1 ;;; calc-alg.el --- algebraic functions for Calc
3 ;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.
5 ;; Author: David Gillespie <daveg@synaptics.com>
6 ;; Maintainer: Jay Belanger <belanger@truman.edu>
8 ;; This file is part of GNU Emacs.
10 ;; GNU Emacs is distributed in the hope that it will be useful,
11 ;; but WITHOUT ANY WARRANTY. No author or distributor
12 ;; accepts responsibility to anyone for the consequences of using it
13 ;; or for whether it serves any particular purpose or works at all,
14 ;; unless he says so in writing. Refer to the GNU Emacs General Public
15 ;; License for full details.
17 ;; Everyone is granted permission to copy, modify and redistribute
18 ;; GNU Emacs, but only under the conditions described in the
19 ;; GNU Emacs General Public License. A copy of this license is
20 ;; supposed to have been given to you along with GNU Emacs so you
21 ;; can know your rights and responsibilities. It should be in a
22 ;; file named COPYING. Among other things, the copyright notice
23 ;; and this notice must be preserved on all copies.
29 ;; This file is autoloaded from calc-ext.el.
36 (defun calc-alg-evaluate (arg)
39 (calc-with-default-simplification
40 (let ((math-simplify-only nil
))
41 (calc-modify-simplify-mode arg
)
42 (calc-enter-result 1 "dsmp" (calc-top 1))))))
44 (defun calc-modify-simplify-mode (arg)
45 (if (= (math-abs arg
) 2)
46 (setq calc-simplify-mode
'alg
)
47 (if (>= (math-abs arg
) 3)
48 (setq calc-simplify-mode
'ext
)))
50 (setq calc-simplify-mode
(list calc-simplify-mode
))))
52 (defun calc-simplify ()
55 (calc-with-default-simplification
56 (calc-enter-result 1 "simp" (math-simplify (calc-top-n 1))))))
58 (defun calc-simplify-extended ()
61 (calc-with-default-simplification
62 (calc-enter-result 1 "esmp" (math-simplify-extended (calc-top-n 1))))))
64 (defun calc-expand-formula (arg)
67 (calc-with-default-simplification
68 (let ((math-simplify-only nil
))
69 (calc-modify-simplify-mode arg
)
70 (calc-enter-result 1 "expf"
72 (let ((math-expand-formulas t
))
74 (let ((top (calc-top-n 1)))
75 (or (math-expand-formula top
)
78 (defun calc-factor (arg)
81 (calc-unary-op "fctr" (if (calc-is-hyperbolic)
82 'calcFunc-factors
'calcFunc-factor
)
85 (defun calc-expand (n)
88 (calc-enter-result 1 "expa"
89 (append (list 'calcFunc-expand
91 (and n
(list (prefix-numeric-value n
)))))))
93 (defun calc-collect (&optional var
)
94 (interactive "sCollect terms involving: ")
96 (if (or (equal var
"") (equal var
"$") (null var
))
97 (calc-enter-result 2 "clct" (cons 'calcFunc-collect
99 (let ((var (math-read-expr var
)))
100 (if (eq (car-safe var
) 'error
)
101 (error "Bad format in expression: %s" (nth 1 var
)))
102 (calc-enter-result 1 "clct" (list 'calcFunc-collect
106 (defun calc-apart (arg)
109 (calc-unary-op "aprt" 'calcFunc-apart arg
)))
111 (defun calc-normalize-rat (arg)
114 (calc-unary-op "nrat" 'calcFunc-nrat arg
)))
116 (defun calc-poly-gcd (arg)
119 (calc-binary-op "pgcd" 'calcFunc-pgcd arg
)))
122 (defun calc-poly-div (arg)
125 (let ((calc-poly-div-remainder nil
))
126 (calc-binary-op "pdiv" 'calcFunc-pdiv arg
)
127 (if (and calc-poly-div-remainder
(null arg
))
129 (calc-clear-command-flag 'clear-message
)
130 (calc-record calc-poly-div-remainder
"prem")
131 (if (not (Math-zerop calc-poly-div-remainder
))
132 (message "(Remainder was %s)"
133 (math-format-flat-expr calc-poly-div-remainder
0))
134 (message "(No remainder)")))))))
136 (defun calc-poly-rem (arg)
139 (calc-binary-op "prem" 'calcFunc-prem arg
)))
141 (defun calc-poly-div-rem (arg)
144 (if (calc-is-hyperbolic)
145 (calc-binary-op "pdvr" 'calcFunc-pdivide arg
)
146 (calc-binary-op "pdvr" 'calcFunc-pdivrem arg
))))
148 (defun calc-substitute (&optional oldname newname
)
149 (interactive "sSubstitute old: ")
151 (let (old new
(num 1) expr
)
152 (if (or (equal oldname
"") (equal oldname
"$") (null oldname
))
153 (setq new
(calc-top-n 1)
158 (progn (calc-unread-command ?\C-a
)
159 (setq newname
(read-string (concat "Substitute old: "
163 (if (or (equal newname
"") (equal newname
"$") (null newname
))
164 (setq new
(calc-top-n 1)
167 (setq new
(if (stringp newname
) (math-read-expr newname
) newname
))
168 (if (eq (car-safe new
) 'error
)
169 (error "Bad format in expression: %s" (nth 1 new
)))
170 (setq expr
(calc-top-n 1)))
171 (setq old
(if (stringp oldname
) (math-read-expr oldname
) oldname
))
172 (if (eq (car-safe old
) 'error
)
173 (error "Bad format in expression: %s" (nth 1 old
)))
174 (or (math-expr-contains expr old
)
175 (error "No occurrences found")))
176 (calc-enter-result num
"sbst" (math-expr-subst expr old new
)))))
179 (defun calc-has-rules (name)
180 (setq name
(calc-var-value name
))
182 (memq (car name
) '(vec calcFunc-assign calcFunc-condition
))
185 ;; math-eval-rules-cache and math-eval-rules-cache-other are
186 ;; declared in calc.el, but are used here by math-recompile-eval-rules.
187 (defvar math-eval-rules-cache
)
188 (defvar math-eval-rules-cache-other
)
190 (defun math-recompile-eval-rules ()
191 (setq math-eval-rules-cache
(and (calc-has-rules 'var-EvalRules
)
192 (math-compile-rewrites
193 '(var EvalRules var-EvalRules
)))
194 math-eval-rules-cache-other
(assq nil math-eval-rules-cache
)
195 math-eval-rules-cache-tag
(calc-var-value 'var-EvalRules
)))
198 ;;; Try to expand a formula according to its definition.
199 (defun math-expand-formula (expr)
202 (or (get (car expr
) 'calc-user-defn
)
203 (get (car expr
) 'math-expandable
))
204 (let ((res (let ((math-expand-formulas t
))
205 (apply (car expr
) (cdr expr
)))))
206 (and (not (eq (car-safe res
) (car expr
)))
212 ;;; True if A comes before B in a canonical ordering of expressions. [P X X]
213 (defun math-beforep (a b
) ; [Public]
214 (cond ((and (Math-realp a
) (Math-realp b
))
215 (let ((comp (math-compare a b
)))
219 (> (length (memq (car-safe a
)
220 '(bigneg nil bigpos frac float
)))
221 (length (memq (car-safe b
)
222 '(bigneg nil bigpos frac float
))))))))
223 ((equal b
'(neg (var inf var-inf
))) nil
)
224 ((equal a
'(neg (var inf var-inf
))) t
)
225 ((equal a
'(var inf var-inf
)) nil
)
226 ((equal b
'(var inf var-inf
)) t
)
228 (if (and (eq (car-safe b
) 'intv
) (math-intv-constp b
))
229 (if (or (math-beforep a
(nth 2 b
)) (Math-equal a
(nth 2 b
)))
234 (if (and (eq (car-safe a
) 'intv
) (math-intv-constp a
))
235 (if (math-beforep (nth 2 a
) b
)
239 ((and (eq (car a
) 'intv
) (eq (car b
) 'intv
)
240 (math-intv-constp a
) (math-intv-constp b
))
241 (let ((comp (math-compare (nth 2 a
) (nth 2 b
))))
242 (cond ((eq comp -
1) t
)
244 ((and (memq (nth 1 a
) '(2 3)) (memq (nth 1 b
) '(0 1))) t
)
245 ((and (memq (nth 1 a
) '(0 1)) (memq (nth 1 b
) '(2 3))) nil
)
246 ((eq (setq comp
(math-compare (nth 3 a
) (nth 3 b
))) -
1) t
)
248 ((and (memq (nth 1 a
) '(0 2)) (memq (nth 1 b
) '(1 3))) t
)
250 ((not (eq (not (Math-objectp a
)) (not (Math-objectp b
))))
253 (if (eq (car b
) 'var
)
254 (string-lessp (symbol-name (nth 1 a
)) (symbol-name (nth 1 b
)))
255 (not (Math-numberp b
))))
256 ((eq (car b
) 'var
) (Math-numberp a
))
257 ((eq (car a
) (car b
))
258 (while (and (setq a
(cdr a
) b
(cdr b
)) a
259 (equal (car a
) (car b
))))
262 (math-beforep (car a
) (car b
)))))
263 (t (string-lessp (symbol-name (car a
)) (symbol-name (car b
))))))
266 (defsubst math-simplify-extended
(a)
267 (let ((math-living-dangerously t
))
270 (defalias 'calcFunc-esimplify
'math-simplify-extended
)
272 ;; math-top-only is local to math-simplify, but is used by
273 ;; math-simplify-step, which is called by math-simplify.
274 (defvar math-top-only
)
276 (defun math-simplify (top-expr)
277 (let ((math-simplifying t
)
278 (math-top-only (consp calc-simplify-mode
))
279 (simp-rules (append (and (calc-has-rules 'var-AlgSimpRules
)
280 '((var AlgSimpRules var-AlgSimpRules
)))
281 (and math-living-dangerously
282 (calc-has-rules 'var-ExtSimpRules
)
283 '((var ExtSimpRules var-ExtSimpRules
)))
284 (and math-simplifying-units
285 (calc-has-rules 'var-UnitSimpRules
)
286 '((var UnitSimpRules var-UnitSimpRules
)))
287 (and math-integrating
288 (calc-has-rules 'var-IntegSimpRules
)
289 '((var IntegSimpRules var-IntegSimpRules
)))))
292 (let ((r simp-rules
))
293 (setq res
(math-simplify-step (math-normalize top-expr
))
294 calc-simplify-mode
'(nil)
295 top-expr
(math-normalize res
))
297 (setq top-expr
(math-rewrite top-expr
(car r
)
298 '(neg (var inf var-inf
)))
300 (calc-with-default-simplification
301 (while (let ((r simp-rules
))
302 (setq res
(math-normalize top-expr
))
304 (setq res
(math-rewrite res
(car r
))
306 (not (equal top-expr
(setq res
(math-simplify-step res
)))))
307 (setq top-expr res
)))))
310 (defalias 'calcFunc-simplify
'math-simplify
)
312 ;;; The following has a "bug" in that if any recursive simplifications
313 ;;; occur only the first handler will be tried; this doesn't really
314 ;;; matter, since math-simplify-step is iterated to a fixed point anyway.
315 (defun math-simplify-step (a)
318 (let ((aa (if (or math-top-only
319 (memq (car a
) '(calcFunc-quote calcFunc-condition
322 (cons (car a
) (mapcar 'math-simplify-step
(cdr a
))))))
323 (and (symbolp (car aa
))
324 (let ((handler (get (car aa
) 'math-simplify
)))
327 (equal (setq aa
(or (funcall (car handler
) aa
)
330 (setq handler
(cdr handler
))))))
334 (defmacro math-defsimplify
(funcs &rest code
)
338 (list 'put
(list 'quote func
) ''math-simplify
340 (list 'get
(list 'quote func
) ''math-simplify
)
343 (append '(lambda (math-simplify-expr))
345 (if (symbolp funcs
) (list funcs
) funcs
))))
346 (put 'math-defsimplify
'lisp-indent-hook
1)
348 ;; The function created by math-defsimplify uses the variable
349 ;; math-simplify-expr, and so is used by functions in math-defsimplify
350 (defvar math-simplify-expr
)
352 (math-defsimplify (+ -
)
353 (math-simplify-plus))
355 (defun math-simplify-plus ()
356 (cond ((and (memq (car-safe (nth 1 math-simplify-expr
)) '(+ -
))
357 (Math-numberp (nth 2 (nth 1 math-simplify-expr
)))
358 (not (Math-numberp (nth 2 math-simplify-expr
))))
359 (let ((x (nth 2 math-simplify-expr
))
360 (op (car math-simplify-expr
)))
361 (setcar (cdr (cdr math-simplify-expr
)) (nth 2 (nth 1 math-simplify-expr
)))
362 (setcar math-simplify-expr
(car (nth 1 math-simplify-expr
)))
363 (setcar (cdr (cdr (nth 1 math-simplify-expr
))) x
)
364 (setcar (nth 1 math-simplify-expr
) op
)))
365 ((and (eq (car math-simplify-expr
) '+)
366 (Math-numberp (nth 1 math-simplify-expr
))
367 (not (Math-numberp (nth 2 math-simplify-expr
))))
368 (let ((x (nth 2 math-simplify-expr
)))
369 (setcar (cdr (cdr math-simplify-expr
)) (nth 1 math-simplify-expr
))
370 (setcar (cdr math-simplify-expr
) x
))))
371 (let ((aa math-simplify-expr
)
373 (while (memq (car-safe (setq aaa
(nth 1 aa
))) '(+ -
))
374 (if (setq temp
(math-combine-sum (nth 2 aaa
) (nth 2 math-simplify-expr
)
376 (eq (car math-simplify-expr
) '-
) t
))
378 (setcar (cdr (cdr math-simplify-expr
)) temp
)
379 (setcar math-simplify-expr
'+)
380 (setcar (cdr (cdr aaa
)) 0)))
381 (setq aa
(nth 1 aa
)))
382 (if (setq temp
(math-combine-sum aaa
(nth 2 math-simplify-expr
)
383 nil
(eq (car math-simplify-expr
) '-
) t
))
385 (setcar (cdr (cdr math-simplify-expr
)) temp
)
386 (setcar math-simplify-expr
'+)
387 (setcar (cdr aa
) 0)))
391 (math-simplify-times))
393 (defun math-simplify-times ()
394 (if (eq (car-safe (nth 2 math-simplify-expr
)) '*)
395 (and (math-beforep (nth 1 (nth 2 math-simplify-expr
)) (nth 1 math-simplify-expr
))
396 (or (math-known-scalarp (nth 1 math-simplify-expr
) t
)
397 (math-known-scalarp (nth 1 (nth 2 math-simplify-expr
)) t
))
398 (let ((x (nth 1 math-simplify-expr
)))
399 (setcar (cdr math-simplify-expr
) (nth 1 (nth 2 math-simplify-expr
)))
400 (setcar (cdr (nth 2 math-simplify-expr
)) x
)))
401 (and (math-beforep (nth 2 math-simplify-expr
) (nth 1 math-simplify-expr
))
402 (or (math-known-scalarp (nth 1 math-simplify-expr
) t
)
403 (math-known-scalarp (nth 2 math-simplify-expr
) t
))
404 (let ((x (nth 2 math-simplify-expr
)))
405 (setcar (cdr (cdr math-simplify-expr
)) (nth 1 math-simplify-expr
))
406 (setcar (cdr math-simplify-expr
) x
))))
407 (let ((aa math-simplify-expr
)
409 (safe t
) (scalar (math-known-scalarp (nth 1 math-simplify-expr
))))
410 (if (and (Math-ratp (nth 1 math-simplify-expr
))
411 (setq temp
(math-common-constant-factor (nth 2 math-simplify-expr
))))
413 (setcar (cdr (cdr math-simplify-expr
))
414 (math-cancel-common-factor (nth 2 math-simplify-expr
) temp
))
415 (setcar (cdr math-simplify-expr
) (math-mul (nth 1 math-simplify-expr
) temp
))))
416 (while (and (eq (car-safe (setq aaa
(nth 2 aa
))) '*)
418 (if (setq temp
(math-combine-prod (nth 1 math-simplify-expr
)
419 (nth 1 aaa
) nil nil t
))
421 (setcar (cdr math-simplify-expr
) temp
)
422 (setcar (cdr aaa
) 1)))
423 (setq safe
(or scalar
(math-known-scalarp (nth 1 aaa
) t
))
425 (if (and (setq temp
(math-combine-prod aaa
(nth 1 math-simplify-expr
) nil nil t
))
428 (setcar (cdr math-simplify-expr
) temp
)
429 (setcar (cdr (cdr aa
)) 1)))
430 (if (and (eq (car-safe (nth 1 math-simplify-expr
)) 'frac
)
431 (memq (nth 1 (nth 1 math-simplify-expr
)) '(1 -
1)))
432 (math-div (math-mul (nth 2 math-simplify-expr
)
433 (nth 1 (nth 1 math-simplify-expr
)))
434 (nth 2 (nth 1 math-simplify-expr
)))
435 math-simplify-expr
)))
438 (math-simplify-divide))
440 (defun math-simplify-divide ()
441 (let ((np (cdr math-simplify-expr
))
443 (nn (and (or (eq (car math-simplify-expr
) '/)
444 (not (Math-realp (nth 2 math-simplify-expr
))))
445 (math-common-constant-factor (nth 2 math-simplify-expr
))))
449 (setq n
(and (or (eq (car math-simplify-expr
) '/)
450 (not (Math-realp (nth 1 math-simplify-expr
))))
451 (math-common-constant-factor (nth 1 math-simplify-expr
))))
452 (if (and (eq (car-safe nn
) 'frac
) (eq (nth 1 nn
) 1) (not n
))
454 (setcar (cdr math-simplify-expr
)
455 (math-mul (nth 2 nn
) (nth 1 math-simplify-expr
)))
456 (setcar (cdr (cdr math-simplify-expr
))
457 (math-cancel-common-factor (nth 2 math-simplify-expr
) nn
))
458 (if (and (math-negp nn
)
459 (setq op
(assq (car math-simplify-expr
) calc-tweak-eqn-table
)))
460 (setcar math-simplify-expr
(nth 1 op
))))
461 (if (and n
(not (eq (setq n
(math-frac-gcd n nn
)) 1)))
463 (setcar (cdr math-simplify-expr
)
464 (math-cancel-common-factor (nth 1 math-simplify-expr
) n
))
465 (setcar (cdr (cdr math-simplify-expr
))
466 (math-cancel-common-factor (nth 2 math-simplify-expr
) n
))
467 (if (and (math-negp n
)
468 (setq op
(assq (car math-simplify-expr
)
469 calc-tweak-eqn-table
)))
470 (setcar math-simplify-expr
(nth 1 op
))))))))
471 (if (and (eq (car-safe (car np
)) '/)
472 (math-known-scalarp (nth 2 math-simplify-expr
) t
))
474 (setq np
(cdr (nth 1 math-simplify-expr
)))
475 (while (eq (car-safe (setq n
(car np
))) '*)
476 (and (math-known-scalarp (nth 2 n
) t
)
477 (math-simplify-divisor (cdr n
) (cdr (cdr math-simplify-expr
)) nil t
))
478 (setq np
(cdr (cdr n
))))
479 (math-simplify-divisor np
(cdr (cdr math-simplify-expr
)) nil t
)
481 np
(cdr (cdr (nth 1 math-simplify-expr
))))))
482 (while (eq (car-safe (setq n
(car np
))) '*)
483 (and (math-known-scalarp (nth 2 n
) t
)
484 (math-simplify-divisor (cdr n
) (cdr (cdr math-simplify-expr
)) nover t
))
485 (setq np
(cdr (cdr n
))))
486 (math-simplify-divisor np
(cdr (cdr math-simplify-expr
)) nover t
)
489 ;; The variables math-simplify-divisor-nover and math-simplify-divisor-dover
490 ;; are local variables for math-simplify-divisor, but are used by
491 ;; math-simplify-one-divisor.
492 (defvar math-simplify-divisor-nover
)
493 (defvar math-simplify-divisor-dover
)
495 (defun math-simplify-divisor (np dp math-simplify-divisor-nover
496 math-simplify-divisor-dover
)
497 (cond ((eq (car-safe (car dp
)) '/)
498 (math-simplify-divisor np
(cdr (car dp
))
499 math-simplify-divisor-nover
500 math-simplify-divisor-dover
)
501 (and (math-known-scalarp (nth 1 (car dp
)) t
)
502 (math-simplify-divisor np
(cdr (cdr (car dp
)))
503 math-simplify-divisor-nover
504 (not math-simplify-divisor-dover
))))
505 ((or (or (eq (car math-simplify-expr
) '/)
506 (let ((signs (math-possible-signs (car np
))))
507 (or (memq signs
'(1 4))
508 (and (memq (car math-simplify-expr
) '(calcFunc-eq calcFunc-neq
))
510 math-living-dangerously
)))
511 (math-numberp (car np
)))
514 (scalar (math-known-scalarp (car np
))))
515 (while (and (eq (car-safe (setq d
(car dp
))) '*)
517 (math-simplify-one-divisor np
(cdr d
))
518 (setq safe
(or scalar
(math-known-scalarp (nth 1 d
) t
))
521 (math-simplify-one-divisor np dp
))))))
523 (defun math-simplify-one-divisor (np dp
)
524 (let ((temp (math-combine-prod (car np
) (car dp
) math-simplify-divisor-nover
525 math-simplify-divisor-dover t
))
529 (and (not (memq (car math-simplify-expr
) '(/ calcFunc-eq calcFunc-neq
)))
530 (math-known-negp (car dp
))
531 (setq op
(assq (car math-simplify-expr
) calc-tweak-eqn-table
))
532 (setcar math-simplify-expr
(nth 1 op
)))
533 (setcar np
(if math-simplify-divisor-nover
(math-div 1 temp
) temp
))
535 (and math-simplify-divisor-dover
(not math-simplify-divisor-nover
)
536 (eq (car math-simplify-expr
) '/)
537 (eq (car-safe (car dp
)) 'calcFunc-sqrt
)
538 (Math-integerp (nth 1 (car dp
)))
540 (setcar np
(math-mul (car np
)
541 (list 'calcFunc-sqrt
(nth 1 (car dp
)))))
542 (setcar dp
(nth 1 (car dp
))))))))
544 (defun math-common-constant-factor (expr)
545 (if (Math-realp expr
)
547 (and (not (memq expr
'(0 1 -
1)))
549 (if (math-ratp (setq expr
(math-to-simple-fraction expr
)))
550 (math-common-constant-factor expr
)))
551 (if (memq (car expr
) '(+ - cplx sdev
))
552 (let ((f1 (math-common-constant-factor (nth 1 expr
)))
553 (f2 (math-common-constant-factor (nth 2 expr
))))
555 (not (eq (setq f1
(math-frac-gcd f1 f2
)) 1))
557 (if (memq (car expr
) '(* polar
))
558 (math-common-constant-factor (nth 1 expr
))
559 (if (eq (car expr
) '/)
560 (or (math-common-constant-factor (nth 1 expr
))
561 (and (Math-integerp (nth 2 expr
))
562 (list 'frac
1 (math-abs (nth 2 expr
))))))))))
564 (defun math-cancel-common-factor (expr val
)
565 (if (memq (car-safe expr
) '(+ - cplx sdev
))
567 (setcar (cdr expr
) (math-cancel-common-factor (nth 1 expr
) val
))
568 (setcar (cdr (cdr expr
)) (math-cancel-common-factor (nth 2 expr
) val
))
570 (if (eq (car-safe expr
) '*)
571 (math-mul (math-cancel-common-factor (nth 1 expr
) val
) (nth 2 expr
))
572 (math-div expr val
))))
574 (defun math-frac-gcd (a b
)
579 (if (and (Math-integerp a
)
582 (and (Math-integerp a
) (setq a
(list 'frac a
1)))
583 (and (Math-integerp b
) (setq b
(list 'frac b
1)))
584 (math-make-frac (math-gcd (nth 1 a
) (nth 1 b
))
585 (math-gcd (nth 2 a
) (nth 2 b
)))))))
590 (defun math-simplify-mod ()
591 (and (Math-realp (nth 2 math-simplify-expr
))
592 (Math-posp (nth 2 math-simplify-expr
))
593 (let ((lin (math-is-linear (nth 1 math-simplify-expr
)))
596 (or (math-negp (car lin
))
597 (not (Math-lessp (car lin
) (nth 2 math-simplify-expr
))))
600 (math-mul (nth 1 lin
) (nth 2 lin
))
601 (math-mod (car lin
) (nth 2 math-simplify-expr
)))
602 (nth 2 math-simplify-expr
)))
604 (not (math-equal-int (nth 1 lin
) 1))
605 (math-num-integerp (nth 1 lin
))
606 (math-num-integerp (nth 2 math-simplify-expr
))
607 (setq t1
(calcFunc-gcd (nth 1 lin
) (nth 2 math-simplify-expr
)))
608 (not (math-equal-int t1
1))
613 (math-mul (math-div (nth 1 lin
) t1
)
615 (let ((calc-prefer-frac t
))
616 (math-div (car lin
) t1
)))
617 (math-div (nth 2 math-simplify-expr
) t1
))))
618 (and (math-equal-int (nth 2 math-simplify-expr
) 1)
619 (math-known-integerp (if lin
620 (math-mul (nth 1 lin
) (nth 2 lin
))
621 (nth 1 math-simplify-expr
)))
622 (if lin
(math-mod (car lin
) 1) 0))))))
624 (math-defsimplify (calcFunc-eq calcFunc-neq calcFunc-lt
625 calcFunc-gt calcFunc-leq calcFunc-geq
)
626 (if (= (length math-simplify-expr
) 3)
627 (math-simplify-ineq)))
629 (defun math-simplify-ineq ()
630 (let ((np (cdr math-simplify-expr
))
632 (while (memq (car-safe (setq n
(car np
))) '(+ -
))
633 (math-simplify-add-term (cdr (cdr n
)) (cdr (cdr math-simplify-expr
))
636 (math-simplify-add-term np
(cdr (cdr math-simplify-expr
)) nil
637 (eq np
(cdr math-simplify-expr
)))
638 (math-simplify-divide)
639 (let ((signs (math-possible-signs (cons '-
(cdr math-simplify-expr
)))))
640 (or (cond ((eq (car math-simplify-expr
) 'calcFunc-eq
)
641 (or (and (eq signs
2) 1)
642 (and (memq signs
'(1 4 5)) 0)))
643 ((eq (car math-simplify-expr
) 'calcFunc-neq
)
644 (or (and (eq signs
2) 0)
645 (and (memq signs
'(1 4 5)) 1)))
646 ((eq (car math-simplify-expr
) 'calcFunc-lt
)
647 (or (and (eq signs
1) 1)
648 (and (memq signs
'(2 4 6)) 0)))
649 ((eq (car math-simplify-expr
) 'calcFunc-gt
)
650 (or (and (eq signs
4) 1)
651 (and (memq signs
'(1 2 3)) 0)))
652 ((eq (car math-simplify-expr
) 'calcFunc-leq
)
653 (or (and (eq signs
4) 0)
654 (and (memq signs
'(1 2 3)) 1)))
655 ((eq (car math-simplify-expr
) 'calcFunc-geq
)
656 (or (and (eq signs
1) 0)
657 (and (memq signs
'(2 4 6)) 1))))
658 math-simplify-expr
))))
660 (defun math-simplify-add-term (np dp minus lplain
)
661 (or (math-vectorp (car np
))
664 (while (memq (car-safe (setq n
(car np
) d
(car dp
))) '(+ -
))
666 (if (setq temp
(math-combine-sum n
(nth 2 d
)
667 minus
(eq (car d
) '+) t
))
668 (if (or lplain
(eq (math-looks-negp temp
) minus
))
670 (setcar np
(setq n
(if minus
(math-neg temp
) temp
)))
671 (setcar (cdr (cdr d
)) 0))
674 (setcar (cdr (cdr d
)) (setq n
(if (eq (car d
) '+)
678 (if (setq temp
(math-combine-sum n d minus t t
))
681 (eq (math-looks-negp temp
) minus
)))
683 (setcar np
(setq n
(if minus
(math-neg temp
) temp
)))
687 (setcar dp
(setq n
(math-neg temp
)))))))))
689 (math-defsimplify calcFunc-sin
690 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
691 (nth 1 (nth 1 math-simplify-expr
)))
692 (and (math-looks-negp (nth 1 math-simplify-expr
))
693 (math-neg (list 'calcFunc-sin
(math-neg (nth 1 math-simplify-expr
)))))
694 (and (eq calc-angle-mode
'rad
)
695 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
697 (math-known-sin (car n
) (nth 1 n
) 120 0))))
698 (and (eq calc-angle-mode
'deg
)
699 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
701 (math-known-sin (car n
) (nth 1 n
) '(frac 2 3) 0))))
702 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
703 (list 'calcFunc-sqrt
(math-sub 1 (math-sqr
704 (nth 1 (nth 1 math-simplify-expr
))))))
705 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
706 (math-div (nth 1 (nth 1 math-simplify-expr
))
708 (math-add 1 (math-sqr
709 (nth 1 (nth 1 math-simplify-expr
)))))))
710 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
))))
711 (and m
(integerp (car m
))
712 (let ((n (car m
)) (a (nth 1 m
)))
714 (list '* (list 'calcFunc-sin
(list '* (1- n
) a
))
715 (list 'calcFunc-cos a
))
716 (list '* (list 'calcFunc-cos
(list '* (1- n
) a
))
717 (list 'calcFunc-sin a
))))))))
719 (math-defsimplify calcFunc-cos
720 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
721 (nth 1 (nth 1 math-simplify-expr
)))
722 (and (math-looks-negp (nth 1 math-simplify-expr
))
723 (list 'calcFunc-cos
(math-neg (nth 1 math-simplify-expr
))))
724 (and (eq calc-angle-mode
'rad
)
725 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
727 (math-known-sin (car n
) (nth 1 n
) 120 300))))
728 (and (eq calc-angle-mode
'deg
)
729 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
731 (math-known-sin (car n
) (nth 1 n
) '(frac 2 3) 300))))
732 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
734 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
))))))
735 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
739 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
740 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
))))
741 (and m
(integerp (car m
))
742 (let ((n (car m
)) (a (nth 1 m
)))
744 (list '* (list 'calcFunc-cos
(list '* (1- n
) a
))
745 (list 'calcFunc-cos a
))
746 (list '* (list 'calcFunc-sin
(list '* (1- n
) a
))
747 (list 'calcFunc-sin a
))))))))
749 (defun math-should-expand-trig (x &optional hyperbolic
)
750 (let ((m (math-is-multiple x
)))
751 (and math-living-dangerously
752 m
(or (and (integerp (car m
)) (> (car m
) 1))
753 (equal (car m
) '(frac 1 2)))
755 (memq (car-safe (nth 1 m
))
757 '(calcFunc-arcsinh calcFunc-arccosh calcFunc-arctanh
)
758 '(calcFunc-arcsin calcFunc-arccos calcFunc-arctan
)))
759 (and (eq (car-safe (nth 1 m
)) 'calcFunc-ln
)
760 (eq hyperbolic
'exp
)))
763 (defun math-known-sin (plus n mul off
)
764 (setq n
(math-mul n mul
))
765 (and (math-num-integerp n
)
766 (setq n
(math-mod (math-add (math-trunc n
) off
) 240))
768 (and (setq n
(math-known-sin plus
(- n
120) 1 0))
772 (if (math-zerop plus
)
773 (and (or calc-symbolic-mode
777 (10 .
(/ (calcFunc-sqrt
778 (- 2 (calcFunc-sqrt 3))) 2))
779 (12 .
(/ (- (calcFunc-sqrt 5) 1) 4))
780 (15 .
(/ (calcFunc-sqrt
781 (- 2 (calcFunc-sqrt 2))) 2))
783 (24 .
(* (^
(/ 1 2) (/ 3 2))
785 (- 5 (calcFunc-sqrt 5)))))
786 (30 .
(/ (calcFunc-sqrt 2) 2))
787 (36 .
(/ (+ (calcFunc-sqrt 5) 1) 4))
788 (40 .
(/ (calcFunc-sqrt 3) 2))
789 (45 .
(/ (calcFunc-sqrt
790 (+ 2 (calcFunc-sqrt 2))) 2))
791 (48 .
(* (^
(/ 1 2) (/ 3 2))
793 (+ 5 (calcFunc-sqrt 5)))))
794 (50 .
(/ (calcFunc-sqrt
795 (+ 2 (calcFunc-sqrt 3))) 2))
797 (cond ((eq n
0) (math-normalize (list 'calcFunc-sin plus
)))
798 ((eq n
60) (math-normalize (list 'calcFunc-cos plus
)))
801 (math-defsimplify calcFunc-tan
802 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctan
)
803 (nth 1 (nth 1 math-simplify-expr
)))
804 (and (math-looks-negp (nth 1 math-simplify-expr
))
805 (math-neg (list 'calcFunc-tan
(math-neg (nth 1 math-simplify-expr
)))))
806 (and (eq calc-angle-mode
'rad
)
807 (let ((n (math-linear-in (nth 1 math-simplify-expr
) '(var pi var-pi
))))
809 (math-known-tan (car n
) (nth 1 n
) 120))))
810 (and (eq calc-angle-mode
'deg
)
811 (let ((n (math-integer-plus (nth 1 math-simplify-expr
))))
813 (math-known-tan (car n
) (nth 1 n
) '(frac 2 3)))))
814 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsin
)
815 (math-div (nth 1 (nth 1 math-simplify-expr
))
817 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
818 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccos
)
819 (math-div (list 'calcFunc-sqrt
820 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))
821 (nth 1 (nth 1 math-simplify-expr
))))
822 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
))))
824 (if (equal (car m
) '(frac 1 2))
825 (math-div (math-sub 1 (list 'calcFunc-cos
(nth 1 m
)))
826 (list 'calcFunc-sin
(nth 1 m
)))
827 (math-div (list 'calcFunc-sin
(nth 1 math-simplify-expr
))
828 (list 'calcFunc-cos
(nth 1 math-simplify-expr
))))))))
830 (defun math-known-tan (plus n mul
)
831 (setq n
(math-mul n mul
))
832 (and (math-num-integerp n
)
833 (setq n
(math-mod (math-trunc n
) 120))
835 (and (setq n
(math-known-tan plus
(- 120 n
) 1))
837 (if (math-zerop plus
)
838 (and (or calc-symbolic-mode
840 (cdr (assq n
'( (0 .
0)
841 (10 .
(- 2 (calcFunc-sqrt 3)))
843 (- 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
844 (15 .
(- (calcFunc-sqrt 2) 1))
845 (20 .
(/ (calcFunc-sqrt 3) 3))
847 (- 5 (* 2 (calcFunc-sqrt 5)))))
850 (+ 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
851 (40 .
(calcFunc-sqrt 3))
852 (45 .
(+ (calcFunc-sqrt 2) 1))
854 (+ 5 (* 2 (calcFunc-sqrt 5)))))
855 (50 .
(+ 2 (calcFunc-sqrt 3)))
856 (60 .
(var uinf var-uinf
))))))
857 (cond ((eq n
0) (math-normalize (list 'calcFunc-tan plus
)))
858 ((eq n
60) (math-normalize (list '/ -
1
859 (list 'calcFunc-tan plus
))))
862 (math-defsimplify calcFunc-sinh
863 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
864 (nth 1 (nth 1 math-simplify-expr
)))
865 (and (math-looks-negp (nth 1 math-simplify-expr
))
866 (math-neg (list 'calcFunc-sinh
(math-neg (nth 1 math-simplify-expr
)))))
867 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
868 math-living-dangerously
870 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1)))
871 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
872 math-living-dangerously
873 (math-div (nth 1 (nth 1 math-simplify-expr
))
875 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
876 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
) t
)))
877 (and m
(integerp (car m
))
878 (let ((n (car m
)) (a (nth 1 m
)))
881 (list '* (list 'calcFunc-sinh
(list '* (1- n
) a
))
882 (list 'calcFunc-cosh a
))
883 (list '* (list 'calcFunc-cosh
(list '* (1- n
) a
))
884 (list 'calcFunc-sinh a
)))))))))
886 (math-defsimplify calcFunc-cosh
887 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
888 (nth 1 (nth 1 math-simplify-expr
)))
889 (and (math-looks-negp (nth 1 math-simplify-expr
))
890 (list 'calcFunc-cosh
(math-neg (nth 1 math-simplify-expr
))))
891 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
892 math-living-dangerously
894 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1)))
895 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
896 math-living-dangerously
899 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr
)))))))
900 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
) t
)))
901 (and m
(integerp (car m
))
902 (let ((n (car m
)) (a (nth 1 m
)))
905 (list '* (list 'calcFunc-cosh
(list '* (1- n
) a
))
906 (list 'calcFunc-cosh a
))
907 (list '* (list 'calcFunc-sinh
(list '* (1- n
) a
))
908 (list 'calcFunc-sinh a
)))))))))
910 (math-defsimplify calcFunc-tanh
911 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arctanh
)
912 (nth 1 (nth 1 math-simplify-expr
)))
913 (and (math-looks-negp (nth 1 math-simplify-expr
))
914 (math-neg (list 'calcFunc-tanh
(math-neg (nth 1 math-simplify-expr
)))))
915 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arcsinh
)
916 math-living-dangerously
917 (math-div (nth 1 (nth 1 math-simplify-expr
))
919 (math-add (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))))
920 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-arccosh
)
921 math-living-dangerously
922 (math-div (list 'calcFunc-sqrt
923 (math-sub (math-sqr (nth 1 (nth 1 math-simplify-expr
))) 1))
924 (nth 1 (nth 1 math-simplify-expr
))))
925 (let ((m (math-should-expand-trig (nth 1 math-simplify-expr
) t
)))
927 (if (equal (car m
) '(frac 1 2))
928 (math-div (math-sub (list 'calcFunc-cosh
(nth 1 m
)) 1)
929 (list 'calcFunc-sinh
(nth 1 m
)))
930 (math-div (list 'calcFunc-sinh
(nth 1 math-simplify-expr
))
931 (list 'calcFunc-cosh
(nth 1 math-simplify-expr
))))))))
933 (math-defsimplify calcFunc-arcsin
934 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
935 (math-neg (list 'calcFunc-arcsin
(math-neg (nth 1 math-simplify-expr
)))))
936 (and (eq (nth 1 math-simplify-expr
) 1)
937 (math-quarter-circle t
))
938 (and (equal (nth 1 math-simplify-expr
) '(frac 1 2))
939 (math-div (math-half-circle t
) 6))
940 (and math-living-dangerously
941 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sin
)
942 (nth 1 (nth 1 math-simplify-expr
)))
943 (and math-living-dangerously
944 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cos
)
945 (math-sub (math-quarter-circle t
)
946 (nth 1 (nth 1 math-simplify-expr
))))))
948 (math-defsimplify calcFunc-arccos
949 (or (and (eq (nth 1 math-simplify-expr
) 0)
950 (math-quarter-circle t
))
951 (and (eq (nth 1 math-simplify-expr
) -
1)
952 (math-half-circle t
))
953 (and (equal (nth 1 math-simplify-expr
) '(frac 1 2))
954 (math-div (math-half-circle t
) 3))
955 (and (equal (nth 1 math-simplify-expr
) '(frac -
1 2))
956 (math-div (math-mul (math-half-circle t
) 2) 3))
957 (and math-living-dangerously
958 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cos
)
959 (nth 1 (nth 1 math-simplify-expr
)))
960 (and math-living-dangerously
961 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sin
)
962 (math-sub (math-quarter-circle t
)
963 (nth 1 (nth 1 math-simplify-expr
))))))
965 (math-defsimplify calcFunc-arctan
966 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
967 (math-neg (list 'calcFunc-arctan
(math-neg (nth 1 math-simplify-expr
)))))
968 (and (eq (nth 1 math-simplify-expr
) 1)
969 (math-div (math-half-circle t
) 4))
970 (and math-living-dangerously
971 (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-tan
)
972 (nth 1 (nth 1 math-simplify-expr
)))))
974 (math-defsimplify calcFunc-arcsinh
975 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
976 (math-neg (list 'calcFunc-arcsinh
(math-neg (nth 1 math-simplify-expr
)))))
977 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sinh
)
978 (or math-living-dangerously
979 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
980 (nth 1 (nth 1 math-simplify-expr
)))))
982 (math-defsimplify calcFunc-arccosh
983 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cosh
)
984 (or math-living-dangerously
985 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
986 (nth 1 (nth 1 math-simplify-expr
))))
988 (math-defsimplify calcFunc-arctanh
989 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
990 (math-neg (list 'calcFunc-arctanh
(math-neg (nth 1 math-simplify-expr
)))))
991 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-tanh
)
992 (or math-living-dangerously
993 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
994 (nth 1 (nth 1 math-simplify-expr
)))))
996 (math-defsimplify calcFunc-sqrt
997 (math-simplify-sqrt))
999 (defun math-simplify-sqrt ()
1000 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'frac
)
1001 (math-div (list 'calcFunc-sqrt
1002 (math-mul (nth 1 (nth 1 math-simplify-expr
))
1003 (nth 2 (nth 1 math-simplify-expr
))))
1004 (nth 2 (nth 1 math-simplify-expr
))))
1005 (let ((fac (if (math-objectp (nth 1 math-simplify-expr
))
1006 (math-squared-factor (nth 1 math-simplify-expr
))
1007 (math-common-constant-factor (nth 1 math-simplify-expr
)))))
1008 (and fac
(not (eq fac
1))
1009 (math-mul (math-normalize (list 'calcFunc-sqrt fac
))
1011 (list 'calcFunc-sqrt
1012 (math-cancel-common-factor
1013 (nth 1 math-simplify-expr
) fac
))))))
1014 (and math-living-dangerously
1015 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) '-
)
1016 (math-equal-int (nth 1 (nth 1 math-simplify-expr
)) 1)
1017 (eq (car-safe (nth 2 (nth 1 math-simplify-expr
))) '^
)
1018 (math-equal-int (nth 2 (nth 2 (nth 1 math-simplify-expr
))) 2)
1019 (or (and (eq (car-safe (nth 1 (nth 2 (nth 1 math-simplify-expr
))))
1022 (nth 1 (nth 1 (nth 2 (nth 1 math-simplify-expr
))))))
1023 (and (eq (car-safe (nth 1 (nth 2 (nth 1 math-simplify-expr
))))
1026 (nth 1 (nth 1 (nth 2
1027 (nth 1 math-simplify-expr
))))))))
1028 (and (eq (car-safe (nth 1 math-simplify-expr
)) '-
)
1029 (math-equal-int (nth 2 (nth 1 math-simplify-expr
)) 1)
1030 (eq (car-safe (nth 1 (nth 1 math-simplify-expr
))) '^
)
1031 (math-equal-int (nth 2 (nth 1 (nth 1 math-simplify-expr
))) 2)
1032 (and (eq (car-safe (nth 1 (nth 1 (nth 1 math-simplify-expr
))))
1034 (list 'calcFunc-sinh
1035 (nth 1 (nth 1 (nth 1 (nth 1 math-simplify-expr
)))))))
1036 (and (eq (car-safe (nth 1 math-simplify-expr
)) '+)
1037 (let ((a (nth 1 (nth 1 math-simplify-expr
)))
1038 (b (nth 2 (nth 1 math-simplify-expr
))))
1039 (and (or (and (math-equal-int a
1)
1040 (setq a b b
(nth 1 (nth 1 math-simplify-expr
))))
1041 (math-equal-int b
1))
1042 (eq (car-safe a
) '^
)
1043 (math-equal-int (nth 2 a
) 2)
1044 (or (and (eq (car-safe (nth 1 a
)) 'calcFunc-sinh
)
1045 (list 'calcFunc-cosh
(nth 1 (nth 1 a
))))
1046 (and (eq (car-safe (nth 1 a
)) 'calcFunc-tan
)
1047 (list '/ 1 (list 'calcFunc-cos
1048 (nth 1 (nth 1 a
)))))))))
1049 (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1051 (nth 1 (nth 1 math-simplify-expr
))
1052 (math-div (nth 2 (nth 1 math-simplify-expr
)) 2)))
1053 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sqrt
)
1054 (list '^
(nth 1 (nth 1 math-simplify-expr
)) (math-div 1 4)))
1055 (and (memq (car-safe (nth 1 math-simplify-expr
)) '(* /))
1056 (list (car (nth 1 math-simplify-expr
))
1057 (list 'calcFunc-sqrt
(nth 1 (nth 1 math-simplify-expr
)))
1058 (list 'calcFunc-sqrt
(nth 2 (nth 1 math-simplify-expr
)))))
1059 (and (memq (car-safe (nth 1 math-simplify-expr
)) '(+ -
))
1060 (not (math-any-floats (nth 1 math-simplify-expr
)))
1061 (let ((f (calcFunc-factors (calcFunc-expand
1062 (nth 1 math-simplify-expr
)))))
1063 (and (math-vectorp f
)
1064 (or (> (length f
) 2)
1065 (> (nth 2 (nth 1 f
)) 1))
1066 (let ((out 1) (rest 1) (sums 1) fac pow
)
1067 (while (setq f
(cdr f
))
1068 (setq fac
(nth 1 (car f
))
1069 pow
(nth 2 (car f
)))
1071 (setq out
(math-mul out
(math-pow
1075 (if (memq (car-safe fac
) '(+ -
))
1076 (setq sums
(math-mul-thru sums fac
))
1077 (setq rest
(math-mul rest fac
)))))
1078 (and (not (and (eq out
1) (memq rest
'(1 -
1))))
1081 (list 'calcFunc-sqrt
1082 (math-mul sums rest
))))))))))))
1084 ;;; Rather than factoring x into primes, just check for the first ten primes.
1085 (defun math-squared-factor (x)
1086 (if (Math-integerp x
)
1087 (let ((prsqr '(4 9 25 49 121 169 289 361 529 841))
1091 (if (eq (cdr (setq res
(math-idivmod x
(car prsqr
)))) 0)
1093 fac
(math-mul fac
(car prsqr
)))
1094 (setq prsqr
(cdr prsqr
))))
1097 (math-defsimplify calcFunc-exp
1098 (math-simplify-exp (nth 1 math-simplify-expr
)))
1100 (defun math-simplify-exp (x)
1101 (or (and (eq (car-safe x
) 'calcFunc-ln
)
1103 (and math-living-dangerously
1104 (or (and (eq (car-safe x
) 'calcFunc-arcsinh
)
1106 (list 'calcFunc-sqrt
1107 (math-add (math-sqr (nth 1 x
)) 1))))
1108 (and (eq (car-safe x
) 'calcFunc-arccosh
)
1110 (list 'calcFunc-sqrt
1111 (math-sub (math-sqr (nth 1 x
)) 1))))
1112 (and (eq (car-safe x
) 'calcFunc-arctanh
)
1113 (math-div (list 'calcFunc-sqrt
(math-add 1 (nth 1 x
)))
1114 (list 'calcFunc-sqrt
(math-sub 1 (nth 1 x
)))))
1115 (let ((m (math-should-expand-trig x
'exp
)))
1116 (and m
(integerp (car m
))
1117 (list '^
(list 'calcFunc-exp
(nth 1 m
)) (car m
))))))
1118 (and calc-symbolic-mode
1119 (math-known-imagp x
)
1120 (let* ((ip (calcFunc-im x
))
1121 (n (math-linear-in ip
'(var pi var-pi
)))
1124 (setq s
(math-known-sin (car n
) (nth 1 n
) 120 0))
1125 (setq c
(math-known-sin (car n
) (nth 1 n
) 120 300))
1126 (list '+ c
(list '* s
'(var i var-i
))))))))
1128 (math-defsimplify calcFunc-ln
1129 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-exp
)
1130 (or math-living-dangerously
1131 (math-known-realp (nth 1 (nth 1 math-simplify-expr
))))
1132 (nth 1 (nth 1 math-simplify-expr
)))
1133 (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1134 (equal (nth 1 (nth 1 math-simplify-expr
)) '(var e var-e
))
1135 (or math-living-dangerously
1136 (math-known-realp (nth 2 (nth 1 math-simplify-expr
))))
1137 (nth 2 (nth 1 math-simplify-expr
)))
1138 (and calc-symbolic-mode
1139 (math-known-negp (nth 1 math-simplify-expr
))
1140 (math-add (list 'calcFunc-ln
(math-neg (nth 1 math-simplify-expr
)))
1141 '(* (var pi var-pi
) (var i var-i
))))
1142 (and calc-symbolic-mode
1143 (math-known-imagp (nth 1 math-simplify-expr
))
1144 (let* ((ip (calcFunc-im (nth 1 math-simplify-expr
)))
1145 (ips (math-possible-signs ip
)))
1146 (or (and (memq ips
'(4 6))
1147 (math-add (list 'calcFunc-ln ip
)
1148 '(/ (* (var pi var-pi
) (var i var-i
)) 2)))
1149 (and (memq ips
'(1 3))
1150 (math-sub (list 'calcFunc-ln
(math-neg ip
))
1151 '(/ (* (var pi var-pi
) (var i var-i
)) 2))))))))
1154 (math-simplify-pow))
1156 (defun math-simplify-pow ()
1157 (or (and math-living-dangerously
1158 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1160 (nth 1 (nth 1 math-simplify-expr
))
1161 (math-mul (nth 2 math-simplify-expr
)
1162 (nth 2 (nth 1 math-simplify-expr
)))))
1163 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-sqrt
)
1165 (nth 1 (nth 1 math-simplify-expr
))
1166 (math-div (nth 2 math-simplify-expr
) 2)))
1167 (and (memq (car-safe (nth 1 math-simplify-expr
)) '(* /))
1168 (list (car (nth 1 math-simplify-expr
))
1169 (list '^
(nth 1 (nth 1 math-simplify-expr
))
1170 (nth 2 math-simplify-expr
))
1171 (list '^
(nth 2 (nth 1 math-simplify-expr
))
1172 (nth 2 math-simplify-expr
))))))
1173 (and (math-equal-int (nth 1 math-simplify-expr
) 10)
1174 (eq (car-safe (nth 2 math-simplify-expr
)) 'calcFunc-log10
)
1175 (nth 1 (nth 2 math-simplify-expr
)))
1176 (and (equal (nth 1 math-simplify-expr
) '(var e var-e
))
1177 (math-simplify-exp (nth 2 math-simplify-expr
)))
1178 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-exp
)
1179 (not math-integrating
)
1180 (list 'calcFunc-exp
(math-mul (nth 1 (nth 1 math-simplify-expr
))
1181 (nth 2 math-simplify-expr
))))
1182 (and (equal (nth 1 math-simplify-expr
) '(var i var-i
))
1184 (math-num-integerp (nth 2 math-simplify-expr
))
1185 (let ((x (math-mod (math-trunc (nth 2 math-simplify-expr
)) 4)))
1187 ((eq x
1) (nth 1 math-simplify-expr
))
1189 ((eq x
3) (math-neg (nth 1 math-simplify-expr
))))))
1190 (and math-integrating
1191 (integerp (nth 2 math-simplify-expr
))
1192 (>= (nth 2 math-simplify-expr
) 2)
1193 (or (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cos
)
1194 (math-mul (math-pow (nth 1 math-simplify-expr
)
1195 (- (nth 2 math-simplify-expr
) 2))
1199 (nth 1 (nth 1 math-simplify-expr
)))))))
1200 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-cosh
)
1201 (math-mul (math-pow (nth 1 math-simplify-expr
)
1202 (- (nth 2 math-simplify-expr
) 2))
1205 (list 'calcFunc-sinh
1206 (nth 1 (nth 1 math-simplify-expr
)))))))))
1207 (and (eq (car-safe (nth 2 math-simplify-expr
)) 'frac
)
1208 (Math-ratp (nth 1 math-simplify-expr
))
1209 (Math-posp (nth 1 math-simplify-expr
))
1210 (if (equal (nth 2 math-simplify-expr
) '(frac 1 2))
1211 (list 'calcFunc-sqrt
(nth 1 math-simplify-expr
))
1212 (let ((flr (math-floor (nth 2 math-simplify-expr
))))
1213 (and (not (Math-zerop flr
))
1214 (list '* (list '^
(nth 1 math-simplify-expr
) flr
)
1215 (list '^
(nth 1 math-simplify-expr
)
1216 (math-sub (nth 2 math-simplify-expr
) flr
)))))))
1217 (and (eq (math-quarter-integer (nth 2 math-simplify-expr
)) 2)
1218 (let ((temp (math-simplify-sqrt)))
1220 (list '^ temp
(math-mul (nth 2 math-simplify-expr
) 2)))))))
1222 (math-defsimplify calcFunc-log10
1223 (and (eq (car-safe (nth 1 math-simplify-expr
)) '^
)
1224 (math-equal-int (nth 1 (nth 1 math-simplify-expr
)) 10)
1225 (or math-living-dangerously
1226 (math-known-realp (nth 2 (nth 1 math-simplify-expr
))))
1227 (nth 2 (nth 1 math-simplify-expr
))))
1230 (math-defsimplify calcFunc-erf
1231 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1232 (math-neg (list 'calcFunc-erf
(math-neg (nth 1 math-simplify-expr
)))))
1233 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-conj
)
1234 (list 'calcFunc-conj
1235 (list 'calcFunc-erf
(nth 1 (nth 1 math-simplify-expr
)))))))
1237 (math-defsimplify calcFunc-erfc
1238 (or (and (math-looks-negp (nth 1 math-simplify-expr
))
1239 (math-sub 2 (list 'calcFunc-erfc
(math-neg (nth 1 math-simplify-expr
)))))
1240 (and (eq (car-safe (nth 1 math-simplify-expr
)) 'calcFunc-conj
)
1241 (list 'calcFunc-conj
1242 (list 'calcFunc-erfc
(nth 1 (nth 1 math-simplify-expr
)))))))
1245 (defun math-linear-in (expr term
&optional always
)
1246 (if (math-expr-contains expr term
)
1247 (let* ((calc-prefer-frac t
)
1248 (p (math-is-polynomial expr term
1)))
1251 (and always
(list expr
0))))
1253 (defun math-multiple-of (expr term
)
1254 (let ((p (math-linear-in expr term
)))
1256 (math-zerop (car p
))
1259 ; not perfect, but it'll do
1260 (defun math-integer-plus (expr)
1261 (cond ((Math-integerp expr
)
1263 ((and (memq (car expr
) '(+ -
))
1264 (Math-integerp (nth 1 expr
)))
1265 (list (if (eq (car expr
) '+) (nth 2 expr
) (math-neg (nth 2 expr
)))
1267 ((and (memq (car expr
) '(+ -
))
1268 (Math-integerp (nth 2 expr
)))
1270 (if (eq (car expr
) '+) (nth 2 expr
) (math-neg (nth 2 expr
)))))
1273 (defun math-is-linear (expr &optional always
)
1276 (if (eq (car-safe expr
) '+)
1277 (if (Math-objectp (nth 1 expr
))
1278 (setq offset
(nth 1 expr
)
1280 (if (Math-objectp (nth 2 expr
))
1281 (setq offset
(nth 2 expr
)
1282 expr
(nth 1 expr
))))
1283 (if (eq (car-safe expr
) '-
)
1284 (if (Math-objectp (nth 1 expr
))
1285 (setq offset
(nth 1 expr
)
1286 expr
(math-neg (nth 2 expr
)))
1287 (if (Math-objectp (nth 2 expr
))
1288 (setq offset
(math-neg (nth 2 expr
))
1289 expr
(nth 1 expr
))))))
1290 (setq coef
(math-is-multiple expr always
))
1292 (list offset
(or (car coef
) 1) (or (nth 1 coef
) expr
))
1296 (defun math-is-multiple (expr &optional always
)
1297 (or (if (eq (car-safe expr
) '*)
1298 (if (Math-objectp (nth 1 expr
))
1299 (list (nth 1 expr
) (nth 2 expr
)))
1300 (if (eq (car-safe expr
) '/)
1301 (if (and (Math-objectp (nth 1 expr
))
1302 (not (math-equal-int (nth 1 expr
) 1)))
1303 (list (nth 1 expr
) (math-div 1 (nth 2 expr
)))
1304 (if (Math-objectp (nth 2 expr
))
1305 (list (math-div 1 (nth 2 expr
)) (nth 1 expr
))
1306 (let ((res (math-is-multiple (nth 1 expr
))))
1309 (math-div (nth 2 (nth 1 expr
)) (nth 2 expr
)))
1310 (setq res
(math-is-multiple (nth 2 expr
)))
1312 (list (math-div 1 (car res
))
1313 (math-div (nth 1 expr
)
1314 (nth 2 (nth 2 expr
)))))))))
1315 (if (eq (car-safe expr
) 'neg
)
1316 (list -
1 (nth 1 expr
)))))
1317 (if (Math-objvecp expr
)
1323 (defun calcFunc-lin (expr &optional var
)
1325 (let ((res (math-linear-in expr var t
)))
1326 (or res
(math-reject-arg expr
"Linear term expected"))
1327 (list 'vec
(car res
) (nth 1 res
) var
))
1328 (let ((res (math-is-linear expr t
)))
1329 (or res
(math-reject-arg expr
"Linear term expected"))
1332 (defun calcFunc-linnt (expr &optional var
)
1334 (let ((res (math-linear-in expr var
)))
1335 (or res
(math-reject-arg expr
"Linear term expected"))
1336 (list 'vec
(car res
) (nth 1 res
) var
))
1337 (let ((res (math-is-linear expr
)))
1338 (or res
(math-reject-arg expr
"Linear term expected"))
1341 (defun calcFunc-islin (expr &optional var
)
1342 (if (and (Math-objvecp expr
) (not var
))
1344 (calcFunc-lin expr var
)
1347 (defun calcFunc-islinnt (expr &optional var
)
1348 (if (Math-objvecp expr
)
1350 (calcFunc-linnt expr var
)
1356 ;;; Simple operations on expressions.
1358 ;;; Return number of occurrences of thing in expr, or nil if none.
1359 (defun math-expr-contains-count (expr thing
)
1360 (cond ((equal expr thing
) 1)
1361 ((Math-primp expr
) nil
)
1364 (while (setq expr
(cdr expr
))
1365 (setq num
(+ num
(or (math-expr-contains-count
1366 (car expr
) thing
) 0))))
1370 (defun math-expr-contains (expr thing
)
1371 (cond ((equal expr thing
) 1)
1372 ((Math-primp expr
) nil
)
1374 (while (and (setq expr
(cdr expr
))
1375 (not (math-expr-contains (car expr
) thing
))))
1378 ;;; Return non-nil if any variable of thing occurs in expr.
1379 (defun math-expr-depends (expr thing
)
1380 (if (Math-primp thing
)
1381 (and (eq (car-safe thing
) 'var
)
1382 (math-expr-contains expr thing
))
1383 (while (and (setq thing
(cdr thing
))
1384 (not (math-expr-depends expr
(car thing
)))))
1387 ;;; Substitute all occurrences of old for new in expr (non-destructive).
1389 ;; The variables math-expr-subst-old and math-expr-subst-new are local
1390 ;; for math-expr-subst, but used by math-expr-subst-rec.
1391 (defvar math-expr-subst-old
)
1392 (defvar math-expr-subst-new
)
1394 (defun math-expr-subst (expr math-expr-subst-old math-expr-subst-new
)
1395 (math-expr-subst-rec expr
))
1397 (defalias 'calcFunc-subst
'math-expr-subst
)
1399 (defun math-expr-subst-rec (expr)
1400 (cond ((equal expr math-expr-subst-old
) math-expr-subst-new
)
1401 ((Math-primp expr
) expr
)
1402 ((memq (car expr
) '(calcFunc-deriv
1404 (if (= (length expr
) 2)
1405 (if (equal (nth 1 expr
) math-expr-subst-old
)
1406 (append expr
(list math-expr-subst-new
))
1408 (list (car expr
) (nth 1 expr
)
1409 (math-expr-subst-rec (nth 2 expr
)))))
1412 (mapcar 'math-expr-subst-rec
(cdr expr
))))))
1414 ;;; Various measures of the size of an expression.
1415 (defun math-expr-weight (expr)
1416 (if (Math-primp expr
)
1419 (while (setq expr
(cdr expr
))
1420 (setq w
(+ w
(math-expr-weight (car expr
)))))
1423 (defun math-expr-height (expr)
1424 (if (Math-primp expr
)
1427 (while (setq expr
(cdr expr
))
1428 (setq h
(max h
(math-expr-height (car expr
)))))
1434 ;;; Polynomial operations (to support the integrator and solve-for).
1436 (defun calcFunc-collect (expr base
)
1437 (let ((p (math-is-polynomial expr base
50 t
)))
1439 (math-normalize ; fix selection bug
1440 (math-build-polynomial-expr p base
))
1443 ;;; If expr is of the form "a + bx + cx^2 + ...", return the list (a b c ...),
1444 ;;; else return nil if not in polynomial form. If "loose" (math-is-poly-loose),
1445 ;;; coefficients may contain x, e.g., sin(x) + cos(x) x^2 is a loose polynomial in x.
1447 ;; The variables math-is-poly-degree and math-is-poly-loose are local to
1448 ;; math-is-polynomial, but are used by math-is-poly-rec
1449 (defvar math-is-poly-degree
)
1450 (defvar math-is-poly-loose
)
1452 (defun math-is-polynomial (expr var
&optional math-is-poly-degree math-is-poly-loose
)
1453 (let* ((math-poly-base-variable (if math-is-poly-loose
1454 (if (eq math-is-poly-loose
'gen
) var
'(var XXX XXX
))
1455 math-poly-base-variable
))
1456 (poly (math-is-poly-rec expr math-poly-neg-powers
)))
1457 (and (or (null math-is-poly-degree
)
1458 (<= (length poly
) (1+ math-is-poly-degree
)))
1461 (defun math-is-poly-rec (expr negpow
)
1463 (or (cond ((or (equal expr var
)
1464 (eq (car-safe expr
) '^
))
1467 (or (equal expr var
)
1468 (setq pow
(nth 2 expr
)
1470 (or (eq math-poly-mult-powers
1)
1471 (setq pow
(let ((m (math-is-multiple pow
1)))
1472 (and (eq (car-safe (car m
)) 'cplx
)
1473 (Math-zerop (nth 1 (car m
)))
1474 (setq m
(list (nth 2 (car m
))
1477 (and (if math-poly-mult-powers
1478 (equal math-poly-mult-powers
1480 (setq math-poly-mult-powers
(nth 1 m
)))
1481 (or (equal expr var
)
1482 (eq math-poly-mult-powers
1))
1486 (setq pow
(math-to-simple-fraction pow
))
1487 (and (eq (car-safe pow
) 'frac
)
1488 math-poly-frac-powers
1490 (setq math-poly-frac-powers
1491 (calcFunc-lcm math-poly-frac-powers
1493 (or (memq math-poly-frac-powers
'(1 nil
))
1494 (setq pow
(math-mul pow math-poly-frac-powers
)))
1500 (let ((p1 (if (equal expr var
)
1502 (math-is-poly-rec expr nil
)))
1506 (or (null math-is-poly-degree
)
1507 (<= (* (1- (length p1
)) n
) math-is-poly-degree
))
1510 (setq accum
(math-poly-mul accum p1
)
1514 (math-is-poly-rec expr nil
)
1515 (setq math-poly-neg-powers
1516 (cons (math-pow expr
(- pow
))
1517 math-poly-neg-powers
))
1518 (list (list '^ expr pow
))))))))
1519 ((Math-objectp expr
)
1521 ((memq (car expr
) '(+ -
))
1522 (let ((p1 (math-is-poly-rec (nth 1 expr
) negpow
)))
1524 (let ((p2 (math-is-poly-rec (nth 2 expr
) negpow
)))
1526 (math-poly-mix p1
1 p2
1527 (if (eq (car expr
) '+) 1 -
1)))))))
1528 ((eq (car expr
) 'neg
)
1529 (mapcar 'math-neg
(math-is-poly-rec (nth 1 expr
) negpow
)))
1531 (let ((p1 (math-is-poly-rec (nth 1 expr
) negpow
)))
1533 (let ((p2 (math-is-poly-rec (nth 2 expr
) negpow
)))
1535 (or (null math-is-poly-degree
)
1536 (<= (- (+ (length p1
) (length p2
)) 2)
1537 math-is-poly-degree
))
1538 (math-poly-mul p1 p2
))))))
1540 (and (or (not (math-poly-depends (nth 2 expr
) var
))
1542 (math-is-poly-rec (nth 2 expr
) nil
)
1543 (setq math-poly-neg-powers
1544 (cons (nth 2 expr
) math-poly-neg-powers
))))
1545 (not (Math-zerop (nth 2 expr
)))
1546 (let ((p1 (math-is-poly-rec (nth 1 expr
) negpow
)))
1547 (mapcar (function (lambda (x) (math-div x
(nth 2 expr
))))
1549 ((and (eq (car expr
) 'calcFunc-exp
)
1550 (equal var
'(var e var-e
)))
1551 (math-is-poly-rec (list '^ var
(nth 1 expr
)) negpow
))
1552 ((and (eq (car expr
) 'calcFunc-sqrt
)
1553 math-poly-frac-powers
)
1554 (math-is-poly-rec (list '^
(nth 1 expr
) '(frac 1 2)) negpow
))
1556 (and (or (not (math-poly-depends expr var
))
1558 (not (eq (car expr
) 'vec
))
1561 ;;; Check if expr is a polynomial in var; if so, return its degree.
1562 (defun math-polynomial-p (expr var
)
1563 (cond ((equal expr var
) 1)
1564 ((Math-primp expr
) 0)
1565 ((memq (car expr
) '(+ -
))
1566 (let ((p1 (math-polynomial-p (nth 1 expr
) var
))
1568 (and p1
(setq p2
(math-polynomial-p (nth 2 expr
) var
))
1571 (let ((p1 (math-polynomial-p (nth 1 expr
) var
))
1573 (and p1
(setq p2
(math-polynomial-p (nth 2 expr
) var
))
1575 ((eq (car expr
) 'neg
)
1576 (math-polynomial-p (nth 1 expr
) var
))
1577 ((and (eq (car expr
) '/)
1578 (not (math-poly-depends (nth 2 expr
) var
)))
1579 (math-polynomial-p (nth 1 expr
) var
))
1580 ((and (eq (car expr
) '^
)
1581 (natnump (nth 2 expr
)))
1582 (let ((p1 (math-polynomial-p (nth 1 expr
) var
)))
1583 (and p1
(* p1
(nth 2 expr
)))))
1584 ((math-poly-depends expr var
) nil
)
1587 (defun math-poly-depends (expr var
)
1588 (if math-poly-base-variable
1589 (math-expr-contains expr math-poly-base-variable
)
1590 (math-expr-depends expr var
)))
1592 ;;; Find the variable (or sub-expression) which is the base of polynomial expr.
1593 ;; The variables math-poly-base-const-ok and math-poly-base-pred are
1594 ;; local to math-polynomial-base, but are used by math-polynomial-base-rec.
1595 (defvar math-poly-base-const-ok
)
1596 (defvar math-poly-base-pred
)
1598 ;; The variable math-poly-base-top-expr is local to math-polynomial-base,
1599 ;; but is used by math-polynomial-p1 in calc-poly.el, which is called
1600 ;; by math-polynomial-base.
1602 (defun math-polynomial-base (math-poly-base-top-expr &optional math-poly-base-pred
)
1603 (or math-poly-base-pred
1604 (setq math-poly-base-pred
(function (lambda (base) (math-polynomial-p
1605 math-poly-base-top-expr base
)))))
1606 (or (let ((math-poly-base-const-ok nil
))
1607 (math-polynomial-base-rec math-poly-base-top-expr
))
1608 (let ((math-poly-base-const-ok t
))
1609 (math-polynomial-base-rec math-poly-base-top-expr
))))
1611 (defun math-polynomial-base-rec (mpb-expr)
1612 (and (not (Math-objvecp mpb-expr
))
1613 (or (and (memq (car mpb-expr
) '(+ -
*))
1614 (or (math-polynomial-base-rec (nth 1 mpb-expr
))
1615 (math-polynomial-base-rec (nth 2 mpb-expr
))))
1616 (and (memq (car mpb-expr
) '(/ neg
))
1617 (math-polynomial-base-rec (nth 1 mpb-expr
)))
1618 (and (eq (car mpb-expr
) '^
)
1619 (math-polynomial-base-rec (nth 1 mpb-expr
)))
1620 (and (eq (car mpb-expr
) 'calcFunc-exp
)
1621 (math-polynomial-base-rec '(var e var-e
)))
1622 (and (or math-poly-base-const-ok
(math-expr-contains-vars mpb-expr
))
1623 (funcall math-poly-base-pred mpb-expr
)
1626 ;;; Return non-nil if expr refers to any variables.
1627 (defun math-expr-contains-vars (expr)
1628 (or (eq (car-safe expr
) 'var
)
1629 (and (not (Math-primp expr
))
1631 (while (and (setq expr
(cdr expr
))
1632 (not (math-expr-contains-vars (car expr
)))))
1635 ;;; Simplify a polynomial in list form by stripping off high-end zeros.
1636 ;;; This always leaves the constant part, i.e., nil->nil and nonnil->nonnil.
1637 (defun math-poly-simplify (p)
1639 (if (Math-zerop (nth (1- (length p
)) p
))
1640 (let ((pp (copy-sequence p
)))
1641 (while (and (cdr pp
)
1642 (Math-zerop (nth (1- (length pp
)) pp
)))
1643 (setcdr (nthcdr (- (length pp
) 2) pp
) nil
))
1647 ;;; Compute ac*a + bc*b for polynomials in list form a, b and
1648 ;;; coefficients ac, bc. Result may be unsimplified.
1649 (defun math-poly-mix (a ac b bc
)
1651 (cons (math-add (math-mul (or (car a
) 0) ac
)
1652 (math-mul (or (car b
) 0) bc
))
1653 (math-poly-mix (cdr a
) ac
(cdr b
) bc
))))
1655 (defun math-poly-zerop (a)
1657 (and (null (cdr a
)) (Math-zerop (car a
)))))
1659 ;;; Multiply two polynomials in list form.
1660 (defun math-poly-mul (a b
)
1662 (math-poly-mix b
(car a
)
1663 (math-poly-mul (cdr a
) (cons 0 b
)) 1)))
1665 ;;; Build an expression from a polynomial list.
1666 (defun math-build-polynomial-expr (p var
)
1668 (if (Math-numberp var
)
1669 (math-with-extra-prec 1
1670 (let* ((rp (reverse p
))
1672 (while (setq rp
(cdr rp
))
1673 (setq accum
(math-add (car rp
) (math-mul accum var
))))
1675 (let* ((rp (reverse p
))
1676 (n (1- (length rp
)))
1677 (accum (math-mul (car rp
) (math-pow var n
)))
1679 (while (setq rp
(cdr rp
))
1681 (or (math-zerop (car rp
))
1682 (setq accum
(list (if (math-looks-negp (car rp
)) '-
'+)
1684 (math-mul (if (math-looks-negp (car rp
))
1687 (math-pow var n
))))))
1692 (defun math-to-simple-fraction (f)
1693 (or (and (eq (car-safe f
) 'float
)
1694 (or (and (>= (nth 2 f
) 0)
1695 (math-scale-int (nth 1 f
) (nth 2 f
)))
1696 (and (integerp (nth 1 f
))
1699 (math-make-frac (nth 1 f
)
1700 (math-scale-int 1 (- (nth 2 f
)))))))
1705 ;;; arch-tag: 52e7dcdf-9688-464d-a02b-4bbe789348d0
1706 ;;; calc-alg.el ends here