math-simplify-expr
))
1424 (Math-posp (nth 1 math-simplify-expr
))
1425 (if (equal (nth 2 math-simplify-expr
) '(frac 1 2))
1426 (list 'calcFunc-sqrt
(nth 1 math-simplify-expr
))
1427 (let ((flr (math-floor (nth 2 math-simplify-expr
))))
1428 (and (not (Math-zerop flr
))
1429 (list '* (list '^
(nth 1 math-simplify-expr
) flr
)
1430 (list '^
(nth 1 math-simplify-expr
)
1431 (math-sub (nth 2 math-simplify-expr
) flr
)))))))
1432 (and (eq (math-quarter-integer (nth 2 math-simplify-expr
)) 2)
1433 (let ((temp (math-simplify-sqrt)))
1435 (list '^ temp
(math-mul (nth 2 math-simplify-expr
) 2)))))))
1437 (math-defsimplify calcFunc-log10
1438 (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1439 (math-equal-int (nth 1 (nth 1 math-simplify-expr
)) 10)
1440 (or math-living-dangerously
1441 (math-known-realp (nth 2 (nth 1 math-simplify-expr
))))
1442 (nth 2 (nth 1 math-simplify-expr
))))
1445 (math-defsimplify calcFunc-erf
1446 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1447 (math-neg (list 'calcFunc-erf
(math-neg (nth 1 math-simplify-expr
)))))
1448 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-conj
)
1449 (list 'calcFunc-conj
1450 (list 'calcFunc-erf
(nth 1 (nth 1 math-simplify-expr
)))))))
1452 (math-defsimplify calcFunc-erfc
1453 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1454 (math-sub 2 (list 'calcFunc-erfc
(math-neg (nth 1 math-simplify-expr
)))))
1455 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-conj
)
1456 (list 'calcFunc-conj
1457 (list 'calcFunc-erfc
(nth 1 (nth 1 math-simplify-expr
)))))))
1460 (defun math-linear-in (expr term
&optional always
)
1461 (if (math-expr-contains expr term
)
1462 (let* ((calc-prefer-frac t
)
1463 (p (math-is-polynomial expr term
1)))
1466 (and always
(list expr
0))))
1468 (defun math-multiple-of (expr term
)
1469 (let ((p (math-linear-in expr term
)))
1471 (math-zerop (car p
))
1474 ; not perfect, but it'll do
1475 (defun math-integer-plus (expr)
1476 (cond ((Math-integerp expr
)
1478 ((and (memq (car expr
) '(+ -
))
1479 (Math-integerp (nth 1 expr
)))
1480 (list (if (eq (car expr
) '+) (nth 2 expr
) (math-neg (nth 2 expr
)))
1482 ((and (memq (car expr
) '(+ -
))
1483 (Math-integerp (nth 2 expr
)))
1485 (if (eq (car expr
) '+) (nth 2 expr
) (math-neg (nth 2 expr
)))))
1488 (defun math-is-linear (expr &optional always
)
1491 (if (eq (car-safe expr
) '+)
1492 (if (Math-objectp (nth 1 expr
))
1493 (setq offset
(nth 1 expr
)
1495 (if (Math-objectp (nth 2 expr
))
1496 (setq offset
(nth 2 expr
)
1497 expr
(nth 1 expr
))))
1498 (if (eq (car-safe expr
) '-
)
1499 (if (Math-objectp (nth 1 expr
))
1500 (setq offset
(nth 1 expr
)
1501 expr
(math-neg (nth 2 expr
)))
1502 (if (Math-objectp (nth 2 expr
))
1503 (setq offset
(math-neg (nth 2 expr
))
1504 expr
(nth 1 expr
))))))
1505 (setq coef
(math-is-multiple expr always
))
1507 (list offset
(or (car coef
) 1) (or (nth 1 coef
) expr
))
1511 (defun math-is-multiple (expr &optional always
)
1512 (or (if (eq (car-safe expr
) '*)
1513 (if (Math-objectp (nth 1 expr
))
1514 (list (nth 1 expr
) (nth 2 expr
)))
1515 (if (eq (car-safe expr
) '/)
1516 (if (and (Math-objectp (nth 1 expr
))
1517 (not (math-equal-int (nth 1 expr
) 1)))
1518 (list (nth 1 expr
) (math-div 1 (nth 2 expr
)))
1519 (if (Math-objectp (nth 2 expr
))
1520 (list (math-div 1 (nth 2 expr
)) (nth 1 expr
))
1521 (let ((res (math-is-multiple (nth 1 expr
))))
1524 (math-div (nth 2 (nth 1 expr
)) (nth 2 expr
)))
1525 (setq res
(math-is-multiple (nth 2 expr
)))
1527 (list (math-div 1 (car res
))
1528 (math-div (nth 1 expr
)
1529 (nth 2 (nth 2 expr
)))))))))
1530 (if (eq (car-safe expr
) 'neg
)
1531 (list -
1 (nth 1 expr
)))))
1532 (if (Math-objvecp expr
)
1538 (defun calcFunc-lin (expr &optional var
)
1540 (let ((res (math-linear-in expr var t
)))
1541 (or res
(math-reject-arg expr
"Linear term expected"))
1542 (list 'vec
(car res
) (nth 1 res
) var
))
1543 (let ((res (math-is-linear expr t
)))
1544 (or res
(math-reject-arg expr
"Linear term expected"))
1547 (defun calcFunc-linnt (expr &optional var
)
1549 (let ((res (math-linear-in expr var
)))
1550 (or res
(math-reject-arg expr
"Linear term expected"))
1551 (list 'vec
(car res
) (nth 1 res
) var
))
1552 (let ((res (math-is-linear expr
)))
1553 (or res
(math-reject-arg expr
"Linear term expected"))
1556 (defun calcFunc-islin (expr &optional var
)
1557 (if (and (Math-objvecp expr
) (not var
))
1559 (calcFunc-lin expr var
)
1562 (defun calcFunc-islinnt (expr &optional var
)
1563 (if (Math-objvecp expr
)
1565 (calcFunc-linnt expr var
)
1571 ;;; Simple operations on expressions.
1573 ;;; Return number of occurrences of thing in expr, or nil if none.
1574 (defun math-expr-contains-count (expr thing
)
1575 (cond ((equal expr thing
) 1)
1576 ((Math-primp expr
) nil
)
1579 (while (setq expr
(cdr expr
))
1580 (setq num
(+ num
(or (math-expr-contains-count
1581 (car expr
) thing
) 0))))
1585 (defun math-expr-contains (expr thing
)
1586 (cond ((equal expr thing
) 1)
1587 ((Math-primp expr
) nil
)
1589 (while (and (setq expr
(cdr expr
))
1590 (not (math-expr-contains (car expr
) thing
))))
1593 ;;; Return non-nil if any variable of thing occurs in expr.
1594 (defun math-expr-depends (expr thing
)
1595 (if (Math-primp thing
)
1596 (and (eq (car-safe thing
) 'var
)
1597 (math-expr-contains expr thing
))
1598 (while (and (setq thing
(cdr thing
))
1599 (not (math-expr-depends expr
(car thing
)))))
1602 ;;; Substitute all occurrences of old for new in expr (non-destructive).
1604 ;; The variables math-expr-subst-old and math-expr-subst-new are local
1605 ;; for math-expr-subst, but used by math-expr-subst-rec.
1606 (defvar math-expr-subst-old
)
1607 (defvar math-expr-subst-new
)
1609 (defun math-expr-subst (expr math-expr-subst-old math-expr-subst-new
)
1610 (math-expr-subst-rec expr
))
1612 (defalias 'calcFunc-subst
'math-expr-subst
)
1614 (defun math-expr-subst-rec (expr)
1615 (cond ((equal expr math-expr-subst-old
) math-expr-subst-new
)
1616 ((Math-primp expr
) expr
)
1617 ((memq (car expr
) '(calcFunc-deriv
1619 (if (= (length expr
) 2)
1620 (if (equal (nth 1 expr
) math-expr-subst-old
)
1621 (append expr
(list math-expr-subst-new
))
1623 (list (car expr
) (nth 1 expr
)
1624 (math-expr-subst-rec (nth 2 expr
)))))
1627 (mapcar 'math-expr-subst-rec
(cdr expr
))))))
1629 ;;; Various measures of the size of an expression.
1630 (defun math-expr-weight (expr)
1631 (if (Math-primp expr
)
1634 (while (setq expr
(cdr expr
))
1635 (setq w
(+ w
(math-expr-weight (car expr
)))))
1638 (defun math-expr-height (expr)
1639 (if (Math-primp expr
)
1642 (while (setq expr
(cdr expr
))
1643 (setq h
(max h
(math-expr-height (car expr
)))))
1649 ;;; Polynomial operations (to support the integrator and solve-for).
1651 (defun calcFunc-collect (expr base
)
1652 (let ((p (math-is-polynomial expr base
50 t
)))
1654 (math-build-polynomial-expr (mapcar 'math-normalize p
) base
)
1657 ;;; If expr is of the form "a + bx + cx^2 + ...", return the list (a b c ...),
1658 ;;; else return nil if not in polynomial form. If "loose" (math-is-poly-loose),
1659 ;;; coefficients may contain x, e.g., sin(x) + cos(x) x^2 is a loose polynomial in x.
1661 ;; These variables are local to math-is-polynomial, but are used by
1662 ;; math-is-poly-rec.
1663 (defvar math-is-poly-degree
)
1664 (defvar math-is-poly-loose
)
1667 (defun math-is-polynomial (expr math-var
&optional math-is-poly-degree math-is-poly-loose
)
1668 (let* ((math-poly-base-variable (if math-is-poly-loose
1669 (if (eq math-is-poly-loose
'gen
) math-var
'(var XXX XXX
))
1670 math-poly-base-variable
))
1671 (poly (math-is-poly-rec expr math-poly-neg-powers
)))
1672 (and (or (null math-is-poly-degree
)
1673 (<= (length poly
) (1+ math-is-poly-degree
)))
1676 (defun math-is-poly-rec (expr negpow
)
1678 (or (cond ((or (equal expr math-var
)
1679 (eq (car-safe expr
) '^
))
1682 (or (equal expr math-var
)
1683 (setq pow
(nth 2 expr
)
1685 (or (eq math-poly-mult-powers
1)
1686 (setq pow
(let ((m (math-is-multiple pow
1)))
1687 (and (eq (car-safe (car m
)) 'cplx
)
1688 (Math-zerop (nth 1 (car m
)))
1689 (setq m
(list (nth 2 (car m
))
1692 (and (if math-poly-mult-powers
1693 (equal math-poly-mult-powers
1695 (setq math-poly-mult-powers
(nth 1 m
)))
1696 (or (equal expr math-var
)
1697 (eq math-poly-mult-powers
1))
1701 (setq pow
(math-to-simple-fraction pow
))
1702 (and (eq (car-safe pow
) 'frac
)
1703 math-poly-frac-powers
1704 (equal expr math-var
)
1705 (setq math-poly-frac-powers
1706 (calcFunc-lcm math-poly-frac-powers
1708 (or (memq math-poly-frac-powers
'(1 nil
))
1709 (setq pow
(math-mul pow math-poly-frac-powers
)))
1712 (equal expr math-var
))
1715 (let ((p1 (if (equal expr math-var
)
1717 (math-is-poly-rec expr nil
)))
1721 (or (null math-is-poly-degree
)
1722 (<= (* (1- (length p1
)) n
) math-is-poly-degree
))
1725 (setq accum
(math-poly-mul accum p1
)
1729 (math-is-poly-rec expr nil
)
1730 (setq math-poly-neg-powers
1731 (cons (math-pow expr
(- pow
))
1732 math-poly-neg-powers
))
1733 (list (list '^ expr pow
))))))))
1734 ((Math-objectp expr
)
1736 ((memq (car expr
) '(+ -
))
1737 (let ((p1 (math-is-poly-rec (nth 1 expr
) negpow
)))
1739 (let ((p2 (math-is-poly-rec (nth 2 expr
) negpow
)))
1741 (math-poly-mix p1
1 p2
1742 (if (eq (car expr
) '+) 1 -
1)))))))
1743 ((eq (car expr
) 'neg
)
1744 (mapcar 'math-neg
(math-is-poly-rec (nth 1 expr
) negpow
)))
1746 (let ((p1 (math-is-poly-rec (nth 1 expr
) negpow
)))
1748 (let ((p2 (math-is-poly-rec (nth 2 expr
) negpow
)))
1750 (or (null math-is-poly-degree
)
1751 (<= (- (+ (length p1
) (length p2
)) 2)
1752 math-is-poly-degree
))
1753 (math-poly-mul p1 p2
))))))
1755 (and (or (not (math-poly-depends (nth 2 expr
) math-var
))
1757 (math-is-poly-rec (nth 2 expr
) nil
)
1758 (setq math-poly-neg-powers
1759 (cons (nth 2 expr
) math-poly-neg-powers
))))
1760 (not (Math-zerop (nth 2 expr
)))
1761 (let ((p1 (math-is-poly-rec (nth 1 expr
) negpow
)))
1762 (mapcar (function (lambda (x) (math-div x
(nth 2 expr
))))
1764 ((and (eq (car expr
) 'calcFunc-exp
)
1765 (equal math-var
'(var e var-e
)))
1766 (math-is-poly-rec (list '^ math-var
(nth 1 expr
)) negpow
))
1767 ((and (eq (car expr
) 'calcFunc-sqrt
)
1768 math-poly-frac-powers
)
1769 (math-is-poly-rec (list '^
(nth 1 expr
) '(frac 1 2)) negpow
))
1771 (and (or (not (math-poly-depends expr math-var
))
1773 (not (eq (car expr
) 'vec
))
1776 ;;; Check if expr is a polynomial in var; if so, return its degree.
1777 (defun math-polynomial-p (expr var
)
1778 (cond ((equal expr var
) 1)
1779 ((Math-primp expr
) 0)
1780 ((memq (car expr
) '(+ -
))
1781 (let ((p1 (math-polynomial-p (nth 1 expr
) var
))
1783 (and p1
(setq p2
(math-polynomial-p (nth 2 expr
) var
))
1786 (let ((p1 (math-polynomial-p (nth 1 expr
) var
))
1788 (and p1
(setq p2
(math-polynomial-p (nth 2 expr
) var
))
1790 ((eq (car expr
) 'neg
)
1791 (math-polynomial-p (nth 1 expr
) var
))
1792 ((and (eq (car expr
) '/)
1793 (not (math-poly-depends (nth 2 expr
) var
)))
1794 (math-polynomial-p (nth 1 expr
) var
))
1795 ((and (eq (car expr
) '^
)
1796 (natnump (nth 2 expr
)))
1797 (let ((p1 (math-polynomial-p (nth 1 expr
) var
)))
1798 (and p1
(* p1
(nth 2 expr
)))))
1799 ((math-poly-depends expr var
) nil
)
1802 (defun math-poly-depends (expr var
)
1803 (if math-poly-base-variable
1804 (math-expr-contains expr math-poly-base-variable
)
1805 (math-expr-depends expr var
)))
1807 ;;; Find the variable (or sub-expression) which is the base of polynomial expr.
1808 ;; The variables math-poly-base-const-ok and math-poly-base-pred are
1809 ;; local to math-polynomial-base, but are used by math-polynomial-base-rec.
1810 (defvar math-poly-base-const-ok
)
1811 (defvar math-poly-base-pred
)
1813 ;; The variable math-poly-base-top-expr is local to math-polynomial-base,
1814 ;; but is used by math-polynomial-p1 in calc-poly.el, which is called
1815 ;; by math-polynomial-base.
1817 (defun math-polynomial-base (math-poly-base-top-expr &optional math-poly-base-pred
)
1818 (or math-poly-base-pred
1819 (setq math-poly-base-pred
(function (lambda (base) (math-polynomial-p
1820 math-poly-base-top-expr base
)))))
1821 (or (let ((math-poly-base-const-ok nil
))
1822 (math-polynomial-base-rec math-poly-base-top-expr
))
1823 (let ((math-poly-base-const-ok t
))
1824 (math-polynomial-base-rec math-poly-base-top-expr
))))
1826 (defun math-polynomial-base-rec (mpb-expr)
1827 (and (not (Math-objvecp mpb-expr
))
1828 (or (and (memq (car mpb-expr
) '(+ -
*))
1829 (or (math-polynomial-base-rec (nth 1 mpb-expr
))
1830 (math-polynomial-base-rec (nth 2 mpb-expr
))))
1831 (and (memq (car mpb-expr
) '(/ neg
))
1832 (math-polynomial-base-rec (nth 1 mpb-expr
)))
1833 (and (eq (car mpb-expr
) '^
)
1834 (math-polynomial-base-rec (nth 1 mpb-expr
)))
1835 (and (eq (car mpb-expr
) 'calcFunc-exp
)
1836 (math-polynomial-base-rec '(var e var-e
)))
1837 (and (or math-poly-base-const-ok
(math-expr-contains-vars mpb-expr
))
1838 (funcall math-poly-base-pred mpb-expr
)
1841 ;;; Return non-nil if expr refers to any variables.
1842 (defun math-expr-contains-vars (expr)
1843 (or (eq (car-safe expr
) 'var
)
1844 (and (not (Math-primp expr
))
1846 (while (and (setq expr
(cdr expr
))
1847 (not (math-expr-contains-vars (car expr
)))))
1850 ;;; Simplify a polynomial in list form by stripping off high-end zeros.
1851 ;;; This always leaves the constant part, i.e., nil->nil and non-nil->non-nil.
1852 (defun math-poly-simplify (p)
1854 (if (Math-zerop (nth (1- (length p
)) p
))
1855 (let ((pp (copy-sequence p
)))
1856 (while (and (cdr pp
)
1857 (Math-zerop (nth (1- (length pp
)) pp
)))
1858 (setcdr (nthcdr (- (length pp
) 2) pp
) nil
))
1862 ;;; Compute ac*a + bc*b for polynomials in list form a, b and
1863 ;;; coefficients ac, bc. Result may be unsimplified.
1864 (defun math-poly-mix (a ac b bc
)
1866 (cons (math-add (math-mul (or (car a
) 0) ac
)
1867 (math-mul (or (car b
) 0) bc
))
1868 (math-poly-mix (cdr a
) ac
(cdr b
) bc
))))
1870 (defun math-poly-zerop (a)
1872 (and (null (cdr a
)) (Math-zerop (car a
)))))
1874 ;;; Multiply two polynomials in list form.
1875 (defun math-poly-mul (a b
)
1877 (math-poly-mix b
(car a
)
1878 (math-poly-mul (cdr a
) (cons 0 b
)) 1)))
1880 ;;; Build an expression from a polynomial list.
1881 (defun math-build-polynomial-expr (p var
)
1883 (if (Math-numberp var
)
1884 (math-with-extra-prec 1
1885 (let* ((rp (reverse p
))
1887 (while (setq rp
(cdr rp
))
1888 (setq accum
(math-add (car rp
) (math-mul accum var
))))
1890 (let* ((rp (reverse p
))
1891 (n (1- (length rp
)))
1892 (accum (math-mul (car rp
) (math-pow var n
)))
1894 (while (setq rp
(cdr rp
))
1896 (or (math-zerop (car rp
))
1897 (setq accum
(list (if (math-looks-negp (car rp
)) '-
'+)
1899 (math-mul (if (math-looks-negp (car rp
))
1902 (math-pow var n
))))))
1907 (defun math-to-simple-fraction (f)
1908 (or (and (eq (car-safe f
) 'float
)
1909 (or (and (>= (nth 2 f
) 0)
1910 (math-scale-int (nth 1 f
) (nth 2 f
)))
1911 (and (integerp (nth 1 f
))
1914 (math-make-frac (nth 1 f
)
1915 (math-scale-int 1 (- (nth 2 f
)))))))
1920 ;;; calc-alg.el ends here
