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1 ;;; -*- Mode: Lisp; Package: Macsyma -*- ;;;
2 ;;; (c) Copyright 1984 the Regents of the University of California. ;;;
3 ;;; All Rights Reserved. ;;;
4 ;;; This work was produced under the sponsorship of the ;;;
5 ;;; U.S. Department of Energy. The Government retains ;;;
6 ;;; certain rights therein. ;;;
7 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
8
9 (macsyma-module seqopt)
10
11 (defmvar $sequence_optim_prefix '$opt
12 "String used to prefix all optimized temporaries arising from a
13 call to SEQUENCE_OPTIMIZE."
14 modified-commands '$sequence_optimize)
15
16 (defmvar $sequence_optim_counter 1
17 "Integer index used to uniquely identify all optimized temporaries
18 arising from a call to SEQUENCE_OPTIMIZE."
19 fixnum
20 modified-commands '$sequence_optimize)
21
22 (defmvar $sequence_optim_suffix 's
23 "String used to suffix all optimized temporaries arising from a
24 call to SEQUENCE_OPTIMIZE, as well as names generated by CRAY_FORTRAN for
25 subexpressions which have been broken out of an expression which is too
26 large for the CFT compiler."
27 modified-commands '($sequence_optimize $cray_fortran))
28
29 (defmvar $save_optim_info nil
30 "Flag which, if TRUE, causes the common subexpressions which
31 SEQUENCE_OPTIMIZE finds to be saved as equations on the MACSYMA list
32 OPTIM_EQUIVS."
33 boolean
34 modified-commands '$sequence_optimize)
35
36 (defmvar $optim_equivs (list '(mlist simp))
37 "Macsyma list of equations for the common subexpressions which
38 SEQUENCE_OPTIMIZE finds when SAVE_OPTIM_INFO is TRUE."
39 modified-commands '$sequence_optimize)
40
41 (defmvar $optim_additions (list '(mlist simp))
42 "Macsyma list of equations for the subexpressions which it is known
43 a priori will occur more than once in a sequence of code to be optimized."
44 modified-commands '$pre_optimize)
45
46 (defmvar $merge_ops (list '(mlist simp) '$cvmgp '$cvmgt)
47 "A MACSYMA list of currently known CRAY-1 vector merge operations."
48 modified-commands '($sequence_optimize $expense))
49
50 (defmvar $cost_float_power (+ $cost_exp $cost_sin_cos_log)
51 "The expense of computing a floating point power in terms of scalar
52 floating point additions on the CRAY-1(For further discussion do:
53 DESCRIBE(COST_RECIPROCAL) )."
54 fixnum
55 modified-commands '($expense $gather_exponents))
56
57 (defvar optim-vars nil
58 "MACSYMA list of generated names for common subexpressions(Not used if
59 a list equations is passed to SEQUENCE_OPTIMIZE).")
60
61 (array subexp t 64.)
62
63 (defmacro make-expt (base exponent) ``((mexpt simp) ,,base ,,exponent))
64
65 (defmacro base (x) `(cadr ,x))
66
67 (defmacro exponent (x) `(caddr ,x))
68
69 (defmacro mquotientp (x) `(and (not (atom ,x)) (eq (caar ,x) 'mquotient)))
70
71 ;; $SEQUENCE_OPTIMIZE takes a Macsyma expression or list of simple equations
72 ;; and returns a LIST which contains a series of equivalences for the common
73 ;; subexpressions and the reduced equations or expression.
74 ;; These subexpressions are found by hashing them.
75
76 (defun alike1-hash (exp)
77 (\ (if (atom exp)
78 (sxhash exp)
79 (do ((n (alike1-hash (caar exp))
80 (+ n (alike1-hash (car arg_list))))
81 (arg_list (cdr exp) (cdr arg_list)))
82 ((null arg_list) n)))
83 27449.)) ; a prime number < 2^15 = PRIME(3000)
84
85 (defun $sequence_optimize (x)
86 (prog (setqs)
87 (fillarray 'subexp (list nil))
88 (if ($listp x)
89 (do ((chk (cdr x) (cdr chk)))
90 ((null chk))
91 (or (and (not (atom (car chk)))
92 (eq (caaar chk) 'mequal)
93 ($mapatom (cadar chk)))
94 (merror "List passed to SEQUENCE_OPTIMIZE of incorrect form. Bad element is ~%~M" (car chk))))
95 (setq optim-vars (append (list '(mlist)) nil)))
96 (setq x (collapse (fix-unary-minus (optim-format ($gather_exponents (copy-tree x))))))
97 (if (atom x) (return x))
98 (comexp x)
99 (setq x (optim x))
100 (and $save_optim_info
101 (setq $optim_equivs (append $optim_equivs (copy-tree setqs))))
102 (return (prog1 (cond ((null setqs) x)
103 (($listp x)
104 (let ((scan x))
105 (do ((opt-con setqs (cdr opt-con)))
106 ((null opt-con) x)
107 (let ((rhs-eqn (caddar opt-con)))
108 (do ((equivs scan (cdr equivs)))
109 ((null (cdr equivs))
110 (rplacd scan (append (ncons (car opt-con)) (cdr scan)))
111 (setq scan (cdr scan)))
112 (or (freeof (cadadr equivs) rhs-eqn)
113 (setq scan (cdr equivs))))))))
114 ((or (not (eq 'mprog (caar x)))
115 (and ($listp (cadr x)) (cdadr x)))
116 `((mprog) ,optim-vars ,.setqs ,x))
117 (t `((mprog) ,optim-vars ,.(nconc setqs (cddr x)))))
118 (setq optim-vars nil)
119 (fillarray 'subexp (list nil))))))
120
121 (defun copy-to-pntr (x y)
122 (do ((redo x (cdr redo))
123 (new nil `(,.new ,(car redo))))
124 ((eq redo y) new)))
125
126 (defun recip-1 (expon)
127 (or (and (numberp expon) (minusp expon))
128 (and (not (atom expon))
129 (let ((op (caar expon)))
130 (or (and (eq op 'mtimes) (equal (cadr expon) -1))
131 (and (eq op 'rat) (minusp (cadr expon)))
132 (eq op 'mminus))))))
133
134 (defun reciprocalp (x)
135 (and (mexptp x)
136 (let ((expon (exponent x)))
137 (cond ((mquotientp expon) (recip-1 (cadr expon)))
138 (t (recip-1 expon))))))
139
140 (defun gen-negative (x)
141 (cond ((mmminusp x) (cadr x))
142 ((mquotientp x) `((mquotient) ,(mul -1 (cadr x)) ,(caddr x)))
143 (t (mul -1 x))))
144
145 (defun mul-list (x)
146 (cond ((cdr x) `((mtimes) ,@x))
147 (t (car x))))
148
149 (defun optim-format (x)
150 (cond ((atom x) x)
151 ((and (eq 'rat (caar x)) (minusp (cadr x)))
152 `((mminus) ((rat) ,(- (cadr x)) ,(caddr x))))
153 ((and (eq 'mquotient (caar x)) (not (equal 1 (cadr x))))
154 (let ((nmr (cadr x)))
155 (optim-format `((mtimes simp) ,@(cond ((mtimesp nmr) (cdr nmr))
156 (t (list nmr)))
157 ((mquotient) 1 ,(caddr x))))))
158 ((eq 'mexpt (caar x)) (opt-expt x))
159 ((eq 'mtimes (caar x))
160 (do ((next (cdr x) (cdr next))
161 (denominator)
162 (numerator))
163 ((null next)
164 (cond (denominator
165 (let ((recip `((mquotient) 1 ,(mul-list denominator))))
166 (cond (numerator
167 (let ((prod? (mul-list numerator)))
168 (cond ((mtimesp prod?)
169 (nconc prod? (ncons recip)))
170 (t `((mtimes) ,prod? ,recip)))))
171 (t recip))))
172 (numerator (mul-list numerator))
173 (t x)))
174 (let ((obj (car next)))
175 (cond ((reciprocalp obj)
176 (let* ((expon (exponent obj))
177 (optim-expt (let ((mbase (base obj)))
178 (cond ((equal expon -1)
179 (optim-format mbase))
180 (t
181 (opt-expt (make-expt mbase (gen-negative expon))))))))
182 (setq denominator
183 (nconc denominator
184 (cond ((mtimesp optim-expt) (cdr optim-expt))
185 (t (ncons optim-expt)))))
186 (or numerator
187 (do ((seplist (cdr x) (cdr seplist)))
188 ((eq seplist next))
189 (let ((element (car seplist)))
190 (or (reciprocalp element)
191 (setq numerator `(,.numerator ,element))))))))
192 (t
193 (let ((result (optim-format obj)))
194 (or (eq obj result)
195 numerator
196 (do ((seplist (cdr x) (cdr seplist)))
197 ((eq seplist next))
198 (let ((element (car seplist)))
199 (or (reciprocalp element)
200 (setq numerator `(,.numerator ,element))))))
201 (and (or numerator denominator (not (eq obj result)))
202 (setq numerator (nconc numerator
203 (cond ((and (mexptp obj) (mtimesp result))
204 (copy-tree (cdr result)))
205 (t (ncons result))))))))))))
206 (t
207 (do ((next (cdr x) (cdr next))
208 (new))
209 ((null next)
210 (cond (new new)
211 (t x)))
212 (let* ((obj (car next))
213 (result (optim-format obj)))
214 (or (eq obj result)
215 new
216 (setq new (copy-to-pntr x next)))
217 (and (or new (not (eq obj result)))
218 (setq new `(,.new ,result))))))))
219
220 (defun opt-expt (x)
221 (let ((osym-base (base x)) (oexp (exponent x)))
222 (let ((sym-base (optim-format osym-base)) (exp (optim-format oexp)))
223 (cond ((reciprocalp x)
224 `((mquotient) 1 ,(cond ((equal -1 exp) sym-base)
225 (t (opt-expt (make-expt sym-base (gen-negative exp)))))))
226 ((and (ratnump exp) (equal 2 (caddr exp)))
227 (setq exp (cadr exp))
228 (cond ((equal 1 exp) `((%sqrt) ,sym-base))
229 (t (let ((int-exp (quotient exp 2)))
230 `((mtimes) ((%sqrt) ,sym-base)
231 ,(cond ((equal int-exp 1) sym-base)
232 (t (make-expt sym-base int-exp))))))))
233 (t
234 (cond ((and (eq osym-base sym-base) (eq oexp exp)) x)
235 (t (make-expt sym-base exp))))))))
236
237 ;; the following two functions were motivated by an inability of the
238 ;; cray merge functions to cope with a unary minus.
239
240 (defun disp-negate (x)
241 (cond ((mtimesp x)
242 (let ((coeff (cadr x)))
243 (cond ((and (fixnump coeff) (minusp coeff))
244 (append `((mtimes) ,(- coeff)) (cddr x)))
245 (t `((mminus) ,x)))))
246 ((mnump x) (mul -1 x))
247 ((or (atom x) (not (eq (caar x) 'mminus))) `((mminus) ,x))
248 (t (cadr x))))
249
250 (defun fix-unary-minus (x)
251 (cond (($mapatom x) x)
252 ((eq (caar x) 'mtimes)
253 (mapc 'fix-unary-minus (cdr x))
254 (let ((sign (cadr x)))
255 (cond ((and (fixnump sign) (minusp sign))
256 (cond ((equal sign -1)
257 (let ((chk-merge (caddr x)))
258 (cond ((and (not (atom chk-merge))
259 (member (caar chk-merge) $merge_ops :test #'eq))
260 (rplacd (cdr x) (append `(((,(caar chk-merge)) ,(disp-negate (cadr chk-merge))
261 ,(disp-negate (caddr chk-merge))
262 ,(cadddr chk-merge)))
263 (cdddr x)))
264 (cond ((cdddr x) (rplacd x (cddr x)) x)
265 (t (caddr x))))
266 (t `((mminus) ,(cond ((cdddr x)
267 (rplacd x (cddr x)) x)
268 (t (caddr x))))))))
269 (t `((mminus) ,(append `((mtimes) ,(- sign)) (cddr x))))))
270 (t x))))
271 (t (do ((search (cdr x) (cdr search)))
272 ((null search) x)
273 (let* ((obj (car search)) (new (fix-unary-minus obj)))
274 (or (eq new obj) (rplaca search new)))))))
275
276 (defun collapse (x)
277 (if (atom x)
278 x
279 (let ((n (logand 63. (alike1-hash x))))
280 (do ((l (cdr x) (cdr l)))
281 ((null l))
282 (let* ((carl (car l)) (res (collapse carl)))
283 (or (eq carl res) (rplaca l res))))
284 (do ((l (subexp n) (cdr l)))
285 ((null l) (setf (subexp n) (cons (list x) (subexp n))) x)
286 (if (alike1 x (caar l)) (return (caar l)))))))
287
288 (defun comexp (x)
289 (cond ((atom x))
290 ((eq 'rat (caar x)))
291 (t
292 (setq x (assoc x (subexp (logand 63. (alike1-hash x))) :test #'eq))
293 (cond ((null (cdr x))
294 (rplacd x 1)
295 (mapc 'comexp (cdar x)))
296 (t (rplacd x (1+ (cdr x))))))))
297
298 (defun optim (x)
299 (cond ((atom x) x)
300 ((and (member 'array (cdar x) :test #'eq) (not (mget (caar x) 'arrayfun-mode))) x)
301 ((eq 'rat (caar x)) x)
302 (t
303 (let ((xpair (assoc x (subexp (logand 63. (alike1-hash x))) :test #'eq))
304 (nx (do ((l (cdr x) (cdr l))
305 (c (list (car x)) (cons (optim (car l)) c)))
306 ((null l) (nreverse c)))))
307 (let ((tmp (cdr xpair))
308 (sym (do ((lk (cdr $optim_equivs) (cdr lk)))
309 ((null lk))
310 (and (alike1 nx (caddar lk))
311 (return (cadar lk))))))
312 (cond ((fixnump tmp)
313 (cond (sym
314 (rplacd xpair sym)
315 (mformat nil "c - earlier opt-vect, ~M, occurs ~M time(s)" sym tmp)
316 sym)
317 ((= tmp 1) nx)
318 (t
319 (let ((sym (getvar)))
320 (rplacd xpair sym)
321 (setq setqs `(,.setqs ,(list (cond (optim-vars (list 'msetq))
322 (t (list 'mequal)))
323 sym nx)))
324 (mformat nil "c - there are ~M occurrences of ~M" tmp sym)
325 sym))))
326 (t tmp))))))) ;;; Should this be an error?
327
328 (defun getvar ()
329 (let ((newvar (implode (nconc (exploden $sequence_optim_prefix)
330 (exploden $sequence_optim_counter)
331 (exploden $sequence_optim_suffix)))))
332 (incf $sequence_optim_counter)
333 (if optim-vars (setq optim-vars `(,.optim-vars ,newvar)))
334 newvar))
335
336 ;;; The following will not PRE_OPTIMIZE top-level forms.
337
338 (defun $pre_optimize (x)
339 (cond ((atom x))
340 ((eq (caar x) '$cvmgp)
341 (let ((term3 (cadddr x))
342 (opt-list (append $optim_equivs (cdr $optim_additions))))
343 (or ($mapatom term3)
344 (and (eq (caar term3) 'mtimes)
345 (equal (cadr term3) -1)
346 (let ((obj (caddr term3))
347 (two-term (= (length term3) 3)))
348 (or (and two-term ($mapatom obj))
349 (do ((l (cdr opt-list) (cdr l)))
350 ((null l))
351 (let ((rhs (caddar l)))
352 (cond ((and two-term (alike1 rhs obj))
353 (rplaca (cdddr x) (mul -1 (cadar l)))
354 (return t))
355 ((alike1 rhs term3)
356 (rplaca (cdddr x) (cadar l))
357 (return t))))))))
358 (do ((l (cdr opt-list) (cdr l)))
359 ((null l))
360 (let ((rhs (caddar l)))
361 (cond ((alike1 rhs term3)
362 (rplaca (cdddr x) (cadar l))
363 (return t))
364 ((and (eq (caar rhs) 'mtimes)
365 (equal (cadr rhs) -1)
366 (null (cdddr rhs))
367 (alike1 (caddr rhs) term3))
368 (rplaca (cdddr x) (mul (cadar l) -1))
369 (return t)))))
370 (let ((name (getvar)))
371 (setq $optim_additions
372 `(,@$optim_additions ((mequal simp) ,name ,term3)))
373 (rplaca (cdddr x) name)))))
374 (t (do ((terms (cdr x) (cdr terms)))
375 ((null terms))
376 (let ((obj (car terms)))
377 ($pre_optimize obj)
378 (do ((lk (cdr $optim_equivs) (cdr lk)))
379 ((null lk))
380 (and (alike1 obj (caddar lk))
381 (rplaca terms (cadar lk))
382 (return t))))))))
383
384 (defun $collapse_pre_optims (x)
385 (cond ((atom x) x)
386 ((do ((lk (cdr $optim_equivs) (cdr lk)))
387 ((null lk))
388 (and (alike1 x (caddar lk))
389 (return (cadar lk)))))
390 (t (do ((terms (cdr x) (cdr terms))
391 (success))
392 ((null terms)
393 (cond ((or success (not (eq (caar x) 'mtimes))) x)
394 (t (do ((l (cdr x) (cdr l))
395 (follow x l))
396 ((null l) x)
397 (let ((saved (car l)))
398 (cond ((atom saved))
399 ((eq (caar saved) '$cvmgp)
400 (rplacd follow (cdr l))
401 (let* ((pminus (equal (cadr x) -1))
402 (new (do ((lk (cdr $optim_equivs) (cdr lk)))
403 ((null lk) x)
404 (let ((rhs (caddar lk)))
405 (cond ((alike1 x rhs)
406 (return (cadar lk)))
407 (t (and (eq (caar rhs) 'mtimes)
408 (cond (pminus
409 (alike1 (cddr x) (cdr rhs)))
410 ((equal (cadr rhs) -1)
411 (alike1 (cdr x) (cddr rhs))))
412 (return (mul -1 (cadar lk))))))))))
413 (return (cond ((eq new x)
414 (rplacd follow `(,saved ,@(cdr follow)))
415 x)
416 (t (mul new saved))))))))))))
417 (let* ((obj (car terms)) (new-obj ($collapse_pre_optims obj)))
418 (or (eq obj new-obj)
419 (and (setq success t)
420 (rplaca terms new-obj))))))))
421
422 (defun product-base (x y)
423 (muln (append (cond ((mtimesp x) (cdr x))
424 (t (ncons x)))
425 (cond ((mtimesp y) (cdr y))
426 (t (ncons y))))
427 nil))
428
429 (defun floating-exponent-gather (x)
430 (cond ((atom x) x)
431 ((mtimesp x)
432 (do ((next (cdr x) (cdr next))
433 (xfol (cdr x) (cdr xfol))
434 (modified)
435 (new))
436 ((null next)
437 (cond ((null new) x)
438 (t (muln new nil))))
439 (let* ((obj (car next)) (result obj))
440 (and (mexptp result)
441 (let ((expon (exponent result)))
442 (and (not (fixnump expon))
443 (do ((remain (cdr next) (cdr remain)))
444 ((null remain))
445 (let ((powered? (car remain)))
446 (and (mexptp powered?)
447 (let ((expon-2 (exponent powered?)))
448 (and (not (fixnump expon-2))
449 (let ((intdif (sub expon expon-2)))
450 (and (fixnump intdif)
451 (let ((pf (> intdif 0))
452 (ab intdif))
453 (declare (fixnum ab))
454 (cond ((or (zerop ab)
455 (> (+ $cost_float_power
456 $cost_float_power
457 (cond (pf ($expense expon-2))
458 (t ($expense expon))))
459 (+ (cond (pf
460 (let ((mbase (base result)))
461 (cond ((mtimesp mbase) ($expense mbase))
462 (t 0))))
463 (t
464 (let ((mbase (base powered?)))
465 (cond ((mtimesp mbase) ($expense mbase))
466 (t 0)))))
467 (multiplies-in-nth-power (abs ab)))))
468 (cond ((not modified)
469 (setq modified t
470 next (append next nil))
471 (setq remain (member powered? next :test #'eq))
472 (setq powered? (car remain))))
473 (cond (pf
474 (let ((mbase (base result)))
475 (setq result
476 (cond ((equal ab 1) mbase)
477 (t (make-expt mbase ab))))
478 (rplaca remain (make-expt (product-base mbase (base powered?)) (exponent powered?))))
479 (return t))
480 (t
481 (setq result (make-expt (product-base (base result) (base powered?)) (exponent result)))
482 (cond ((zerop ab)
483 (setq next (delete powered? next :test #'eq)))
484 (t
485 (let ((pabs (- ab))
486 (mbase (base powered?)))
487 (cond ((equal pabs 1)
488 (cond ((mtimesp mbase)
489 (setq next (nconc next (cdr mbase)))
490 (setq remain (member powered? next :test #'eq))
491 (setq next (delete powered? next :test #'eq)))
492 (t (rplaca remain mbase))))
493 (t (rplaca remain (make-expt mbase pabs))))))))))))))))))))))
494 (setq result (floating-exponent-gather result))
495 (or (eq obj result)
496 new
497 (setq new (copy-to-pntr (cdr x) xfol)))
498 (and (or new (not (eq obj result)))
499 (setq new (nconc new
500 (cond ((mtimesp result)
501 (copy-tree (cdr result)))
502 (t (ncons result)))))))))
503 (t
504 (do ((next (cdr x) (cdr next))
505 (new))
506 ((null next)
507 (cond (new new)
508 (t x)))
509 (let* ((obj (car next))
510 (result (floating-exponent-gather obj)))
511 (or (eq obj result)
512 new
513 (setq new (copy-to-pntr x next)))
514 (and (or new (not (eq obj result)))
515 (setq new `(,.new ,result))))))))
516
517 (defmacro div-q (x y) `(div (simplify ,x) (simplify ,y)))
518
519 (defun fgcd-exponent-gather (x)
520 (cond ((atom x) x)
521 ((mtimesp x)
522 (do ((next (cdr x) (cdr next))
523 (xfol (cdr x) (cdr xfol))
524 (modified)
525 (new))
526 ((null next)
527 (cond ((null new) x)
528 (t (muln new nil))))
529 (let* ((obj (car next))
530 (result (fgcd-exponent-gather obj)))
531 (and (mexptp result)
532 (let ((expon (exponent result)))
533 (and (not (fixnump expon))
534 (do ((allow-fix t nil)
535 (repeat t))
536 ((null repeat))
537 (do ((remain (cdr next) (cdr remain))
538 (current-gcd expon)
539 (pntrs))
540 ((null remain)
541 (or allow-fix (setq repeat nil))
542 (and pntrs
543 (if (fixnump current-gcd)
544 (<= (multiplies-in-nth-power current-gcd)
545 (1+ (length pntrs)))
546 t)
547 (let* ((leadiv (gen-quotients (div-q expon current-gcd)))
548 (a-single (equal leadiv 1))
549 (ints (and (not a-single) (fixnump leadiv))))
550 (do ((scan pntrs (cdr scan))
551 (save (cond (ints
552 (- $cost_float_power
553 (multiplies-in-nth-power leadiv)))
554 (t 0)))
555 (interms (cond (ints (ncons (make-expt (base result) leadiv)))
556 (t ())))
557 (others (cond (ints ())
558 (t
559 (let ((mbase (base result)))
560 (cond (a-single
561 (cond ((mtimesp mbase) (cdr mbase))
562 (t (ncons mbase))))
563 (t (ncons (make-expt mbase leadiv)))))))))
564 ((null scan)
565 (cond (interms
566 (let* ((prod-ints (muln interms nil))
567 (try-ints-gather (integer-gathering prod-ints))
568 (savings (- (+ (1+ (length pntrs))
569 (- ($expense prod-ints)
570 ($expense try-ints-gather))
571 save)
572 $cost_float_power)))
573 (declare (fixnum savings))
574 (if (< savings 0) (return nil))
575 (setq result (make-expt (muln (nconc others
576 (cond ((mtimesp try-ints-gather)
577 (cdr try-ints-gather))
578 (t (ncons try-ints-gather))))
579 nil)
580 current-gcd))))
581 (t
582 (if (not a-single) (return (setq repeat nil)))
583 (setq result (make-expt (muln others nil) current-gcd))))
584 (do ((rescan pntrs (cdr rescan)))
585 ((null rescan) (setq repeat nil))
586 (setq next (delete (car rescan) next :test #'eq))))
587 (declare (fixnum save))
588 (let* ((expt (car scan))
589 (expon-2 (exponent expt))
590 (nxdiv (gen-quotients (div-q expon-2 current-gcd))))
591 (cond ((equal nxdiv 1)
592 (setq a-single t
593 save (+ save $cost_float_power)
594 others (nconc others (let ((mbase (base expt)))
595 (cond ((mtimesp mbase) (cdr mbase))
596 (t (ncons mbase)))))))
597 ((fixnump nxdiv)
598 (setq save (+ save (- $cost_float_power
599 (multiplies-in-nth-power nxdiv)))
600 interms `(,.interms ,(make-expt (base expt) nxdiv))))
601 (t
602 (setq others `(,.others ,(make-expt (base expt) nxdiv))))))))))
603 (let ((powered? (car remain)))
604 (and (mexptp powered?)
605 (let ((expon-2 (exponent powered?)))
606 (and (not (fixnump expon-2))
607 (let ((fgcd (gen-quotients ($gcd current-gcd expon-2))))
608 (cond ((equal fgcd 1))
609 ((or pntrs
610 (alike1 fgcd expon)
611 (alike1 fgcd expon-2)
612 (and allow-fix
613 (or (fixnump (div-q expon fgcd))
614 (fixnump (div-q expon-2 fgcd)))))
615 (cond ((not modified)
616 (setq modified t
617 next (append next nil))
618 (setq remain (member powered? next :test #'eq))
619 (setq powered? (car remain))))
620 (setq current-gcd fgcd
621 pntrs `(,.pntrs ,powered?))))))))))))))
622 (or (eq obj result)
623 new
624 (setq new (copy-to-pntr (cdr x) xfol)))
625 (and (or new (not (eq obj result)))
626 (setq new `(,.new ,result))))))
627 (t
628 (do ((next (cdr x) (cdr next))
629 (new))
630 ((null next)
631 (cond (new new)
632 (t x)))
633 (let* ((obj (car next))
634 (result (fgcd-exponent-gather obj)))
635 (or (eq obj result)
636 new
637 (setq new (copy-to-pntr x next)))
638 (and (or new (not (eq obj result)))
639 (setq new `(,.new ,result))))))))
640
641 (defun integer-exponent-gather (x)
642 (cond ((atom x) x)
643 ((mtimesp x)
644 (do ((top x (or new top))
645 (new 0 new))
646 ((null new) top)
647 (setq new nil)
648 (do ((next (cdr top) (cdr next))
649 (xfol (cdr top) (cdr xfol))
650 (modified))
651 ((null next)
652 (and new (setq new (muln new nil))))
653 (let* ((obj (car next)) (result obj))
654 (and (mexptp result)
655 (let ((expon (exponent result)))
656 (and (fixnump expon)
657 (do ((remain (cdr next) (cdr remain)))
658 ((null remain))
659 (let ((powered? (car remain)))
660 (and (mexptp powered?)
661 (let ((expon-2 (exponent powered?)))
662 (and (fixnump expon-2)
663 (let* ((intdif (- expon expon-2))
664 (pf (plusp intdif)))
665 (declare (fixnum intdif))
666 (cond ((or (zerop intdif)
667 (< (+ (cond (pf
668 (let ((mbase (base result)))
669 (cond ((mtimesp mbase) ($expense mbase))
670 (t 0))))
671 (t
672 (let ((mbase (base powered?)))
673 (cond ((mtimesp mbase) ($expense mbase))
674 (t 0)))))
675 1
676 (multiplies-in-nth-power (abs intdif)))
677 (multiplies-in-nth-power (max expon expon-2))))
678 (cond ((not modified)
679 (setq modified t
680 next (append next nil))
681 (setq remain (member powered? next :test #'eq))
682 (setq powered? (car remain))))
683 (cond (pf
684 (let ((mbase (base result)))
685 (setq result
686 (cond ((equal intdif 1) mbase)
687 (t (make-expt mbase intdif))))
688 (rplaca remain (make-expt (product-base mbase (base powered?)) (exponent powered?))))
689 (return t))
690 (t
691 (setq result (make-expt (product-base (base result) (base powered?)) (exponent result)))
692 (cond ((zerop intdif)
693 (setq next (delete powered? next :test #'eq)))
694 (t
695 (let ((pabs (- intdif))
696 (mbase (base powered?)))
697 (cond ((equal pabs 1)
698 (cond ((mtimesp mbase)
699 (setq next (nconc next (cdr mbase)))
700 (setq remain (member powered? next :test #'eq))
701 (setq next (delete powered? next :test #'eq)))
702 (t (rplaca remain mbase))))
703 (t (rplaca remain (make-expt mbase pabs))))))))))))))))))))
704 (setq result (integer-exponent-gather result))
705 (or (eq obj result)
706 new
707 (setq new (copy-to-pntr (cdr top) xfol)))
708 (and (or new (not (eq obj result)))
709 (setq new (nconc new
710 (cond ((mtimesp result)
711 (copy-tree (cdr result)))
712 (t (ncons result))))))))))
713 (t
714 (do ((next (cdr x) (cdr next))
715 (new))
716 ((null next)
717 (cond (new new)
718 (t x)))
719 (let* ((obj (car next))
720 (result (integer-exponent-gather obj)))
721 (or (eq obj result)
722 new
723 (setq new (copy-to-pntr x next)))
724 (and (or new (not (eq obj result)))
725 (setq new `(,.new ,result))))))))
726
727 (defun igcd-exponent-gather (x)
728 (cond ((atom x) x)
729 ((mtimesp x)
730 (do ((next (cdr x) (cdr next))
731 (xfol (cdr x) (cdr xfol))
732 (modified)
733 (new))
734 ((null next)
735 (cond ((null new) x)
736 (t (muln new nil))))
737 (let* ((obj (car next))
738 (result (igcd-exponent-gather obj)))
739 (and (mexptp result)
740 (let ((expon (exponent result)))
741 (and (fixnump expon)
742 (do ((remain (cdr next) (cdr remain))
743 (current-gcd expon)
744 (pntrs))
745 ((null remain)
746 (and pntrs
747 (do ((scan pntrs (cdr scan))
748 (newbase (let ((mbase (base result)))
749 (cond ((equal expon current-gcd)
750 (cond ((mtimesp mbase) (cdr mbase))
751 (t (ncons mbase))))
752 (t (ncons (make-expt mbase (quotient expon current-gcd))))))))
753 ((null scan)
754 (setq result (make-expt (muln newbase nil) current-gcd)))
755 (let* ((expt (car scan))
756 (expon-2 (exponent expt)))
757 (setq newbase (nconc newbase (let ((mbase (base expt)))
758 (cond ((equal expon-2 current-gcd)
759 (cond ((mtimesp mbase) (cdr mbase))
760 (t (ncons mbase))))
761 (t (ncons (make-expt mbase (quotient expon-2 current-gcd)))))))
762 next (delete expt next :test #'eq))))))
763 (declare (fixnum current-gcd))
764 (let ((powered? (car remain)))
765 (and (mexptp powered?)
766 (let ((expon-2 (exponent powered?)))
767 (and (fixnump expon-2)
768 (let ((intgcd (gcd current-gcd expon-2)))
769 (cond ((not (equal intgcd 1))
770 (cond ((not modified)
771 (setq modified t
772 next (append next nil))
773 (setq remain (member powered? next :test#'eq))
774 (setq powered? (car remain))))
775 (setq current-gcd intgcd
776 pntrs `(,.pntrs ,powered?)))))))))))))
777 (or (eq obj result)
778 new
779 (setq new (copy-to-pntr (cdr x) xfol)))
780 (and (or new (not (eq obj result)))
781 (setq new `(,.new ,result))))))
782 (t
783 (do ((next (cdr x) (cdr next))
784 (new))
785 ((null next)
786 (cond (new new)
787 (t x)))
788 (let* ((obj (car next))
789 (result (igcd-exponent-gather obj)))
790 (or (eq obj result)
791 new
792 (setq new (copy-to-pntr x next)))
793 (and (or new (not (eq obj result)))
794 (setq new `(,.new ,result))))))))
795
796 (defun gen-quotients (x)
797 (cond (($mapatom x) x)
798 ((specrepp x) (gen-quotients (specdisrep x)))
799 ((eq 'mtimes (caar x))
800 (do ((next (cdr x) (cdr next))
801 (denominator)
802 (numerator))
803 ((null next)
804 (cond (denominator
805 (let ((den (mul-list denominator)))
806 (cond (numerator
807 `((mquotient) ,(mul-list numerator) ,den))
808 (t `((mquotient) 1 ,den)))))
809 (numerator (mul-list numerator))
810 (t x)))
811 (let ((obj (car next)))
812 (cond ((reciprocalp obj)
813 (let ((expon (gen-quotients (exponent obj)))
814 (mbase (gen-quotients (base obj))))
815 (setq denominator
816 (nconc denominator
817 (cond ((equal expon -1)
818 (cond ((mtimesp mbase) (cdr mbase))
819 (t (ncons mbase))))
820 (t (ncons (make-expt mbase (gen-negative expon)))))))
821 (or numerator
822 (do ((seplist (cdr x) (cdr seplist)))
823 ((eq seplist next))
824 (let ((element (car seplist)))
825 (or (reciprocalp element)
826 (setq numerator `(,.numerator ,element))))))))
827 (t
828 (let ((result (gen-quotients obj)))
829 (or (eq obj result)
830 numerator
831 (do ((seplist (cdr x) (cdr seplist)))
832 ((eq seplist next))
833 (let ((element (car seplist)))
834 (or (reciprocalp element)
835 (setq numerator `(,.numerator ,element))))))
836 (and (or numerator denominator (not (eq obj result)))
837 (setq numerator `(,.numerator ,result)))))))))
838 ((reciprocalp x)
839 `((mquotient) 1 ,(gen-quotients (let ((exp (exponent x))
840 (mbase (base x)))
841 (cond ((equal -1 exp) mbase)
842 (t (make-expt mbase (gen-negative exp))))))))
843 (t
844 (do ((next (cdr x) (cdr next))
845 (new))
846 ((null next)
847 (cond (new new)
848 (t x)))
849 (let* ((obj (car next))
850 (result (gen-quotients obj)))
851 (or (eq obj result)
852 new
853 (setq new (copy-to-pntr x next)))
854 (and (or new (not (eq obj result)))
855 (setq new `(,.new ,result))))))))
856
857 (defun integer-gathering (x)
858 (do ((new x (igcd-exponent-gather (integer-exponent-gather new)))
859 (onew 0 new))
860 ((eq new onew) new)))
861
862 (defun $gather_exponents (x)
863 (do ((new (gen-quotients x)
864 (fgcd-exponent-gather (floating-exponent-gather new)))
865 (onew 0 new))
866 ((eq new onew) (integer-gathering new))))
Maxima, a Computer Algebra System