1 ;;; calc-arith.el --- arithmetic 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.
34 ;;; The following lists are not exhaustive.
35 (defvar math-scalar-functions
'(calcFunc-det
36 calcFunc-cnorm calcFunc-rnorm
37 calcFunc-vlen calcFunc-vcount
38 calcFunc-vsum calcFunc-vprod
39 calcFunc-vmin calcFunc-vmax
))
41 (defvar math-nonscalar-functions
'(vec calcFunc-idn calcFunc-diag
42 calcFunc-cvec calcFunc-index
45 calcFunc-cons calcFunc-rcons
46 calcFunc-tail calcFunc-rhead
))
48 (defvar math-scalar-if-args-functions
'(+ -
* / neg
))
50 (defvar math-real-functions
'(calcFunc-arg
51 calcFunc-re calcFunc-im
52 calcFunc-floor calcFunc-ceil
53 calcFunc-trunc calcFunc-round
54 calcFunc-rounde calcFunc-roundu
55 calcFunc-ffloor calcFunc-fceil
56 calcFunc-ftrunc calcFunc-fround
57 calcFunc-frounde calcFunc-froundu
))
59 (defvar math-positive-functions
'())
61 (defvar math-nonnegative-functions
'(calcFunc-cnorm calcFunc-rnorm
62 calcFunc-vlen calcFunc-vcount
))
64 (defvar math-real-scalar-functions
'(% calcFunc-idiv calcFunc-abs
65 calcFunc-choose calcFunc-perm
66 calcFunc-eq calcFunc-neq
67 calcFunc-lt calcFunc-gt
68 calcFunc-leq calcFunc-geq
70 calcFunc-max calcFunc-min
))
72 (defvar math-real-if-arg-functions
'(calcFunc-sin calcFunc-cos
73 calcFunc-tan calcFunc-sec
74 calcFunc-csc calcFunc-cot
76 calcFunc-sinh calcFunc-cosh
77 calcFunc-tanh calcFunc-sech
78 calcFunc-csch calcFunc-coth
80 calcFunc-gamma calcFunc-fact
))
82 (defvar math-integer-functions
'(calcFunc-idiv
83 calcFunc-isqrt calcFunc-ilog
84 calcFunc-vlen calcFunc-vcount
))
86 (defvar math-num-integer-functions
'())
88 (defvar math-rounding-functions
'(calcFunc-floor
90 calcFunc-round calcFunc-trunc
91 calcFunc-rounde calcFunc-roundu
))
93 (defvar math-float-rounding-functions
'(calcFunc-ffloor
95 calcFunc-fround calcFunc-ftrunc
96 calcFunc-frounde calcFunc-froundu
))
98 (defvar math-integer-if-args-functions
'(+ -
* % neg calcFunc-abs
99 calcFunc-min calcFunc-max
100 calcFunc-choose calcFunc-perm
))
105 (defun calc-min (arg)
108 (calc-binary-op "min" 'calcFunc-min arg
'(var inf var-inf
))))
110 (defun calc-max (arg)
113 (calc-binary-op "max" 'calcFunc-max arg
'(neg (var inf var-inf
)))))
115 (defun calc-abs (arg)
118 (calc-unary-op "abs" 'calcFunc-abs arg
)))
121 (defun calc-idiv (arg)
124 (calc-binary-op "\\" 'calcFunc-idiv arg
1)))
127 (defun calc-floor (arg)
130 (if (calc-is-inverse)
131 (if (calc-is-hyperbolic)
132 (calc-unary-op "ceil" 'calcFunc-fceil arg
)
133 (calc-unary-op "ceil" 'calcFunc-ceil arg
))
134 (if (calc-is-hyperbolic)
135 (calc-unary-op "flor" 'calcFunc-ffloor arg
)
136 (calc-unary-op "flor" 'calcFunc-floor arg
)))))
138 (defun calc-ceiling (arg)
143 (defun calc-round (arg)
146 (if (calc-is-inverse)
147 (if (calc-is-hyperbolic)
148 (calc-unary-op "trnc" 'calcFunc-ftrunc arg
)
149 (calc-unary-op "trnc" 'calcFunc-trunc arg
))
150 (if (calc-is-hyperbolic)
151 (calc-unary-op "rond" 'calcFunc-fround arg
)
152 (calc-unary-op "rond" 'calcFunc-round arg
)))))
154 (defun calc-trunc (arg)
159 (defun calc-mant-part (arg)
162 (calc-unary-op "mant" 'calcFunc-mant arg
)))
164 (defun calc-xpon-part (arg)
167 (calc-unary-op "xpon" 'calcFunc-xpon arg
)))
169 (defun calc-scale-float (arg)
172 (calc-binary-op "scal" 'calcFunc-scf arg
)))
174 (defun calc-abssqr (arg)
177 (calc-unary-op "absq" 'calcFunc-abssqr arg
)))
179 (defun calc-sign (arg)
182 (calc-unary-op "sign" 'calcFunc-sign arg
)))
184 (defun calc-increment (arg)
187 (calc-enter-result 1 "incr" (list 'calcFunc-incr
(calc-top-n 1) arg
))))
189 (defun calc-decrement (arg)
192 (calc-enter-result 1 "decr" (list 'calcFunc-decr
(calc-top-n 1) arg
))))
195 (defun math-abs-approx (a)
201 (math-add (math-abs (nth 1 a
)) (math-abs (nth 2 a
))))
205 (math-abs-approx (nth 1 a
)))
207 (math-max (math-abs (nth 2 a
)) (math-abs (nth 3 a
))))
211 (math-reduce-vec 'math-add-abs-approx a
))
212 ((eq (car a
) 'calcFunc-abs
)
216 (defun math-add-abs-approx (a b
)
217 (math-add (math-abs-approx a
) (math-abs-approx b
)))
222 (defvar math-decls-cache-tag nil
)
223 (defvar math-decls-cache nil
)
224 (defvar math-decls-all nil
)
226 ;;; Math-decls-cache is an a-list where each entry is a list of the form:
227 ;;; (VAR TYPES RANGE)
228 ;;; where VAR is a variable name (with var- prefix) or function name;
229 ;;; TYPES is a list of type symbols (any, int, frac, ...)
230 ;;; RANGE is a sorted vector of intervals describing the range.
232 (defvar math-super-types
233 '((int numint rat real number
)
235 (frac rat real number
)
245 (defun math-setup-declarations ()
246 (or (eq math-decls-cache-tag
(calc-var-value 'var-Decls
))
247 (let ((p (calc-var-value 'var-Decls
))
249 (setq math-decls-cache-tag p
250 math-decls-cache nil
)
251 (and (eq (car-safe p
) 'vec
)
252 (while (setq p
(cdr p
))
253 (and (eq (car-safe (car p
)) 'vec
)
254 (setq vec
(nth 2 (car p
)))
256 (let ((v (nth 1 (car p
))))
257 (setq type nil range nil
)
258 (or (eq (car-safe vec
) 'vec
)
259 (setq vec
(list 'vec vec
)))
260 (while (and (setq vec
(cdr vec
))
261 (not (Math-objectp (car vec
))))
262 (and (eq (car-safe (car vec
)) 'var
)
263 (let ((st (assq (nth 1 (car vec
))
265 (cond (st (setq type
(append type st
)))
266 ((eq (nth 1 (car vec
)) 'pos
)
267 (setq type
(append type
270 '(intv 1 0 (var inf var-inf
))))
271 ((eq (nth 1 (car vec
)) 'nonneg
)
272 (setq type
(append type
276 (var inf var-inf
))))))))
278 (setq type
(append type
'(real number
))
279 range
(math-prepare-set (cons 'vec vec
))))
280 (setq type
(list type range
))
281 (or (eq (car-safe v
) 'vec
)
282 (setq v
(list 'vec v
)))
283 (while (setq v
(cdr v
))
284 (if (or (eq (car-safe (car v
)) 'var
)
285 (not (Math-primp (car v
))))
286 (setq math-decls-cache
287 (cons (cons (if (eq (car (car v
)) 'var
)
291 math-decls-cache
)))))
293 (setq math-decls-all
(assq 'var-All math-decls-cache
)))))
295 (defun math-known-scalarp (a &optional assume-scalar
)
296 (math-setup-declarations)
297 (if (if calc-matrix-mode
298 (eq calc-matrix-mode
'scalar
)
300 (not (math-check-known-matrixp a
))
301 (math-check-known-scalarp a
)))
303 (defun math-known-matrixp (a)
304 (and (not (Math-scalarp a
))
305 (not (math-known-scalarp a t
))))
307 ;;; Try to prove that A is a scalar (i.e., a non-vector).
308 (defun math-check-known-scalarp (a)
309 (cond ((Math-objectp a
) t
)
310 ((memq (car a
) math-scalar-functions
)
312 ((memq (car a
) math-real-scalar-functions
)
314 ((memq (car a
) math-scalar-if-args-functions
)
315 (while (and (setq a
(cdr a
))
316 (math-check-known-scalarp (car a
))))
319 (math-check-known-scalarp (nth 1 a
)))
320 ((math-const-var a
) t
)
322 (let ((decl (if (eq (car a
) 'var
)
323 (or (assq (nth 2 a
) math-decls-cache
)
325 (assq (car a
) math-decls-cache
))))
326 (memq 'scalar
(nth 1 decl
))))))
328 ;;; Try to prove that A is *not* a scalar.
329 (defun math-check-known-matrixp (a)
330 (cond ((Math-objectp a
) nil
)
331 ((memq (car a
) math-nonscalar-functions
)
333 ((memq (car a
) math-scalar-if-args-functions
)
334 (while (and (setq a
(cdr a
))
335 (not (math-check-known-matrixp (car a
)))))
338 (math-check-known-matrixp (nth 1 a
)))
339 ((math-const-var a
) nil
)
341 (let ((decl (if (eq (car a
) 'var
)
342 (or (assq (nth 2 a
) math-decls-cache
)
344 (assq (car a
) math-decls-cache
))))
345 (memq 'vector
(nth 1 decl
))))))
348 ;;; Try to prove that A is a real (i.e., not complex).
349 (defun math-known-realp (a)
350 (< (math-possible-signs a
) 8))
352 ;;; Try to prove that A is real and positive.
353 (defun math-known-posp (a)
354 (eq (math-possible-signs a
) 4))
356 ;;; Try to prove that A is real and negative.
357 (defun math-known-negp (a)
358 (eq (math-possible-signs a
) 1))
360 ;;; Try to prove that A is real and nonnegative.
361 (defun math-known-nonnegp (a)
362 (memq (math-possible-signs a
) '(2 4 6)))
364 ;;; Try to prove that A is real and nonpositive.
365 (defun math-known-nonposp (a)
366 (memq (math-possible-signs a
) '(1 2 3)))
368 ;;; Try to prove that A is nonzero.
369 (defun math-known-nonzerop (a)
370 (memq (math-possible-signs a
) '(1 4 5 8 9 12 13)))
372 ;;; Return true if A is negative, or looks negative but we don't know.
373 (defun math-guess-if-neg (a)
374 (let ((sgn (math-possible-signs a
)))
375 (if (memq sgn
'(1 3))
377 (if (memq sgn
'(2 4 6))
379 (math-looks-negp a
)))))
381 ;;; Find the possible signs of A, assuming A is a number of some kind.
382 ;;; Returns an integer with bits: 1 may be negative,
384 ;;; 4 may be positive,
385 ;;; 8 may be nonreal.
387 (defun math-possible-signs (a &optional origin
)
388 (cond ((Math-objectp a
)
389 (if origin
(setq a
(math-sub a origin
)))
390 (cond ((Math-posp a
) 4)
395 ((math-known-posp (nth 2 a
)) 4)
396 ((math-known-negp (nth 3 a
)) 1)
397 ((Math-zerop (nth 2 a
)) 6)
398 ((Math-zerop (nth 3 a
)) 3)
401 (if (math-known-realp (nth 1 a
)) 7 15))
403 ((memq (car a
) '(+ -
))
404 (cond ((Math-realp (nth 1 a
))
407 (math-possible-signs (nth 2 a
)
409 (math-add origin
(nth 1 a
))
411 (math-possible-signs (nth 2 a
)
413 (math-sub origin
(nth 1 a
))
414 (math-neg (nth 1 a
))))))
415 ((Math-realp (nth 2 a
))
416 (let ((org (if (eq (car a
) '-
)
418 (math-neg (nth 2 a
)))))
419 (math-possible-signs (nth 1 a
)
421 (math-add origin org
)
424 (let ((s1 (math-possible-signs (nth 1 a
) origin
))
425 (s2 (math-possible-signs (nth 2 a
))))
426 (if (eq (car a
) '-
) (setq s2
(math-neg-signs s2
)))
427 (cond ((eq s1 s2
) s1
)
432 ((and (eq s1
4) (eq s2
6)) 4)
433 ((and (eq s2
4) (eq s1
6)) 4)
434 ((and (eq s1
1) (eq s2
3)) 1)
435 ((and (eq s2
1) (eq s1
3)) 1)
438 (math-neg-signs (math-possible-signs
440 (and origin
(math-neg origin
)))))
441 ((and origin
(Math-zerop origin
) (setq origin nil
)
443 ((and (or (eq (car a
) '*)
444 (and (eq (car a
) '/) origin
))
445 (Math-realp (nth 1 a
)))
446 (let ((s (if (eq (car a
) '*)
447 (if (Math-zerop (nth 1 a
))
448 (math-possible-signs 0 origin
)
449 (math-possible-signs (nth 2 a
)
450 (math-div (or origin
0)
453 (math-possible-signs (nth 2 a
)
456 (if (Math-negp (nth 1 a
)) (math-neg-signs s
) s
)))
457 ((and (memq (car a
) '(* /)) (Math-realp (nth 2 a
)))
458 (let ((s (math-possible-signs (nth 1 a
)
460 (math-mul (or origin
0) (nth 2 a
))
461 (math-div (or origin
0) (nth 2 a
))))))
462 (if (Math-negp (nth 2 a
)) (math-neg-signs s
) s
)))
465 (while (and (setq a
(cdr a
)) (< signs
15))
466 (setq signs
(logior signs
(math-possible-signs
471 ((memq (car a
) '(* /))
472 (let ((s1 (math-possible-signs (nth 1 a
)))
473 (s2 (math-possible-signs (nth 2 a
))))
476 ((and (eq (car a
) '/) (memq s2
'(2 3 6 7))) 15)
478 (logior (if (memq s1
'(4 5 6 7)) s2
0)
479 (if (memq s1
'(2 3 6 7)) 2 0)
480 (if (memq s1
'(1 3 5 7))
481 (math-neg-signs s2
) 0))))))
483 (let ((s1 (math-possible-signs (nth 1 a
)))
484 (s2 (math-possible-signs (nth 2 a
))))
488 ((eq s1
2) (if (eq s2
4) 2 15))
489 ((eq s2
2) (if (memq s1
'(1 5)) 2 15))
490 ((Math-integerp (nth 2 a
))
491 (if (math-evenp (nth 2 a
))
492 (if (memq s1
'(3 6 7)) 6 4)
494 ((eq s1
6) (if (eq s2
4) 6 15))
497 (let ((s2 (math-possible-signs (nth 2 a
))))
503 ((and (memq (car a
) '(calcFunc-abs calcFunc-abssqr
))
505 (let ((s1 (math-possible-signs (nth 1 a
))))
507 ((memq s1
'(1 4 5)) 4)
509 ((and (eq (car a
) 'calcFunc-exp
) (= (length a
) 2))
510 (let ((s1 (math-possible-signs (nth 1 a
))))
513 (if (or (not origin
) (math-negp origin
))
515 (setq origin
(math-sub (or origin
0) 1))
516 (if (Math-zerop origin
) (setq origin nil
))
518 ((or (and (memq (car a
) '(calcFunc-ln calcFunc-log10
))
520 (and (eq (car a
) 'calcFunc-log
)
522 (math-known-posp (nth 2 a
))))
523 (if (math-known-nonnegp (nth 1 a
))
524 (math-possible-signs (nth 1 a
) 1)
526 ((and (eq (car a
) 'calcFunc-sqrt
) (= (length a
) 2))
527 (let ((s1 (math-possible-signs (nth 1 a
))))
528 (if (memq s1
'(2 4 6)) s1
15)))
529 ((memq (car a
) math-nonnegative-functions
) 6)
530 ((memq (car a
) math-positive-functions
) 4)
531 ((memq (car a
) math-real-functions
) 7)
532 ((memq (car a
) math-real-scalar-functions
) 7)
533 ((and (memq (car a
) math-real-if-arg-functions
)
535 (if (math-known-realp (nth 1 a
)) 7 15)))))
539 (if (Math-posp origin
)
540 (if (memq sign
'(1 2 3 8 9 10 11)) 1 7)
541 (if (memq sign
'(2 4 6 8 10 12 14)) 4 7)))
544 (cond ((eq (nth 2 a
) 'var-pi
)
546 (math-possible-signs (math-pi) origin
)
548 ((eq (nth 2 a
) 'var-e
)
550 (math-possible-signs (math-e) origin
)
552 ((eq (nth 2 a
) 'var-inf
) 4)
553 ((eq (nth 2 a
) 'var-uinf
) 13)
554 ((eq (nth 2 a
) 'var-i
) 8)
557 (math-setup-declarations)
558 (let ((decl (if (eq (car a
) 'var
)
559 (or (assq (nth 2 a
) math-decls-cache
)
561 (assq (car a
) math-decls-cache
))))
563 (memq 'int
(nth 1 decl
))
564 (not (Math-num-integerp origin
)))
567 (math-possible-signs (nth 2 decl
) origin
)
568 (if (memq 'real
(nth 1 decl
))
572 (defun math-neg-signs (s1)
574 (+ 8 (math-neg-signs (- s1
8)))
575 (+ (if (memq s1
'(1 3 5 7)) 4 0)
576 (if (memq s1
'(2 3 6 7)) 2 0)
577 (if (memq s1
'(4 5 6 7)) 1 0))))
580 ;;; Try to prove that A is an integer.
581 (defun math-known-integerp (a)
582 (eq (math-possible-types a
) 1))
584 (defun math-known-num-integerp (a)
585 (<= (math-possible-types a t
) 3))
587 (defun math-known-imagp (a)
588 (= (math-possible-types a
) 16))
591 ;;; Find the possible types of A.
592 ;;; Returns an integer with bits: 1 may be integer.
593 ;;; 2 may be integer-valued float.
594 ;;; 4 may be fraction.
595 ;;; 8 may be non-integer-valued float.
596 ;;; 16 may be imaginary.
597 ;;; 32 may be non-real, non-imaginary.
598 ;;; Real infinities count as integers for the purposes of this function.
599 (defun math-possible-types (a &optional num
)
600 (cond ((Math-objectp a
)
601 (cond ((Math-integerp a
) (if num
3 1))
602 ((Math-messy-integerp a
) (if num
3 2))
603 ((eq (car a
) 'frac
) (if num
12 4))
604 ((eq (car a
) 'float
) (if num
12 8))
606 (if (equal (nth 2 a
) (nth 3 a
))
607 (math-possible-types (nth 2 a
))
610 (if (math-known-realp (nth 1 a
)) 15 63))
612 (if (math-zerop (nth 1 a
)) 16 32))
614 (if (or (Math-equal (nth 2 a
) (math-quarter-circle nil
))
615 (Math-equal (nth 2 a
)
616 (math-neg (math-quarter-circle nil
))))
620 (let* ((t1 (math-possible-types (nth 1 a
) num
))
621 (t2 (math-possible-types (nth 2 a
) num
))
622 (t12 (logior t1 t2
)))
624 (if (> (logand t12
10) 0)
626 (if (or (= t1
4) (= t2
4) calc-prefer-frac
)
632 (if (< t2
16) 16 31))
637 ((memq (car a
) '(+ -
* %
))
638 (let* ((t1 (math-possible-types (nth 1 a
) num
))
639 (t2 (math-possible-types (nth 2 a
) num
))
640 (t12 (logior t1 t2
)))
642 (setq t1
(logand t1
15) t2
(logand t2
15) t12
(logand t12
15)))
644 (let ((mask (if (<= t12
3)
646 (if (and (or (and (<= t1
3) (= (logand t2
3) 0))
647 (and (<= t2
3) (= (logand t1
3) 0)))
648 (memq (car a
) '(+ -
)))
653 (logior (if (and (> (logand t1
5) 0) (> (logand t2
5) 0))
655 (if (> (logand t12
10) 0)
661 (if (< t2
16) 16 31))
666 (if (or (and (= t1
16) (< t2
16))
667 (and (= t2
16) (< t1
16))) 32 63)))
670 (math-possible-types (nth 1 a
)))
672 (let* ((t1 (math-possible-types (nth 1 a
) num
))
673 (t2 (math-possible-types (nth 2 a
) num
))
674 (t12 (logior t1 t2
)))
675 (if (and (<= t2
3) (math-known-nonnegp (nth 2 a
)) (< t1
16))
676 (let ((mask (logior (if (> (logand t1
3) 0) 1 0)
678 (if (> (logand t1
12) 0) 5 0))))
681 (logior (if (and (> (logand t1
5) 0) (> (logand t2
5) 0))
683 (if (> (logand t12
10) 0)
685 (if (and (math-known-nonnegp (nth 1 a
))
686 (math-known-posp (nth 2 a
)))
689 ((eq (car a
) 'calcFunc-sqrt
)
690 (let ((t1 (math-possible-signs (nth 1 a
))))
691 (logior (if (> (logand t1
2) 0) 3 0)
692 (if (> (logand t1
1) 0) 16 0)
693 (if (> (logand t1
4) 0) 15 0)
694 (if (> (logand t1
8) 0) 32 0))))
697 (while (and (setq a
(cdr a
)) (< types
63))
698 (setq types
(logior types
(math-possible-types (car a
) t
))))
700 ((or (memq (car a
) math-integer-functions
)
701 (and (memq (car a
) math-rounding-functions
)
702 (math-known-nonnegp (or (nth 2 a
) 0))))
704 ((or (memq (car a
) math-num-integer-functions
)
705 (and (memq (car a
) math-float-rounding-functions
)
706 (math-known-nonnegp (or (nth 2 a
) 0))))
708 ((eq (car a
) 'calcFunc-frac
)
710 ((and (eq (car a
) 'calcFunc-float
) (= (length a
) 2))
711 (let ((t1 (math-possible-types (nth 1 a
))))
712 (logior (if (> (logand t1
3) 0) 2 0)
713 (if (> (logand t1
12) 0) 8 0)
715 ((and (memq (car a
) '(calcFunc-abs calcFunc-abssqr
))
717 (let ((t1 (math-possible-types (nth 1 a
))))
722 (cond ((memq (nth 2 a
) '(var-e var-pi var-phi var-gamma
)) 8)
723 ((eq (nth 2 a
) 'var-inf
) 1)
724 ((eq (nth 2 a
) 'var-i
) 16)
727 (math-setup-declarations)
728 (let ((decl (if (eq (car a
) 'var
)
729 (or (assq (nth 2 a
) math-decls-cache
)
731 (assq (car a
) math-decls-cache
))))
732 (cond ((memq 'int
(nth 1 decl
))
734 ((memq 'numint
(nth 1 decl
))
736 ((memq 'frac
(nth 1 decl
))
738 ((memq 'rat
(nth 1 decl
))
740 ((memq 'float
(nth 1 decl
))
743 (math-possible-types (nth 2 decl
)))
744 ((memq 'real
(nth 1 decl
))
748 (defun math-known-evenp (a)
749 (cond ((Math-integerp a
)
751 ((Math-messy-integerp a
)
753 (math-evenp (math-trunc a
))))
755 (if (math-known-evenp (nth 1 a
))
756 (math-known-num-integerp (nth 2 a
))
757 (if (math-known-num-integerp (nth 1 a
))
758 (math-known-evenp (nth 2 a
)))))
759 ((memq (car a
) '(+ -
))
760 (or (and (math-known-evenp (nth 1 a
))
761 (math-known-evenp (nth 2 a
)))
762 (and (math-known-oddp (nth 1 a
))
763 (math-known-oddp (nth 2 a
)))))
765 (math-known-evenp (nth 1 a
)))))
767 (defun math-known-oddp (a)
768 (cond ((Math-integerp a
)
770 ((Math-messy-integerp a
)
771 (and (<= (nth 2 a
) 0)
772 (math-oddp (math-trunc a
))))
773 ((memq (car a
) '(+ -
))
774 (or (and (math-known-evenp (nth 1 a
))
775 (math-known-oddp (nth 2 a
)))
776 (and (math-known-oddp (nth 1 a
))
777 (math-known-evenp (nth 2 a
)))))
779 (math-known-oddp (nth 1 a
)))))
782 (defun calcFunc-dreal (expr)
783 (let ((types (math-possible-types expr
)))
785 (if (= (logand types
15) 0) 0
786 (math-reject-arg expr
'realp
'quiet
)))))
788 (defun calcFunc-dimag (expr)
789 (let ((types (math-possible-types expr
)))
791 (if (= (logand types
16) 0) 0
792 (math-reject-arg expr
"Expected an imaginary number")))))
794 (defun calcFunc-dpos (expr)
795 (let ((signs (math-possible-signs expr
)))
797 (if (memq signs
'(1 2 3)) 0
798 (math-reject-arg expr
'posp
'quiet
)))))
800 (defun calcFunc-dneg (expr)
801 (let ((signs (math-possible-signs expr
)))
803 (if (memq signs
'(2 4 6)) 0
804 (math-reject-arg expr
'negp
'quiet
)))))
806 (defun calcFunc-dnonneg (expr)
807 (let ((signs (math-possible-signs expr
)))
808 (if (memq signs
'(2 4 6)) 1
810 (math-reject-arg expr
'posp
'quiet
)))))
812 (defun calcFunc-dnonzero (expr)
813 (let ((signs (math-possible-signs expr
)))
814 (if (memq signs
'(1 4 5 8 9 12 13)) 1
816 (math-reject-arg expr
'nonzerop
'quiet
)))))
818 (defun calcFunc-dint (expr)
819 (let ((types (math-possible-types expr
)))
821 (if (= (logand types
1) 0) 0
822 (math-reject-arg expr
'integerp
'quiet
)))))
824 (defun calcFunc-dnumint (expr)
825 (let ((types (math-possible-types expr t
)))
827 (if (= (logand types
3) 0) 0
828 (math-reject-arg expr
'integerp
'quiet
)))))
830 (defun calcFunc-dnatnum (expr)
831 (let ((res (calcFunc-dint expr
)))
833 (calcFunc-dnonneg expr
)
836 (defun calcFunc-deven (expr)
837 (if (math-known-evenp expr
)
839 (if (or (math-known-oddp expr
)
840 (= (logand (math-possible-types expr
) 3) 0))
842 (math-reject-arg expr
"Can't tell if expression is odd or even"))))
844 (defun calcFunc-dodd (expr)
845 (if (math-known-oddp expr
)
847 (if (or (math-known-evenp expr
)
848 (= (logand (math-possible-types expr
) 3) 0))
850 (math-reject-arg expr
"Can't tell if expression is odd or even"))))
852 (defun calcFunc-drat (expr)
853 (let ((types (math-possible-types expr
)))
854 (if (memq types
'(1 4 5)) 1
855 (if (= (logand types
5) 0) 0
856 (math-reject-arg expr
"Rational number expected")))))
858 (defun calcFunc-drange (expr)
859 (math-setup-declarations)
861 (if (Math-realp expr
)
863 (if (eq (car-safe expr
) 'intv
)
865 (if (eq (car-safe expr
) 'var
)
866 (setq range
(nth 2 (or (assq (nth 2 expr
) math-decls-cache
)
868 (setq range
(nth 2 (assq (car-safe expr
) math-decls-cache
))))
870 (math-clean-set (copy-sequence range
))
871 (setq range
(math-possible-signs expr
))
874 (intv 2 (neg (var inf var-inf
)) 0)
876 (intv 3 (neg (var inf var-inf
)) 0)
877 (intv 1 0 (var inf var-inf
))
878 (vec (intv 2 (neg (var inf var-inf
)) 0)
879 (intv 1 0 (var inf var-inf
)))
880 (intv 3 0 (var inf var-inf
))
881 (intv 3 (neg (var inf var-inf
)) (var inf var-inf
))] range
)
882 (math-reject-arg expr
'realp
'quiet
)))))))
884 (defun calcFunc-dscalar (a)
885 (if (math-known-scalarp a
) 1
886 (if (math-known-matrixp a
) 0
887 (math-reject-arg a
'objectp
'quiet
))))
892 (defsubst calcFunc-neg
(a)
893 (math-normalize (list 'neg a
)))
895 (defun math-neg-fancy (a)
896 (cond ((eq (car a
) 'polar
)
899 (if (math-posp (nth 2 a
))
900 (math-sub (nth 2 a
) (math-half-circle nil
))
901 (math-add (nth 2 a
) (math-half-circle nil
)))))
903 (if (math-zerop (nth 1 a
))
905 (list 'mod
(math-sub (nth 2 a
) (nth 1 a
)) (nth 2 a
))))
907 (list 'sdev
(math-neg (nth 1 a
)) (nth 2 a
)))
909 (math-make-intv (aref [0 2 1 3] (nth 1 a
))
911 (math-neg (nth 2 a
))))
912 ((and math-simplify-only
913 (not (equal a math-simplify-only
)))
916 (math-sub (math-neg (nth 1 a
)) (nth 2 a
)))
918 (math-sub (nth 2 a
) (nth 1 a
)))
919 ((and (memq (car a
) '(* /))
920 (math-okay-neg (nth 1 a
)))
921 (list (car a
) (math-neg (nth 1 a
)) (nth 2 a
)))
922 ((and (memq (car a
) '(* /))
923 (math-okay-neg (nth 2 a
)))
924 (list (car a
) (nth 1 a
) (math-neg (nth 2 a
))))
925 ((and (memq (car a
) '(* /))
926 (or (math-objectp (nth 1 a
))
927 (and (eq (car (nth 1 a
)) '*)
928 (math-objectp (nth 1 (nth 1 a
))))))
929 (list (car a
) (math-neg (nth 1 a
)) (nth 2 a
)))
930 ((and (eq (car a
) '/)
931 (or (math-objectp (nth 2 a
))
932 (and (eq (car (nth 2 a
)) '*)
933 (math-objectp (nth 1 (nth 2 a
))))))
934 (list (car a
) (nth 1 a
) (math-neg (nth 2 a
))))
935 ((and (eq (car a
) 'var
) (memq (nth 2 a
) '(var-uinf var-nan
)))
941 (defun math-okay-neg (a)
942 (or (math-looks-negp a
)
943 (eq (car-safe a
) '-
)))
945 (defun math-neg-float (a)
946 (list 'float
(Math-integer-neg (nth 1 a
)) (nth 2 a
)))
949 (defun calcFunc-add (&rest rest
)
951 (let ((a (car rest
)))
952 (while (setq rest
(cdr rest
))
953 (setq a
(list '+ a
(car rest
))))
957 (defun calcFunc-sub (&rest rest
)
959 (let ((a (car rest
)))
960 (while (setq rest
(cdr rest
))
961 (setq a
(list '- a
(car rest
))))
965 (defun math-add-objects-fancy (a b
)
966 (cond ((and (Math-numberp a
) (Math-numberp b
))
967 (let ((aa (math-complex a
))
968 (bb (math-complex b
)))
970 (let ((res (list 'cplx
971 (math-add (nth 1 aa
) (nth 1 bb
))
972 (math-add (nth 2 aa
) (nth 2 bb
)))))
973 (if (math-want-polar a b
)
976 ((or (Math-vectorp a
) (Math-vectorp b
))
977 (math-map-vec-2 'math-add a b
))
978 ((eq (car-safe a
) 'sdev
)
979 (if (eq (car-safe b
) 'sdev
)
980 (math-make-sdev (math-add (nth 1 a
) (nth 1 b
))
981 (math-hypot (nth 2 a
) (nth 2 b
)))
982 (and (or (Math-scalarp b
)
983 (not (Math-objvecp b
)))
984 (math-make-sdev (math-add (nth 1 a
) b
) (nth 2 a
)))))
985 ((and (eq (car-safe b
) 'sdev
)
987 (not (Math-objvecp a
))))
988 (math-make-sdev (math-add a
(nth 1 b
)) (nth 2 b
)))
989 ((eq (car-safe a
) 'intv
)
990 (if (eq (car-safe b
) 'intv
)
991 (math-make-intv (logior (logand (nth 1 a
) (nth 1 b
))
993 '(neg (var inf var-inf
)))
994 (logand (nth 1 a
) 2) 0)
996 '(neg (var inf var-inf
)))
997 (logand (nth 1 b
) 2) 0)
998 (if (equal (nth 3 a
) '(var inf var-inf
))
999 (logand (nth 1 a
) 1) 0)
1000 (if (equal (nth 3 b
) '(var inf var-inf
))
1001 (logand (nth 1 b
) 1) 0))
1002 (math-add (nth 2 a
) (nth 2 b
))
1003 (math-add (nth 3 a
) (nth 3 b
)))
1004 (and (or (Math-anglep b
)
1006 (not (Math-objvecp b
)))
1007 (math-make-intv (nth 1 a
)
1008 (math-add (nth 2 a
) b
)
1009 (math-add (nth 3 a
) b
)))))
1010 ((and (eq (car-safe b
) 'intv
)
1013 (not (Math-objvecp a
))))
1014 (math-make-intv (nth 1 b
)
1015 (math-add a
(nth 2 b
))
1016 (math-add a
(nth 3 b
))))
1017 ((eq (car-safe a
) 'date
)
1018 (cond ((eq (car-safe b
) 'date
)
1019 (math-add (nth 1 a
) (nth 1 b
)))
1020 ((eq (car-safe b
) 'hms
)
1021 (let ((parts (math-date-parts (nth 1 a
))))
1023 (math-add (car parts
) ; this minimizes roundoff
1025 (math-add (nth 1 parts
)
1028 (math-mul (nth 1 b
) 3600)
1029 (math-add (math-mul (nth 2 b
) 60)
1033 (list 'date
(math-add (nth 1 a
) b
)))
1035 ((eq (car-safe b
) 'date
)
1036 (math-add-objects-fancy b a
))
1037 ((and (eq (car-safe a
) 'mod
)
1038 (eq (car-safe b
) 'mod
)
1039 (equal (nth 2 a
) (nth 2 b
)))
1040 (math-make-mod (math-add (nth 1 a
) (nth 1 b
)) (nth 2 a
)))
1041 ((and (eq (car-safe a
) 'mod
)
1043 (math-make-mod (math-add (nth 1 a
) b
) (nth 2 a
)))
1044 ((and (eq (car-safe b
) 'mod
)
1046 (math-make-mod (math-add a
(nth 1 b
)) (nth 2 b
)))
1047 ((and (or (eq (car-safe a
) 'hms
) (eq (car-safe b
) 'hms
))
1048 (and (Math-anglep a
) (Math-anglep b
)))
1049 (or (eq (car-safe a
) 'hms
) (setq a
(math-to-hms a
)))
1050 (or (eq (car-safe b
) 'hms
) (setq b
(math-to-hms b
)))
1053 (math-neg (math-add (math-neg a
) (math-neg b
)))
1055 (let* ((s (math-add (nth 3 a
) (nth 3 b
)))
1056 (m (math-add (nth 2 a
) (nth 2 b
)))
1057 (h (math-add (nth 1 a
) (nth 1 b
))))
1059 (setq s
(math-add s
60)
1062 (setq m
(math-add m
60)
1067 (let* ((s (math-add (nth 3 a
) (nth 3 b
)))
1068 (m (math-add (nth 2 a
) (nth 2 b
)))
1069 (h (math-add (nth 1 a
) (nth 1 b
))))
1070 (list 'hms h m s
))))))
1071 (t (calc-record-why "*Incompatible arguments for +" a b
))))
1073 (defun math-add-symb-fancy (a b
)
1074 (or (and math-simplify-only
1075 (not (equal a math-simplify-only
))
1077 (and (eq (car-safe b
) '+)
1078 (math-add (math-add a
(nth 1 b
))
1080 (and (eq (car-safe b
) '-
)
1081 (math-sub (math-add a
(nth 1 b
))
1083 (and (eq (car-safe b
) 'neg
)
1084 (eq (car-safe (nth 1 b
)) '+)
1085 (math-sub (math-sub a
(nth 1 (nth 1 b
)))
1087 (and (or (and (Math-vectorp a
) (math-known-scalarp b
))
1088 (and (Math-vectorp b
) (math-known-scalarp a
)))
1089 (math-map-vec-2 'math-add a b
))
1090 (let ((inf (math-infinitep a
)))
1093 (let ((inf2 (math-infinitep b
)))
1095 (if (or (memq (nth 2 inf
) '(var-uinf var-nan
))
1096 (memq (nth 2 inf2
) '(var-uinf var-nan
)))
1098 (let ((dir (math-infinite-dir a inf
))
1099 (dir2 (math-infinite-dir b inf2
)))
1100 (if (and (Math-objectp dir
) (Math-objectp dir2
))
1101 (if (Math-equal dir dir2
)
1103 '(var nan var-nan
)))))
1104 (if (and (equal a
'(var inf var-inf
))
1105 (eq (car-safe b
) 'intv
)
1106 (memq (nth 1 b
) '(2 3))
1107 (equal (nth 2 b
) '(neg (var inf var-inf
))))
1108 (list 'intv
3 (nth 2 b
) a
)
1109 (if (and (equal a
'(neg (var inf var-inf
)))
1110 (eq (car-safe b
) 'intv
)
1111 (memq (nth 1 b
) '(1 3))
1112 (equal (nth 3 b
) '(var inf var-inf
)))
1113 (list 'intv
3 a
(nth 3 b
))
1116 (if (eq (car-safe a
) 'intv
)
1119 ((eq (car-safe a
) '+)
1120 (let ((temp (math-combine-sum (nth 2 a
) b nil nil t
)))
1122 (math-add (nth 1 a
) temp
))))
1123 ((eq (car-safe a
) '-
)
1124 (let ((temp (math-combine-sum (nth 2 a
) b t nil t
)))
1126 (math-add (nth 1 a
) temp
))))
1127 ((and (Math-objectp a
) (Math-objectp b
))
1130 (math-combine-sum a b nil nil nil
))))
1131 (and (Math-looks-negp b
)
1132 (list '- a
(math-neg b
)))
1133 (and (Math-looks-negp a
)
1134 (list '- b
(math-neg a
)))
1135 (and (eq (car-safe a
) 'calcFunc-idn
)
1137 (or (and (eq (car-safe b
) 'calcFunc-idn
)
1139 (list 'calcFunc-idn
(math-add (nth 1 a
) (nth 1 b
))))
1140 (and (math-square-matrixp b
)
1141 (math-add (math-mimic-ident (nth 1 a
) b
) b
))
1142 (and (math-known-scalarp b
)
1143 (math-add (nth 1 a
) b
))))
1144 (and (eq (car-safe b
) 'calcFunc-idn
)
1146 (or (and (math-square-matrixp a
)
1147 (math-add a
(math-mimic-ident (nth 1 b
) a
)))
1148 (and (math-known-scalarp a
)
1149 (math-add a
(nth 1 b
)))))
1153 (defun calcFunc-mul (&rest rest
)
1155 (let ((a (car rest
)))
1156 (while (setq rest
(cdr rest
))
1157 (setq a
(list '* a
(car rest
))))
1161 (defun math-mul-objects-fancy (a b
)
1162 (cond ((and (Math-numberp a
) (Math-numberp b
))
1164 (if (math-want-polar a b
)
1165 (let ((a (math-polar a
))
1168 (math-mul (nth 1 a
) (nth 1 b
))
1169 (math-fix-circular (math-add (nth 2 a
) (nth 2 b
)))))
1170 (setq a
(math-complex a
)
1173 (math-sub (math-mul (nth 1 a
) (nth 1 b
))
1174 (math-mul (nth 2 a
) (nth 2 b
)))
1175 (math-add (math-mul (nth 1 a
) (nth 2 b
))
1176 (math-mul (nth 2 a
) (nth 1 b
)))))))
1178 (if (Math-vectorp b
)
1179 (if (math-matrixp a
)
1180 (if (math-matrixp b
)
1181 (if (= (length (nth 1 a
)) (length b
))
1183 (math-dimension-error))
1184 (if (= (length (nth 1 a
)) 2)
1185 (if (= (length a
) (length b
))
1186 (math-mul-mats a
(list 'vec b
))
1187 (math-dimension-error))
1188 (if (= (length (nth 1 a
)) (length b
))
1189 (math-mul-mat-vec a b
)
1190 (math-dimension-error))))
1191 (if (math-matrixp b
)
1192 (if (= (length a
) (length b
))
1193 (nth 1 (math-mul-mats (list 'vec a
) b
))
1194 (math-dimension-error))
1195 (if (= (length a
) (length b
))
1196 (math-dot-product a b
)
1197 (math-dimension-error))))
1198 (math-map-vec-2 'math-mul a b
)))
1200 (math-map-vec-2 'math-mul a b
))
1201 ((eq (car-safe a
) 'sdev
)
1202 (if (eq (car-safe b
) 'sdev
)
1203 (math-make-sdev (math-mul (nth 1 a
) (nth 1 b
))
1204 (math-hypot (math-mul (nth 2 a
) (nth 1 b
))
1205 (math-mul (nth 2 b
) (nth 1 a
))))
1206 (and (or (Math-scalarp b
)
1207 (not (Math-objvecp b
)))
1208 (math-make-sdev (math-mul (nth 1 a
) b
)
1209 (math-mul (nth 2 a
) b
)))))
1210 ((and (eq (car-safe b
) 'sdev
)
1211 (or (Math-scalarp a
)
1212 (not (Math-objvecp a
))))
1213 (math-make-sdev (math-mul a
(nth 1 b
)) (math-mul a
(nth 2 b
))))
1214 ((and (eq (car-safe a
) 'intv
) (Math-anglep b
))
1216 (math-neg (math-mul a
(math-neg b
)))
1217 (math-make-intv (nth 1 a
)
1218 (math-mul (nth 2 a
) b
)
1219 (math-mul (nth 3 a
) b
))))
1220 ((and (eq (car-safe b
) 'intv
) (Math-anglep a
))
1222 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
1223 (eq (car-safe b
) 'intv
) (math-intv-constp b
))
1224 (let ((lo (math-mul a
(nth 2 b
)))
1225 (hi (math-mul a
(nth 3 b
))))
1226 (or (eq (car-safe lo
) 'intv
)
1227 (setq lo
(list 'intv
(if (memq (nth 1 b
) '(2 3)) 3 0) lo lo
)))
1228 (or (eq (car-safe hi
) 'intv
)
1229 (setq hi
(list 'intv
(if (memq (nth 1 b
) '(1 3)) 3 0) hi hi
)))
1230 (math-combine-intervals
1231 (nth 2 lo
) (and (or (memq (nth 1 b
) '(2 3))
1232 (math-infinitep (nth 2 lo
)))
1233 (memq (nth 1 lo
) '(2 3)))
1234 (nth 3 lo
) (and (or (memq (nth 1 b
) '(2 3))
1235 (math-infinitep (nth 3 lo
)))
1236 (memq (nth 1 lo
) '(1 3)))
1237 (nth 2 hi
) (and (or (memq (nth 1 b
) '(1 3))
1238 (math-infinitep (nth 2 hi
)))
1239 (memq (nth 1 hi
) '(2 3)))
1240 (nth 3 hi
) (and (or (memq (nth 1 b
) '(1 3))
1241 (math-infinitep (nth 3 hi
)))
1242 (memq (nth 1 hi
) '(1 3))))))
1243 ((and (eq (car-safe a
) 'mod
)
1244 (eq (car-safe b
) 'mod
)
1245 (equal (nth 2 a
) (nth 2 b
)))
1246 (math-make-mod (math-mul (nth 1 a
) (nth 1 b
)) (nth 2 a
)))
1247 ((and (eq (car-safe a
) 'mod
)
1249 (math-make-mod (math-mul (nth 1 a
) b
) (nth 2 a
)))
1250 ((and (eq (car-safe b
) 'mod
)
1252 (math-make-mod (math-mul a
(nth 1 b
)) (nth 2 b
)))
1253 ((and (eq (car-safe a
) 'hms
) (Math-realp b
))
1254 (math-with-extra-prec 2
1255 (math-to-hms (math-mul (math-from-hms a
'deg
) b
) 'deg
)))
1256 ((and (eq (car-safe b
) 'hms
) (Math-realp a
))
1258 (t (calc-record-why "*Incompatible arguments for *" a b
))))
1260 ;;; Fast function to multiply floating-point numbers.
1261 (defun math-mul-float (a b
) ; [F F F]
1262 (math-make-float (math-mul (nth 1 a
) (nth 1 b
))
1263 (+ (nth 2 a
) (nth 2 b
))))
1265 (defun math-sqr-float (a) ; [F F]
1266 (math-make-float (math-mul (nth 1 a
) (nth 1 a
))
1267 (+ (nth 2 a
) (nth 2 a
))))
1269 (defun math-intv-constp (a &optional finite
)
1270 (and (or (Math-anglep (nth 2 a
))
1271 (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
1273 (memq (nth 1 a
) '(0 1)))))
1274 (or (Math-anglep (nth 3 a
))
1275 (and (equal (nth 3 a
) '(var inf var-inf
))
1277 (memq (nth 1 a
) '(0 2)))))))
1279 (defun math-mul-zero (a b
)
1280 (if (math-known-matrixp b
)
1281 (if (math-vectorp b
)
1282 (math-map-vec-2 'math-mul a b
)
1283 (math-mimic-ident 0 b
))
1284 (if (math-infinitep b
)
1286 (let ((aa nil
) (bb nil
))
1287 (if (and (eq (car-safe b
) 'intv
)
1289 (and (equal (nth 2 b
) '(neg (var inf var-inf
)))
1290 (memq (nth 1 b
) '(2 3))
1291 (setq aa
(nth 2 b
)))
1292 (and (equal (nth 3 b
) '(var inf var-inf
))
1293 (memq (nth 1 b
) '(1 3))
1294 (setq bb
(nth 3 b
)))
1296 (if (or (math-posp a
)
1298 (or (memq calc-infinite-mode
'(-1 1))
1299 (setq aa
'(neg (var inf var-inf
))
1300 bb
'(var inf var-inf
)))))
1301 (list 'intv
3 (or aa
0) (or bb
0))
1303 (math-neg (list 'intv
3 (or aa
0) (or bb
0)))
1304 '(var nan var-nan
)))
1305 (if (or (math-floatp a
) (math-floatp b
)) '(float 0 0) 0))))))
1308 (defun math-mul-symb-fancy (a b
)
1309 (or (and math-simplify-only
1310 (not (equal a math-simplify-only
))
1312 (and (Math-equal-int a
1)
1314 (and (Math-equal-int a -
1)
1316 (and (or (and (Math-vectorp a
) (math-known-scalarp b
))
1317 (and (Math-vectorp b
) (math-known-scalarp a
)))
1318 (math-map-vec-2 'math-mul a b
))
1319 (and (Math-objectp b
) (not (Math-objectp a
))
1321 (and (eq (car-safe a
) 'neg
)
1322 (math-neg (math-mul (nth 1 a
) b
)))
1323 (and (eq (car-safe b
) 'neg
)
1324 (math-neg (math-mul a
(nth 1 b
))))
1325 (and (eq (car-safe a
) '*)
1327 (math-mul (nth 2 a
) b
)))
1328 (and (eq (car-safe a
) '^
)
1329 (Math-looks-negp (nth 2 a
))
1330 (not (and (eq (car-safe b
) '^
) (Math-looks-negp (nth 2 b
))))
1331 (math-known-scalarp b t
)
1332 (math-div b
(math-normalize
1333 (list '^
(nth 1 a
) (math-neg (nth 2 a
))))))
1334 (and (eq (car-safe b
) '^
)
1335 (Math-looks-negp (nth 2 b
))
1336 (not (and (eq (car-safe a
) '^
) (Math-looks-negp (nth 2 a
))))
1337 (math-div a
(math-normalize
1338 (list '^
(nth 1 b
) (math-neg (nth 2 b
))))))
1339 (and (eq (car-safe a
) '/)
1340 (or (math-known-scalarp a t
) (math-known-scalarp b t
))
1341 (let ((temp (math-combine-prod (nth 2 a
) b t nil t
)))
1343 (math-mul (nth 1 a
) temp
)
1344 (math-div (math-mul (nth 1 a
) b
) (nth 2 a
)))))
1345 (and (eq (car-safe b
) '/)
1346 (math-div (math-mul a
(nth 1 b
)) (nth 2 b
)))
1347 (and (eq (car-safe b
) '+)
1349 (or (Math-numberp (nth 1 b
))
1350 (Math-numberp (nth 2 b
)))
1351 (math-add (math-mul a
(nth 1 b
))
1352 (math-mul a
(nth 2 b
))))
1353 (and (eq (car-safe b
) '-
)
1355 (or (Math-numberp (nth 1 b
))
1356 (Math-numberp (nth 2 b
)))
1357 (math-sub (math-mul a
(nth 1 b
))
1358 (math-mul a
(nth 2 b
))))
1359 (and (eq (car-safe b
) '*)
1360 (Math-numberp (nth 1 b
))
1361 (not (Math-numberp a
))
1362 (math-mul (nth 1 b
) (math-mul a
(nth 2 b
))))
1363 (and (eq (car-safe a
) 'calcFunc-idn
)
1365 (or (and (eq (car-safe b
) 'calcFunc-idn
)
1367 (list 'calcFunc-idn
(math-mul (nth 1 a
) (nth 1 b
))))
1368 (and (math-known-scalarp b
)
1369 (list 'calcFunc-idn
(math-mul (nth 1 a
) b
)))
1370 (and (math-known-matrixp b
)
1371 (math-mul (nth 1 a
) b
))))
1372 (and (eq (car-safe b
) 'calcFunc-idn
)
1374 (or (and (math-known-scalarp a
)
1375 (list 'calcFunc-idn
(math-mul a
(nth 1 b
))))
1376 (and (math-known-matrixp a
)
1377 (math-mul a
(nth 1 b
)))))
1378 (and (math-looks-negp b
)
1379 (math-mul (math-neg a
) (math-neg b
)))
1380 (and (eq (car-safe b
) '-
)
1382 (math-mul (math-neg a
) (math-neg b
)))
1384 ((eq (car-safe b
) '*)
1385 (let ((temp (math-combine-prod a
(nth 1 b
) nil nil t
)))
1387 (math-mul temp
(nth 2 b
)))))
1389 (math-combine-prod a b nil nil nil
)))
1390 (and (equal a
'(var nan var-nan
))
1392 (and (equal b
'(var nan var-nan
))
1394 (and (equal a
'(var uinf var-uinf
))
1396 (and (equal b
'(var uinf var-uinf
))
1398 (and (equal b
'(var inf var-inf
))
1399 (let ((s1 (math-possible-signs a
)))
1403 '(intv 3 0 (var inf var-inf
)))
1407 '(intv 3 (neg (var inf var-inf
)) 0))
1408 ((and (eq (car a
) 'intv
) (math-intv-constp a
))
1409 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
)))
1410 ((and (eq (car a
) 'cplx
)
1411 (math-zerop (nth 1 a
)))
1412 (list '* (list 'cplx
0 (calcFunc-sign (nth 2 a
))) b
))
1413 ((eq (car a
) 'polar
)
1414 (list '* (list 'polar
1 (nth 2 a
)) b
)))))
1415 (and (equal a
'(var inf var-inf
))
1420 (defun calcFunc-div (a &rest rest
)
1422 (setq a
(list '/ a
(car rest
))
1426 (defun math-div-objects-fancy (a b
)
1427 (cond ((and (Math-numberp a
) (Math-numberp b
))
1429 (cond ((math-want-polar a b
)
1430 (let ((a (math-polar a
))
1433 (math-div (nth 1 a
) (nth 1 b
))
1434 (math-fix-circular (math-sub (nth 2 a
)
1437 (setq a
(math-complex a
))
1438 (list 'cplx
(math-div (nth 1 a
) b
)
1439 (math-div (nth 2 a
) b
)))
1441 (setq a
(math-complex a
)
1445 (math-add (math-mul (nth 1 a
) (nth 1 b
))
1446 (math-mul (nth 2 a
) (nth 2 b
)))
1447 (math-sub (math-mul (nth 2 a
) (nth 1 b
))
1448 (math-mul (nth 1 a
) (nth 2 b
))))
1449 (math-add (math-sqr (nth 1 b
))
1450 (math-sqr (nth 2 b
))))))))
1452 (if (math-square-matrixp b
)
1453 (let ((n1 (length b
)))
1454 (if (Math-vectorp a
)
1455 (if (math-matrixp a
)
1456 (if (= (length a
) n1
)
1457 (math-lud-solve (math-matrix-lud b
) a b
)
1458 (if (= (length (nth 1 a
)) n1
)
1460 (math-lud-solve (math-matrix-lud
1462 (math-transpose a
) b
))
1463 (math-dimension-error)))
1464 (if (= (length a
) n1
)
1465 (math-mat-col (math-lud-solve (math-matrix-lud b
)
1466 (math-col-matrix a
) b
)
1468 (math-dimension-error)))
1469 (if (Math-equal-int a
1)
1471 (math-mul a
(calcFunc-inv b
)))))
1472 (math-reject-arg b
'square-matrixp
)))
1473 ((and (Math-vectorp a
) (Math-objectp b
))
1474 (math-map-vec-2 'math-div a b
))
1475 ((eq (car-safe a
) 'sdev
)
1476 (if (eq (car-safe b
) 'sdev
)
1477 (let ((x (math-div (nth 1 a
) (nth 1 b
))))
1479 (math-div (math-hypot (nth 2 a
)
1480 (math-mul (nth 2 b
) x
))
1482 (if (or (Math-scalarp b
)
1483 (not (Math-objvecp b
)))
1484 (math-make-sdev (math-div (nth 1 a
) b
) (math-div (nth 2 a
) b
))
1485 (math-reject-arg 'realp b
))))
1486 ((and (eq (car-safe b
) 'sdev
)
1487 (or (Math-scalarp a
)
1488 (not (Math-objvecp a
))))
1489 (let ((x (math-div a
(nth 1 b
))))
1491 (math-div (math-mul (nth 2 b
) x
) (nth 1 b
)))))
1492 ((and (eq (car-safe a
) 'intv
) (Math-anglep b
))
1494 (math-neg (math-div a
(math-neg b
)))
1495 (math-make-intv (nth 1 a
)
1496 (math-div (nth 2 a
) b
)
1497 (math-div (nth 3 a
) b
))))
1498 ((and (eq (car-safe b
) 'intv
) (Math-anglep a
))
1499 (if (or (Math-posp (nth 2 b
))
1500 (and (Math-zerop (nth 2 b
)) (or (memq (nth 1 b
) '(0 1))
1501 calc-infinite-mode
)))
1503 (math-neg (math-div (math-neg a
) b
))
1504 (let ((calc-infinite-mode 1))
1505 (math-make-intv (aref [0 2 1 3] (nth 1 b
))
1506 (math-div a
(nth 3 b
))
1507 (math-div a
(nth 2 b
)))))
1508 (if (or (Math-negp (nth 3 b
))
1509 (and (Math-zerop (nth 3 b
)) (or (memq (nth 1 b
) '(0 2))
1510 calc-infinite-mode
)))
1511 (math-neg (math-div a
(math-neg b
)))
1512 (if calc-infinite-mode
1513 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1514 (math-reject-arg b
"*Division by zero")))))
1515 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
1516 (eq (car-safe b
) 'intv
) (math-intv-constp b
))
1517 (if (or (Math-posp (nth 2 b
))
1518 (and (Math-zerop (nth 2 b
)) (or (memq (nth 1 b
) '(0 1))
1519 calc-infinite-mode
)))
1520 (let* ((calc-infinite-mode 1)
1521 (lo (math-div a
(nth 2 b
)))
1522 (hi (math-div a
(nth 3 b
))))
1523 (or (eq (car-safe lo
) 'intv
)
1524 (setq lo
(list 'intv
(if (memq (nth 1 b
) '(2 3)) 3 0)
1526 (or (eq (car-safe hi
) 'intv
)
1527 (setq hi
(list 'intv
(if (memq (nth 1 b
) '(1 3)) 3 0)
1529 (math-combine-intervals
1530 (nth 2 lo
) (and (or (memq (nth 1 b
) '(2 3))
1531 (and (math-infinitep (nth 2 lo
))
1532 (not (math-zerop (nth 2 b
)))))
1533 (memq (nth 1 lo
) '(2 3)))
1534 (nth 3 lo
) (and (or (memq (nth 1 b
) '(2 3))
1535 (and (math-infinitep (nth 3 lo
))
1536 (not (math-zerop (nth 2 b
)))))
1537 (memq (nth 1 lo
) '(1 3)))
1538 (nth 2 hi
) (and (or (memq (nth 1 b
) '(1 3))
1539 (and (math-infinitep (nth 2 hi
))
1540 (not (math-zerop (nth 3 b
)))))
1541 (memq (nth 1 hi
) '(2 3)))
1542 (nth 3 hi
) (and (or (memq (nth 1 b
) '(1 3))
1543 (and (math-infinitep (nth 3 hi
))
1544 (not (math-zerop (nth 3 b
)))))
1545 (memq (nth 1 hi
) '(1 3)))))
1546 (if (or (Math-negp (nth 3 b
))
1547 (and (Math-zerop (nth 3 b
)) (or (memq (nth 1 b
) '(0 2))
1548 calc-infinite-mode
)))
1549 (math-neg (math-div a
(math-neg b
)))
1550 (if calc-infinite-mode
1551 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1552 (math-reject-arg b
"*Division by zero")))))
1553 ((and (eq (car-safe a
) 'mod
)
1554 (eq (car-safe b
) 'mod
)
1555 (equal (nth 2 a
) (nth 2 b
)))
1556 (math-make-mod (math-div-mod (nth 1 a
) (nth 1 b
) (nth 2 a
))
1558 ((and (eq (car-safe a
) 'mod
)
1560 (math-make-mod (math-div-mod (nth 1 a
) b
(nth 2 a
)) (nth 2 a
)))
1561 ((and (eq (car-safe b
) 'mod
)
1563 (math-make-mod (math-div-mod a
(nth 1 b
) (nth 2 b
)) (nth 2 b
)))
1564 ((eq (car-safe a
) 'hms
)
1565 (if (eq (car-safe b
) 'hms
)
1566 (math-with-extra-prec 1
1567 (math-div (math-from-hms a
'deg
)
1568 (math-from-hms b
'deg
)))
1569 (math-with-extra-prec 2
1570 (math-to-hms (math-div (math-from-hms a
'deg
) b
) 'deg
))))
1571 (t (calc-record-why "*Incompatible arguments for /" a b
))))
1573 (defun math-div-by-zero (a b
)
1574 (if (math-infinitep a
)
1575 (if (or (equal a
'(var nan var-nan
))
1576 (equal b
'(var uinf var-uinf
))
1577 (memq calc-infinite-mode
'(-1 1)))
1579 '(var uinf var-uinf
))
1580 (if calc-infinite-mode
1583 (if (eq calc-infinite-mode
1)
1584 (math-mul a
'(var inf var-inf
))
1585 (if (eq calc-infinite-mode -
1)
1586 (math-mul a
'(neg (var inf var-inf
)))
1587 (if (eq (car-safe a
) 'intv
)
1588 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1589 '(var uinf var-uinf
)))))
1590 (math-reject-arg a
"*Division by zero"))))
1592 (defun math-div-zero (a b
)
1593 (if (math-known-matrixp b
)
1594 (if (math-vectorp b
)
1595 (math-map-vec-2 'math-div a b
)
1596 (math-mimic-ident 0 b
))
1597 (if (equal b
'(var nan var-nan
))
1599 (if (and (eq (car-safe b
) 'intv
) (math-intv-constp b
)
1600 (not (math-posp b
)) (not (math-negp b
)))
1601 (if calc-infinite-mode
1603 (if (and (math-zerop (nth 2 b
))
1604 (memq calc-infinite-mode
'(1 -
1)))
1605 (nth 2 b
) '(neg (var inf var-inf
)))
1606 (if (and (math-zerop (nth 3 b
))
1607 (memq calc-infinite-mode
'(1 -
1)))
1608 (nth 3 b
) '(var inf var-inf
)))
1609 (math-reject-arg b
"*Division by zero"))
1612 ;; For math-div-symb-fancy
1613 (defvar math-trig-inverses
1614 '((calcFunc-sin . calcFunc-csc
)
1615 (calcFunc-cos . calcFunc-sec
)
1616 (calcFunc-tan . calcFunc-cot
)
1617 (calcFunc-sec . calcFunc-cos
)
1618 (calcFunc-csc . calcFunc-sin
)
1619 (calcFunc-cot . calcFunc-tan
)
1620 (calcFunc-sinh . calcFunc-csch
)
1621 (calcFunc-cosh . calcFunc-sech
)
1622 (calcFunc-tanh . calcFunc-coth
)
1623 (calcFunc-sech . calcFunc-cosh
)
1624 (calcFunc-csch . calcFunc-sinh
)
1625 (calcFunc-coth . calcFunc-tanh
)))
1627 (defvar math-div-trig
)
1628 (defvar math-div-non-trig
)
1630 (defun math-div-new-trig (tr)
1633 (list '* tr math-div-trig
))
1634 (setq math-div-trig tr
)))
1636 (defun math-div-new-non-trig (ntr)
1637 (if math-div-non-trig
1638 (setq math-div-non-trig
1639 (list '* ntr math-div-non-trig
))
1640 (setq math-div-non-trig ntr
)))
1642 (defun math-div-isolate-trig (expr)
1643 (if (eq (car-safe expr
) '*)
1645 (math-div-isolate-trig-term (nth 1 expr
))
1646 (math-div-isolate-trig (nth 2 expr
)))
1647 (math-div-isolate-trig-term expr
)))
1649 (defun math-div-isolate-trig-term (term)
1650 (let ((fn (assoc (car-safe term
) math-trig-inverses
)))
1653 (cons (cdr fn
) (cdr term
)))
1654 (math-div-new-non-trig term
))))
1656 (defun math-div-symb-fancy (a b
)
1657 (or (and math-simplify-only
1658 (not (equal a math-simplify-only
))
1660 (and (Math-equal-int b
1) a
)
1661 (and (Math-equal-int b -
1) (math-neg a
))
1662 (and (Math-vectorp a
) (math-known-scalarp b
)
1663 (math-map-vec-2 'math-div a b
))
1664 (and (eq (car-safe b
) '^
)
1665 (or (Math-looks-negp (nth 2 b
)) (Math-equal-int a
1))
1666 (math-mul a
(math-normalize
1667 (list '^
(nth 1 b
) (math-neg (nth 2 b
))))))
1668 (and (eq (car-safe a
) 'neg
)
1669 (math-neg (math-div (nth 1 a
) b
)))
1670 (and (eq (car-safe b
) 'neg
)
1671 (math-neg (math-div a
(nth 1 b
))))
1672 (and (eq (car-safe a
) '/)
1673 (math-div (nth 1 a
) (math-mul (nth 2 a
) b
)))
1674 (and (eq (car-safe b
) '/)
1675 (or (math-known-scalarp (nth 1 b
) t
)
1676 (math-known-scalarp (nth 2 b
) t
))
1677 (math-div (math-mul a
(nth 2 b
)) (nth 1 b
)))
1678 (and (eq (car-safe b
) 'frac
)
1679 (math-mul (math-make-frac (nth 2 b
) (nth 1 b
)) a
))
1680 (and (eq (car-safe a
) '+)
1681 (or (Math-numberp (nth 1 a
))
1682 (Math-numberp (nth 2 a
)))
1684 (math-add (math-div (nth 1 a
) b
)
1685 (math-div (nth 2 a
) b
)))
1686 (and (eq (car-safe a
) '-
)
1687 (or (Math-numberp (nth 1 a
))
1688 (Math-numberp (nth 2 a
)))
1690 (math-sub (math-div (nth 1 a
) b
)
1691 (math-div (nth 2 a
) b
)))
1692 (and (or (eq (car-safe a
) '-
)
1693 (math-looks-negp a
))
1695 (math-div (math-neg a
) (math-neg b
)))
1696 (and (eq (car-safe b
) '-
)
1698 (math-div (math-neg a
) (math-neg b
)))
1699 (and (eq (car-safe a
) 'calcFunc-idn
)
1701 (or (and (eq (car-safe b
) 'calcFunc-idn
)
1703 (list 'calcFunc-idn
(math-div (nth 1 a
) (nth 1 b
))))
1704 (and (math-known-scalarp b
)
1705 (list 'calcFunc-idn
(math-div (nth 1 a
) b
)))
1706 (and (math-known-matrixp b
)
1707 (math-div (nth 1 a
) b
))))
1708 (and (eq (car-safe b
) 'calcFunc-idn
)
1710 (or (and (math-known-scalarp a
)
1711 (list 'calcFunc-idn
(math-div a
(nth 1 b
))))
1712 (and (math-known-matrixp a
)
1713 (math-div a
(nth 1 b
)))))
1714 (and math-simplifying
1715 (let ((math-div-trig nil
)
1716 (math-div-non-trig nil
))
1717 (math-div-isolate-trig b
)
1719 (if math-div-non-trig
1720 (math-div (math-mul a math-div-trig
) math-div-non-trig
)
1721 (math-mul a math-div-trig
))
1723 (if (and calc-matrix-mode
1724 (or (math-known-matrixp a
) (math-known-matrixp b
)))
1725 (math-combine-prod a b nil t nil
)
1726 (if (eq (car-safe a
) '*)
1727 (if (eq (car-safe b
) '*)
1728 (let ((c (math-combine-prod (nth 1 a
) (nth 1 b
) nil t t
)))
1730 (math-div (math-mul c
(nth 2 a
)) (nth 2 b
))))
1731 (let ((c (math-combine-prod (nth 1 a
) b nil t t
)))
1733 (math-mul c
(nth 2 a
)))))
1734 (if (eq (car-safe b
) '*)
1735 (let ((c (math-combine-prod a
(nth 1 b
) nil t t
)))
1737 (math-div c
(nth 2 b
))))
1738 (math-combine-prod a b nil t nil
))))
1739 (and (math-infinitep a
)
1740 (if (math-infinitep b
)
1742 (if (or (equal a
'(var nan var-nan
))
1743 (equal a
'(var uinf var-uinf
)))
1745 (if (equal a
'(var inf var-inf
))
1746 (if (or (math-posp b
)
1747 (and (eq (car-safe b
) 'intv
)
1748 (math-zerop (nth 2 b
))))
1749 (if (and (eq (car-safe b
) 'intv
)
1750 (not (math-intv-constp b t
)))
1751 '(intv 3 0 (var inf var-inf
))
1753 (if (or (math-negp b
)
1754 (and (eq (car-safe b
) 'intv
)
1755 (math-zerop (nth 3 b
))))
1756 (if (and (eq (car-safe b
) 'intv
)
1757 (not (math-intv-constp b t
)))
1758 '(intv 3 (neg (var inf var-inf
)) 0)
1760 (if (and (eq (car-safe b
) 'intv
)
1761 (math-negp (nth 2 b
)) (math-posp (nth 3 b
)))
1762 '(intv 3 (neg (var inf var-inf
))
1763 (var inf var-inf
)))))))))
1764 (and (math-infinitep b
)
1765 (if (equal b
'(var nan var-nan
))
1767 (let ((calc-infinite-mode 1))
1768 (math-mul-zero b a
))))
1772 (defun calcFunc-mod (a b
)
1773 (math-normalize (list '% a b
)))
1775 (defun math-mod-fancy (a b
)
1776 (cond ((equal b
'(var inf var-inf
))
1777 (if (or (math-posp a
) (math-zerop a
))
1781 (if (eq (car-safe a
) 'intv
)
1782 (if (math-negp (nth 2 a
))
1783 '(intv 3 0 (var inf var-inf
))
1786 ((and (eq (car-safe a
) 'mod
) (Math-realp b
) (math-posp b
))
1787 (math-make-mod (nth 1 a
) b
))
1788 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a t
) (math-posp b
))
1789 (math-mod-intv a b
))
1792 (calc-record-why 'anglep b
)
1793 (calc-record-why 'anglep a
))
1797 (defun calcFunc-pow (a b
)
1798 (math-normalize (list '^ a b
)))
1800 (defun math-pow-of-zero (a b
)
1801 "Raise A to the power of B, where A is a form of zero."
1802 (if (math-floatp b
) (setq a
(math-float a
)))
1807 ;; 0^0.0, etc., are undetermined
1809 (if calc-infinite-mode
1811 (math-reject-arg (list '^ a b
) "*Indeterminate form")))
1813 ((math-known-posp b
)
1815 ;; 0^negative is undefined (let math-div handle it)
1816 ((math-known-negp b
)
1818 ;; 0^infinity is undefined
1822 ((and (eq (car b
) 'intv
)
1824 (math-negp (nth 2 b
))
1825 (math-posp (nth 3 b
)))
1826 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
)))
1827 ;; If none of the above, leave it alone.
1831 (defun math-pow-zero (a b
)
1832 (if (eq (car-safe a
) 'mod
)
1833 (math-make-mod 1 (nth 2 a
))
1834 (if (math-known-matrixp a
)
1835 (math-mimic-ident 1 a
)
1836 (if (math-infinitep a
)
1838 (if (and (eq (car a
) 'intv
) (math-intv-constp a
)
1839 (or (and (not (math-posp a
)) (not (math-negp a
)))
1840 (not (math-intv-constp a t
))))
1841 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1842 (if (or (math-floatp a
) (math-floatp b
))
1843 '(float 1 0) 1))))))
1845 (defun math-pow-fancy (a b
)
1846 (cond ((and (Math-numberp a
) (Math-numberp b
))
1847 (or (if (memq (math-quarter-integer b
) '(1 2 3))
1848 (let ((sqrt (math-sqrt (if (math-floatp b
)
1849 (math-float a
) a
))))
1850 (and (Math-numberp sqrt
)
1851 (math-pow sqrt
(math-mul 2 b
))))
1852 (and (eq (car b
) 'frac
)
1853 (integerp (nth 2 b
))
1855 (let ((root (math-nth-root a
(nth 2 b
))))
1856 (and root
(math-ipow root
(nth 1 b
))))))
1857 (and (or (eq a
10) (equal a
'(float 1 1)))
1858 (math-num-integerp b
)
1859 (calcFunc-scf '(float 1 0) b
))
1860 (and calc-symbolic-mode
1862 (math-with-extra-prec 2
1864 (math-float (math-mul b
(math-ln-raw (math-float a
))))))))
1865 ((or (not (Math-objvecp a
))
1866 (not (Math-objectp b
)))
1868 (cond ((and math-simplify-only
1869 (not (equal a math-simplify-only
)))
1871 ((and (eq (car-safe a
) '*)
1872 (or (math-known-num-integerp b
)
1873 (math-known-nonnegp (nth 1 a
))
1874 (math-known-nonnegp (nth 2 a
))))
1875 (math-mul (math-pow (nth 1 a
) b
)
1876 (math-pow (nth 2 a
) b
)))
1877 ((and (eq (car-safe a
) '/)
1878 (or (math-known-num-integerp b
)
1879 (math-known-nonnegp (nth 2 a
))))
1880 (math-div (math-pow (nth 1 a
) b
)
1881 (math-pow (nth 2 a
) b
)))
1882 ((and (eq (car-safe a
) '/)
1883 (math-known-nonnegp (nth 1 a
))
1884 (not (math-equal-int (nth 1 a
) 1)))
1885 (math-mul (math-pow (nth 1 a
) b
)
1886 (math-pow (math-div 1 (nth 2 a
)) b
)))
1887 ((and (eq (car-safe a
) '^
)
1888 (or (math-known-num-integerp b
)
1889 (math-known-nonnegp (nth 1 a
))))
1890 (math-pow (nth 1 a
) (math-mul (nth 2 a
) b
)))
1891 ((and (eq (car-safe a
) 'calcFunc-sqrt
)
1892 (or (math-known-num-integerp b
)
1893 (math-known-nonnegp (nth 1 a
))))
1894 (math-pow (nth 1 a
) (math-div b
2)))
1895 ((and (eq (car-safe a
) '^
)
1896 (math-known-evenp (nth 2 a
))
1897 (memq (math-quarter-integer b
) '(1 2 3))
1898 (math-known-realp (nth 1 a
)))
1899 (math-abs (math-pow (nth 1 a
) (math-mul (nth 2 a
) b
))))
1900 ((and (math-looks-negp a
)
1901 (math-known-integerp b
)
1902 (setq temp
(or (and (math-known-evenp b
)
1903 (math-pow (math-neg a
) b
))
1904 (and (math-known-oddp b
)
1905 (math-neg (math-pow (math-neg a
)
1908 ((and (eq (car-safe a
) 'calcFunc-abs
)
1909 (math-known-realp (nth 1 a
))
1910 (math-known-evenp b
))
1911 (math-pow (nth 1 a
) b
))
1913 (cond ((equal a
'(var nan var-nan
))
1916 (math-mul (math-pow -
1 b
) (math-pow (nth 1 a
) b
)))
1920 (if (math-floatp b
) '(float 0 0) 0))
1921 ((and (eq (car-safe b
) 'intv
)
1922 (math-intv-constp b
))
1923 '(intv 3 0 (var inf var-inf
)))
1925 '(var nan var-nan
))))
1928 (cond ((math-negp b
)
1929 (math-pow (math-div 1 a
) (math-neg b
)))
1930 ((not (math-posp b
))
1932 ((math-equal-int (setq scale
(calcFunc-abssqr a
)) 1)
1934 ((Math-lessp scale
1)
1935 (if (math-floatp a
) '(float 0 0) 0))
1939 '(var uinf var-uinf
))
1940 ((and (eq (car a
) 'intv
)
1941 (math-intv-constp a
))
1942 (if (Math-lessp -
1 a
)
1943 (if (math-equal-int (nth 3 a
) 1)
1945 '(intv 3 0 (var inf var-inf
)))
1946 '(intv 3 (neg (var inf var-inf
))
1947 (var inf var-inf
))))
1948 (t (list '^ a b
)))))
1949 ((and (eq (car-safe a
) 'calcFunc-idn
)
1951 (math-known-num-integerp b
))
1952 (list 'calcFunc-idn
(math-pow (nth 1 a
) b
)))
1953 (t (if (Math-objectp a
)
1954 (calc-record-why 'objectp b
)
1955 (calc-record-why 'objectp a
))
1957 ((and (eq (car-safe a
) 'sdev
) (eq (car-safe b
) 'sdev
))
1958 (if (and (math-constp a
) (math-constp b
))
1959 (math-with-extra-prec 2
1960 (let* ((ln (math-ln-raw (math-float (nth 1 a
))))
1962 (math-float (math-mul (nth 1 b
) ln
)))))
1967 (math-hypot (math-mul (nth 2 a
)
1968 (math-div (nth 1 b
) (nth 1 a
)))
1969 (math-mul (nth 2 b
) ln
))))))
1970 (let ((pow (math-pow (nth 1 a
) (nth 1 b
))))
1974 (math-hypot (math-mul (nth 2 a
)
1975 (math-div (nth 1 b
) (nth 1 a
)))
1976 (math-mul (nth 2 b
) (calcFunc-ln
1978 ((and (eq (car-safe a
) 'sdev
) (Math-numberp b
))
1980 (math-with-extra-prec 2
1981 (let ((pow (math-pow (nth 1 a
) (math-sub b
1))))
1982 (math-make-sdev (math-mul pow
(nth 1 a
))
1983 (math-mul pow
(math-mul (nth 2 a
) b
)))))
1984 (math-make-sdev (math-pow (nth 1 a
) b
)
1985 (math-mul (math-pow (nth 1 a
) (math-add b -
1))
1986 (math-mul (nth 2 a
) b
)))))
1987 ((and (eq (car-safe b
) 'sdev
) (Math-numberp a
))
1988 (math-with-extra-prec 2
1989 (let* ((ln (math-ln-raw (math-float a
)))
1990 (pow (calcFunc-exp (math-mul (nth 1 b
) ln
))))
1991 (math-make-sdev pow
(math-mul pow
(math-mul (nth 2 b
) ln
))))))
1992 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
1994 (or (Math-natnump b
)
1995 (Math-posp (nth 2 a
))
1996 (and (math-zerop (nth 2 a
))
1998 (and (Math-integerp b
) calc-infinite-mode
)))
1999 (Math-negp (nth 3 a
))
2000 (and (math-zerop (nth 3 a
))
2002 (and (Math-integerp b
) calc-infinite-mode
)))))
2004 (setq a
(math-abs a
)))
2005 (let ((calc-infinite-mode (if (math-zerop (nth 3 a
)) -
1 1)))
2006 (math-sort-intv (nth 1 a
)
2007 (math-pow (nth 2 a
) b
)
2008 (math-pow (nth 3 a
) b
))))
2009 ((and (eq (car-safe b
) 'intv
) (math-intv-constp b
)
2010 (Math-realp a
) (Math-posp a
))
2011 (math-sort-intv (nth 1 b
)
2012 (math-pow a
(nth 2 b
))
2013 (math-pow a
(nth 3 b
))))
2014 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
2015 (eq (car-safe b
) 'intv
) (math-intv-constp b
)
2016 (or (and (not (Math-negp (nth 2 a
)))
2017 (not (Math-negp (nth 2 b
))))
2018 (and (Math-posp (nth 2 a
))
2019 (not (Math-posp (nth 3 b
))))))
2020 (let ((lo (math-pow a
(nth 2 b
)))
2021 (hi (math-pow a
(nth 3 b
))))
2022 (or (eq (car-safe lo
) 'intv
)
2023 (setq lo
(list 'intv
(if (memq (nth 1 b
) '(2 3)) 3 0) lo lo
)))
2024 (or (eq (car-safe hi
) 'intv
)
2025 (setq hi
(list 'intv
(if (memq (nth 1 b
) '(1 3)) 3 0) hi hi
)))
2026 (math-combine-intervals
2027 (nth 2 lo
) (and (or (memq (nth 1 b
) '(2 3))
2028 (math-infinitep (nth 2 lo
)))
2029 (memq (nth 1 lo
) '(2 3)))
2030 (nth 3 lo
) (and (or (memq (nth 1 b
) '(2 3))
2031 (math-infinitep (nth 3 lo
)))
2032 (memq (nth 1 lo
) '(1 3)))
2033 (nth 2 hi
) (and (or (memq (nth 1 b
) '(1 3))
2034 (math-infinitep (nth 2 hi
)))
2035 (memq (nth 1 hi
) '(2 3)))
2036 (nth 3 hi
) (and (or (memq (nth 1 b
) '(1 3))
2037 (math-infinitep (nth 3 hi
)))
2038 (memq (nth 1 hi
) '(1 3))))))
2039 ((and (eq (car-safe a
) 'mod
) (eq (car-safe b
) 'mod
)
2040 (equal (nth 2 a
) (nth 2 b
)))
2041 (math-make-mod (math-pow-mod (nth 1 a
) (nth 1 b
) (nth 2 a
))
2043 ((and (eq (car-safe a
) 'mod
) (Math-anglep b
))
2044 (math-make-mod (math-pow-mod (nth 1 a
) b
(nth 2 a
)) (nth 2 a
)))
2045 ((and (eq (car-safe b
) 'mod
) (Math-anglep a
))
2046 (math-make-mod (math-pow-mod a
(nth 1 b
) (nth 2 b
)) (nth 2 b
)))
2047 ((not (Math-numberp a
))
2048 (math-reject-arg a
'numberp
))
2050 (math-reject-arg b
'numberp
))))
2052 (defun math-quarter-integer (x)
2053 (if (Math-integerp x
)
2057 (setq x
(math-quarter-integer (math-neg x
)))
2059 (if (eq (car x
) 'frac
)
2060 (if (eq (nth 2 x
) 2)
2062 (and (eq (nth 2 x
) 4)
2065 (%
(if (consp x
) (nth 1 x
) x
) 4))))
2066 (if (eq (car x
) 'float
)
2067 (if (>= (nth 2 x
) 0)
2069 (if (= (nth 2 x
) -
1)
2072 (and (= (%
(if (consp x
) (nth 1 x
) x
) 10) 5) 2))
2073 (if (= (nth 2 x
) -
2)
2076 x
(%
(if (consp x
) (nth 1 x
) x
) 100))
2078 (if (= x
75) 3)))))))))))
2080 ;;; This assumes A < M and M > 0.
2081 (defun math-pow-mod (a b m
) ; [R R R R]
2082 (if (and (Math-integerp a
) (Math-integerp b
) (Math-integerp m
))
2084 (math-div-mod 1 (math-pow-mod a
(Math-integer-neg b
) m
) m
)
2087 (math-pow-mod-step a b m
)))
2088 (math-mod (math-pow a b
) m
)))
2090 (defun math-pow-mod-step (a n m
) ; [I I I I]
2091 (math-working "pow" a
)
2096 (let ((rest (math-pow-mod-step
2097 (math-imod (math-mul a a
) m
)
2102 (math-mod (math-mul a rest
) m
)))))))
2103 (math-working "pow" val
)
2107 ;;; Compute the minimum of two real numbers. [R R R] [Public]
2108 (defun math-min (a b
)
2109 (if (and (consp a
) (eq (car a
) 'intv
))
2110 (if (and (consp b
) (eq (car b
) 'intv
))
2111 (let ((lo (nth 2 a
))
2112 (lom (memq (nth 1 a
) '(2 3)))
2114 (him (memq (nth 1 a
) '(1 3)))
2116 (if (= (setq res
(math-compare (nth 2 b
) lo
)) -
1)
2117 (setq lo
(nth 2 b
) lom
(memq (nth 1 b
) '(2 3)))
2119 (setq lom
(or lom
(memq (nth 1 b
) '(2 3))))))
2120 (if (= (setq res
(math-compare (nth 3 b
) hi
)) -
1)
2121 (setq hi
(nth 3 b
) him
(memq (nth 1 b
) '(1 3)))
2123 (setq him
(or him
(memq (nth 1 b
) '(1 3))))))
2124 (math-make-intv (+ (if lom
2 0) (if him
1 0)) lo hi
))
2125 (math-min a
(list 'intv
3 b b
)))
2126 (if (and (consp b
) (eq (car b
) 'intv
))
2127 (math-min (list 'intv
3 a a
) b
)
2128 (let ((res (math-compare a b
)))
2135 (defun calcFunc-min (&optional a
&rest b
)
2138 (if (not (or (Math-anglep a
) (eq (car a
) 'date
)
2139 (and (eq (car a
) 'intv
) (math-intv-constp a
))
2140 (math-infinitep a
)))
2141 (math-reject-arg a
'anglep
))
2142 (math-min-list a b
)))
2144 (defun math-min-list (a b
)
2146 (if (or (Math-anglep (car b
)) (eq (car b
) 'date
)
2147 (and (eq (car (car b
)) 'intv
) (math-intv-constp (car b
)))
2148 (math-infinitep (car b
)))
2149 (math-min-list (math-min a
(car b
)) (cdr b
))
2150 (math-reject-arg (car b
) 'anglep
))
2153 ;;; Compute the maximum of two real numbers. [R R R] [Public]
2154 (defun math-max (a b
)
2155 (if (or (and (consp a
) (eq (car a
) 'intv
))
2156 (and (consp b
) (eq (car b
) 'intv
)))
2157 (math-neg (math-min (math-neg a
) (math-neg b
)))
2158 (let ((res (math-compare a b
)))
2165 (defun calcFunc-max (&optional a
&rest b
)
2167 '(neg (var inf var-inf
))
2168 (if (not (or (Math-anglep a
) (eq (car a
) 'date
)
2169 (and (eq (car a
) 'intv
) (math-intv-constp a
))
2170 (math-infinitep a
)))
2171 (math-reject-arg a
'anglep
))
2172 (math-max-list a b
)))
2174 (defun math-max-list (a b
)
2176 (if (or (Math-anglep (car b
)) (eq (car b
) 'date
)
2177 (and (eq (car (car b
)) 'intv
) (math-intv-constp (car b
)))
2178 (math-infinitep (car b
)))
2179 (math-max-list (math-max a
(car b
)) (cdr b
))
2180 (math-reject-arg (car b
) 'anglep
))
2184 ;;; Compute the absolute value of A. [O O; r r] [Public]
2186 (cond ((Math-negp a
)
2191 (math-hypot (nth 1 a
) (nth 2 a
)))
2192 ((eq (car a
) 'polar
)
2195 (if (cdr (cdr (cdr a
)))
2196 (math-sqrt (calcFunc-abssqr a
))
2198 (math-hypot (nth 1 a
) (nth 2 a
))
2200 (math-abs (nth 1 a
))
2203 (list 'sdev
(math-abs (nth 1 a
)) (nth 2 a
)))
2204 ((and (eq (car a
) 'intv
) (math-intv-constp a
))
2207 (let* ((nlo (math-neg (nth 2 a
)))
2208 (res (math-compare nlo
(nth 3 a
))))
2210 (math-make-intv (if (memq (nth 1 a
) '(0 1)) 2 3) 0 nlo
))
2212 (math-make-intv (if (eq (nth 1 a
) 0) 2 3) 0 nlo
))
2214 (math-make-intv (if (memq (nth 1 a
) '(0 2)) 2 3)
2216 ((math-looks-negp a
)
2217 (list 'calcFunc-abs
(math-neg a
)))
2218 ((let ((signs (math-possible-signs a
)))
2219 (or (and (memq signs
'(2 4 6)) a
)
2220 (and (memq signs
'(1 3)) (math-neg a
)))))
2221 ((let ((inf (math-infinitep a
)))
2223 (if (equal inf
'(var nan var-nan
))
2225 '(var inf var-inf
)))))
2226 (t (calc-record-why 'numvecp a
)
2227 (list 'calcFunc-abs a
))))
2229 (defalias 'calcFunc-abs
'math-abs
)
2231 (defun math-float-fancy (a)
2232 (cond ((eq (car a
) 'intv
)
2233 (cons (car a
) (cons (nth 1 a
) (mapcar 'math-float
(nthcdr 2 a
)))))
2234 ((and (memq (car a
) '(* /))
2235 (math-numberp (nth 1 a
)))
2236 (list (car a
) (math-float (nth 1 a
))
2237 (list 'calcFunc-float
(nth 2 a
))))
2238 ((and (eq (car a
) '/)
2239 (eq (car (nth 1 a
)) '*)
2240 (math-numberp (nth 1 (nth 1 a
))))
2241 (list '* (math-float (nth 1 (nth 1 a
)))
2242 (list 'calcFunc-float
(list '/ (nth 2 (nth 1 a
)) (nth 2 a
)))))
2243 ((math-infinitep a
) a
)
2244 ((eq (car a
) 'calcFunc-float
) a
)
2245 ((let ((func (assq (car a
) '((calcFunc-floor . calcFunc-ffloor
)
2246 (calcFunc-ceil . calcFunc-fceil
)
2247 (calcFunc-trunc . calcFunc-ftrunc
)
2248 (calcFunc-round . calcFunc-fround
)
2249 (calcFunc-rounde . calcFunc-frounde
)
2250 (calcFunc-roundu . calcFunc-froundu
)))))
2251 (and func
(cons (cdr func
) (cdr a
)))))
2252 (t (math-reject-arg a
'objectp
))))
2254 (defalias 'calcFunc-float
'math-float
)
2256 ;; The variable math-trunc-prec is local to math-trunc in calc-misc.el,
2257 ;; but used by math-trunc-fancy which is called by math-trunc.
2258 (defvar math-trunc-prec
)
2260 (defun math-trunc-fancy (a)
2261 (cond ((eq (car a
) 'frac
) (math-quotient (nth 1 a
) (nth 2 a
)))
2262 ((eq (car a
) 'cplx
) (math-trunc (nth 1 a
)))
2263 ((eq (car a
) 'polar
) (math-trunc (math-complex a
)))
2264 ((eq (car a
) 'hms
) (list 'hms
(nth 1 a
) 0 0))
2265 ((eq (car a
) 'date
) (list 'date
(math-trunc (nth 1 a
))))
2267 (if (math-messy-integerp (nth 2 a
))
2268 (math-trunc (math-make-mod (nth 1 a
) (math-trunc (nth 2 a
))))
2269 (math-make-mod (math-trunc (nth 1 a
)) (nth 2 a
))))
2271 (math-make-intv (+ (if (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
2272 (memq (nth 1 a
) '(0 1)))
2274 (if (and (equal (nth 3 a
) '(var inf var-inf
))
2275 (memq (nth 1 a
) '(0 2)))
2277 (if (and (Math-negp (nth 2 a
))
2278 (Math-num-integerp (nth 2 a
))
2279 (memq (nth 1 a
) '(0 1)))
2280 (math-add (math-trunc (nth 2 a
)) 1)
2281 (math-trunc (nth 2 a
)))
2282 (if (and (Math-posp (nth 3 a
))
2283 (Math-num-integerp (nth 3 a
))
2284 (memq (nth 1 a
) '(0 2)))
2285 (math-add (math-trunc (nth 3 a
)) -
1)
2286 (math-trunc (nth 3 a
)))))
2287 ((math-provably-integerp a
) a
)
2289 (math-map-vec (function (lambda (x) (math-trunc x math-trunc-prec
))) a
))
2291 (if (or (math-posp a
) (math-negp a
))
2293 '(var nan var-nan
)))
2294 ((math-to-integer a
))
2295 (t (math-reject-arg a
'numberp
))))
2297 (defun math-trunc-special (a prec
)
2298 (if (Math-messy-integerp prec
)
2299 (setq prec
(math-trunc prec
)))
2301 (math-reject-arg prec
'fixnump
))
2302 (if (and (<= prec
0)
2303 (math-provably-integerp a
))
2305 (calcFunc-scf (math-trunc (let ((calc-prefer-frac t
))
2306 (calcFunc-scf a prec
)))
2309 (defun math-to-integer (a)
2310 (let ((func (assq (car-safe a
) '((calcFunc-ffloor . calcFunc-floor
)
2311 (calcFunc-fceil . calcFunc-ceil
)
2312 (calcFunc-ftrunc . calcFunc-trunc
)
2313 (calcFunc-fround . calcFunc-round
)
2314 (calcFunc-frounde . calcFunc-rounde
)
2315 (calcFunc-froundu . calcFunc-roundu
)))))
2316 (and func
(= (length a
) 2)
2317 (cons (cdr func
) (cdr a
)))))
2319 (defun calcFunc-ftrunc (a &optional prec
)
2320 (if (and (Math-messy-integerp a
)
2321 (or (not prec
) (and (integerp prec
)
2324 (math-float (math-trunc a prec
))))
2326 ;; The variable math-floor-prec is local to math-floor in calc-misc.el,
2327 ;; but used by math-floor-fancy which is called by math-floor.
2328 (defvar math-floor-prec
)
2330 (defun math-floor-fancy (a)
2331 (cond ((math-provably-integerp a
) a
)
2333 (if (or (math-posp a
)
2334 (and (math-zerop (nth 2 a
))
2335 (math-zerop (nth 3 a
))))
2337 (math-add (math-trunc a
) -
1)))
2338 ((eq (car a
) 'date
) (list 'date
(math-floor (nth 1 a
))))
2340 (math-make-intv (+ (if (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
2341 (memq (nth 1 a
) '(0 1)))
2343 (if (and (equal (nth 3 a
) '(var inf var-inf
))
2344 (memq (nth 1 a
) '(0 2)))
2346 (math-floor (nth 2 a
))
2347 (if (and (Math-num-integerp (nth 3 a
))
2348 (memq (nth 1 a
) '(0 2)))
2349 (math-add (math-floor (nth 3 a
)) -
1)
2350 (math-floor (nth 3 a
)))))
2352 (math-map-vec (function (lambda (x) (math-floor x math-floor-prec
))) a
))
2354 (if (or (math-posp a
) (math-negp a
))
2356 '(var nan var-nan
)))
2357 ((math-to-integer a
))
2358 (t (math-reject-arg a
'anglep
))))
2360 (defun math-floor-special (a prec
)
2361 (if (Math-messy-integerp prec
)
2362 (setq prec
(math-trunc prec
)))
2364 (math-reject-arg prec
'fixnump
))
2365 (if (and (<= prec
0)
2366 (math-provably-integerp a
))
2368 (calcFunc-scf (math-floor (let ((calc-prefer-frac t
))
2369 (calcFunc-scf a prec
)))
2372 (defun calcFunc-ffloor (a &optional prec
)
2373 (if (and (Math-messy-integerp a
)
2374 (or (not prec
) (and (integerp prec
)
2377 (math-float (math-floor a prec
))))
2379 ;;; Coerce A to be an integer (by truncation toward plus infinity). [I N]
2380 (defun math-ceiling (a &optional prec
) ; [Public]
2382 (if (Math-messy-integerp prec
)
2383 (setq prec
(math-trunc prec
)))
2385 (math-reject-arg prec
'fixnump
))
2386 (if (and (<= prec
0)
2387 (math-provably-integerp a
))
2389 (calcFunc-scf (math-ceiling (let ((calc-prefer-frac t
))
2390 (calcFunc-scf a prec
)))
2392 ((Math-integerp a
) a
)
2393 ((Math-messy-integerp a
) (math-trunc a
))
2396 (math-add (math-trunc a
) 1)
2398 ((math-provably-integerp a
) a
)
2400 (if (or (math-negp a
)
2401 (and (math-zerop (nth 2 a
))
2402 (math-zerop (nth 3 a
))))
2404 (math-add (math-trunc a
) 1)))
2405 ((eq (car a
) 'date
) (list 'date
(math-ceiling (nth 1 a
))))
2407 (math-make-intv (+ (if (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
2408 (memq (nth 1 a
) '(0 1)))
2410 (if (and (equal (nth 3 a
) '(var inf var-inf
))
2411 (memq (nth 1 a
) '(0 2)))
2413 (if (and (Math-num-integerp (nth 2 a
))
2414 (memq (nth 1 a
) '(0 1)))
2415 (math-add (math-floor (nth 2 a
)) 1)
2416 (math-ceiling (nth 2 a
)))
2417 (math-ceiling (nth 3 a
))))
2419 (math-map-vec (function (lambda (x) (math-ceiling x prec
))) a
))
2421 (if (or (math-posp a
) (math-negp a
))
2423 '(var nan var-nan
)))
2424 ((math-to-integer a
))
2425 (t (math-reject-arg a
'anglep
))))
2427 (defalias 'calcFunc-ceil
'math-ceiling
)
2429 (defun calcFunc-fceil (a &optional prec
)
2430 (if (and (Math-messy-integerp a
)
2431 (or (not prec
) (and (integerp prec
)
2434 (math-float (math-ceiling a prec
))))
2436 (defvar math-rounding-mode nil
)
2438 ;;; Coerce A to be an integer (by rounding to nearest integer). [I N] [Public]
2439 (defun math-round (a &optional prec
)
2441 (if (Math-messy-integerp prec
)
2442 (setq prec
(math-trunc prec
)))
2444 (math-reject-arg prec
'fixnump
))
2445 (if (and (<= prec
0)
2446 (math-provably-integerp a
))
2448 (calcFunc-scf (math-round (let ((calc-prefer-frac t
))
2449 (calcFunc-scf a prec
)))
2452 (if (Math-num-integerp a
)
2454 (if (and (Math-negp a
) (not (eq math-rounding-mode
'up
)))
2455 (math-neg (math-round (math-neg a
)))
2456 (setq a
(let ((calc-angle-mode 'deg
)) ; in case of HMS forms
2457 (math-add a
(if (Math-ratp a
)
2460 (if (and (Math-num-integerp a
) (eq math-rounding-mode
'even
))
2462 (setq a
(math-floor a
))
2464 (setq a
(math-sub a
1)))
2467 ((math-provably-integerp a
) a
)
2468 ((eq (car a
) 'date
) (list 'date
(math-round (nth 1 a
))))
2470 (math-floor (math-add a
'(frac 1 2))))
2472 (math-map-vec (function (lambda (x) (math-round x prec
))) a
))
2474 (if (or (math-posp a
) (math-negp a
))
2476 '(var nan var-nan
)))
2477 ((math-to-integer a
))
2478 (t (math-reject-arg a
'anglep
))))
2480 (defalias 'calcFunc-round
'math-round
)
2482 (defsubst calcFunc-rounde
(a &optional prec
)
2483 (let ((math-rounding-mode 'even
))
2484 (math-round a prec
)))
2486 (defsubst calcFunc-roundu
(a &optional prec
)
2487 (let ((math-rounding-mode 'up
))
2488 (math-round a prec
)))
2490 (defun calcFunc-fround (a &optional prec
)
2491 (if (and (Math-messy-integerp a
)
2492 (or (not prec
) (and (integerp prec
)
2495 (math-float (math-round a prec
))))
2497 (defsubst calcFunc-frounde
(a &optional prec
)
2498 (let ((math-rounding-mode 'even
))
2499 (calcFunc-fround a prec
)))
2501 (defsubst calcFunc-froundu
(a &optional prec
)
2502 (let ((math-rounding-mode 'up
))
2503 (calcFunc-fround a prec
)))
2505 ;;; Pull floating-point values apart into mantissa and exponent.
2506 (defun calcFunc-mant (x)
2508 (if (or (Math-ratp x
)
2511 (list 'float
(nth 1 x
) (- 1 (math-numdigs (nth 1 x
)))))
2512 (calc-record-why 'realp x
)
2513 (list 'calcFunc-mant x
)))
2515 (defun calcFunc-xpon (x)
2517 (if (or (Math-ratp x
)
2520 (math-normalize (+ (nth 2 x
) (1- (math-numdigs (nth 1 x
))))))
2521 (calc-record-why 'realp x
)
2522 (list 'calcFunc-xpon x
)))
2524 (defun calcFunc-scf (x n
)
2530 (math-scale-int x n
)
2531 (math-div x
(math-scale-int 1 (- n
)))))
2534 (math-make-frac (math-scale-int (nth 1 x
) n
) (nth 2 x
))
2535 (math-make-frac (nth 1 x
) (math-scale-int (nth 2 x
) (- n
)))))
2536 ((eq (car x
) 'float
)
2537 (math-make-float (nth 1 x
) (+ (nth 2 x
) n
)))
2538 ((memq (car x
) '(cplx sdev
))
2541 (calcFunc-scf (nth 1 x
) n
)
2542 (calcFunc-scf (nth 2 x
) n
))))
2543 ((memq (car x
) '(polar mod
))
2546 (calcFunc-scf (nth 1 x
) n
)
2552 (calcFunc-scf (nth 2 x
) n
)
2553 (calcFunc-scf (nth 3 x
) n
))))
2555 (math-map-vec (function (lambda (x) (calcFunc-scf x n
))) x
))
2559 (calc-record-why 'realp x
)
2560 (list 'calcFunc-scf x n
)))
2561 (if (math-messy-integerp n
)
2562 (if (< (nth 2 n
) 10)
2563 (calcFunc-scf x
(math-trunc n
))
2565 (if (math-integerp n
)
2567 (calc-record-why 'integerp n
)
2568 (list 'calcFunc-scf x n
)))))
2571 (defun calcFunc-incr (x &optional step relative-to
)
2572 (or step
(setq step
1))
2573 (cond ((not (Math-integerp step
))
2574 (math-reject-arg step
'integerp
))
2577 ((eq (car x
) 'float
)
2578 (if (and (math-zerop x
)
2579 (eq (car-safe relative-to
) 'float
))
2581 (calcFunc-scf relative-to
(- 1 calc-internal-prec
)))
2582 (math-add-float x
(math-make-float
2585 (- (math-numdigs (nth 1 x
))
2586 calc-internal-prec
))))))
2588 (if (Math-integerp (nth 1 x
))
2590 (math-add x
(list 'hms
0 0 step
))))
2592 (math-reject-arg x
'realp
))))
2594 (defsubst calcFunc-decr
(x &optional step relative-to
)
2595 (calcFunc-incr x
(math-neg (or step
1)) relative-to
))
2597 (defun calcFunc-percent (x)
2598 (if (math-objectp x
)
2599 (let ((calc-prefer-frac nil
))
2601 (list 'calcFunc-percent x
)))
2603 (defun calcFunc-relch (x y
)
2604 (if (and (math-objectp x
) (math-objectp y
))
2605 (math-div (math-sub y x
) x
)
2606 (list 'calcFunc-relch x y
)))
2608 ;;; Compute the absolute value squared of A. [F N] [Public]
2609 (defun calcFunc-abssqr (a)
2610 (cond ((Math-realp a
)
2613 (math-add (math-sqr (nth 1 a
))
2614 (math-sqr (nth 2 a
))))
2615 ((eq (car a
) 'polar
)
2616 (math-sqr (nth 1 a
)))
2617 ((and (memq (car a
) '(sdev intv
)) (math-constp a
))
2618 (math-sqr (math-abs a
)))
2620 (math-reduce-vec 'math-add
(math-map-vec 'calcFunc-abssqr a
)))
2621 ((math-known-realp a
)
2623 ((let ((inf (math-infinitep a
)))
2625 (math-mul (calcFunc-abssqr (math-infinite-dir a inf
)) inf
))))
2626 (t (calc-record-why 'numvecp a
)
2627 (list 'calcFunc-abssqr a
))))
2629 (defsubst math-sqr
(a)
2634 (defun calcFunc-idiv (a b
) ; [I I I] [Public]
2635 (cond ((and (Math-natnump a
) (Math-natnump b
) (not (eq b
0)))
2636 (math-quotient a b
))
2639 (let ((calc-prefer-frac t
))
2640 (math-floor (math-div a b
)))
2641 (math-reject-arg b
'realp
)))
2642 ((eq (car-safe a
) 'hms
)
2643 (if (eq (car-safe b
) 'hms
)
2644 (let ((calc-prefer-frac t
))
2645 (math-floor (math-div a b
)))
2646 (math-reject-arg b
'hmsp
)))
2647 ((and (or (eq (car-safe a
) 'intv
) (Math-realp a
))
2648 (or (eq (car-safe b
) 'intv
) (Math-realp b
)))
2649 (math-floor (math-div a b
)))
2650 ((or (math-infinitep a
)
2653 (t (math-reject-arg a
'anglep
))))
2656 ;;; Combine two terms being added, if possible.
2657 (defun math-combine-sum (a b nega negb scalar-okay
)
2658 (if (and scalar-okay
(Math-objvecp a
) (Math-objvecp b
))
2659 (math-add-or-sub a b nega negb
)
2660 (let ((amult 1) (bmult 1))
2662 (cond ((and (eq (car a
) '*)
2663 (Math-objectp (nth 1 a
)))
2664 (setq amult
(nth 1 a
)
2666 ((and (eq (car a
) '/)
2667 (Math-objectp (nth 2 a
)))
2668 (setq amult
(if (Math-integerp (nth 2 a
))
2669 (list 'frac
1 (nth 2 a
))
2670 (math-div 1 (nth 2 a
)))
2676 (cond ((and (eq (car b
) '*)
2677 (Math-objectp (nth 1 b
)))
2678 (setq bmult
(nth 1 b
)
2680 ((and (eq (car b
) '/)
2681 (Math-objectp (nth 2 b
)))
2682 (setq bmult
(if (Math-integerp (nth 2 b
))
2683 (list 'frac
1 (nth 2 b
))
2684 (math-div 1 (nth 2 b
)))
2689 (and (if math-simplifying
2693 (if nega
(setq amult
(math-neg amult
)))
2694 (if negb
(setq bmult
(math-neg bmult
)))
2695 (setq amult
(math-add amult bmult
))
2696 (math-mul amult a
))))))
2698 (defun math-add-or-sub (a b aneg bneg
)
2699 (if aneg
(setq a
(math-neg a
)))
2700 (if bneg
(setq b
(math-neg b
)))
2701 (if (or (Math-vectorp a
) (Math-vectorp b
))
2702 (math-normalize (list '+ a b
))
2705 (defvar math-combine-prod-e
'(var e var-e
))
2707 ;;; The following is expanded out four ways for speed.
2709 ;; math-unit-prefixes is defined in calc-units.el,
2711 (defvar math-unit-prefixes
)
2713 (defun math-combine-prod (a b inva invb scalar-okay
)
2715 ((or (and inva
(Math-zerop a
))
2716 (and invb
(Math-zerop b
)))
2718 ((and scalar-okay
(Math-objvecp a
) (Math-objvecp b
))
2719 (setq a
(math-mul-or-div a b inva invb
))
2720 (and (Math-objvecp a
)
2722 ((and (eq (car-safe a
) '^
)
2724 (math-looks-negp (nth 2 a
)))
2725 (math-mul (math-pow (nth 1 a
) (math-neg (nth 2 a
))) b
))
2726 ((and (eq (car-safe b
) '^
)
2728 (math-looks-negp (nth 2 b
)))
2729 (math-mul a
(math-pow (nth 1 b
) (math-neg (nth 2 b
)))))
2730 ((and math-simplifying
2731 (math-combine-prod-trig a b
)))
2732 (t (let ((apow 1) (bpow 1))
2734 (cond ((and (eq (car a
) '^
)
2735 (or math-simplifying
2736 (Math-numberp (nth 2 a
))))
2737 (setq apow
(nth 2 a
)
2739 ((eq (car a
) 'calcFunc-sqrt
)
2740 (setq apow
'(frac 1 2)
2742 ((and (eq (car a
) 'calcFunc-exp
)
2743 (or math-simplifying
2744 (Math-numberp (nth 1 a
))))
2745 (setq apow
(nth 1 a
)
2746 a math-combine-prod-e
))))
2747 (and (consp a
) (eq (car a
) 'frac
)
2748 (Math-lessp (nth 1 a
) (nth 2 a
))
2749 (setq a
(math-div 1 a
) apow
(math-neg apow
)))
2751 (cond ((and (eq (car b
) '^
)
2752 (or math-simplifying
2753 (Math-numberp (nth 2 b
))))
2754 (setq bpow
(nth 2 b
)
2756 ((eq (car b
) 'calcFunc-sqrt
)
2757 (setq bpow
'(frac 1 2)
2759 ((and (eq (car b
) 'calcFunc-exp
)
2760 (or math-simplifying
2761 (Math-numberp (nth 1 b
))))
2762 (setq bpow
(nth 1 b
)
2763 b math-combine-prod-e
))))
2764 (and (consp b
) (eq (car b
) 'frac
)
2765 (Math-lessp (nth 1 b
) (nth 2 b
))
2766 (setq b
(math-div 1 b
) bpow
(math-neg bpow
)))
2767 (if inva
(setq apow
(math-neg apow
)))
2768 (if invb
(setq bpow
(math-neg bpow
)))
2769 (or (and (if math-simplifying
2770 (math-commutative-equal a b
)
2772 (let ((sumpow (math-add apow bpow
)))
2773 (and (or (not (Math-integerp a
))
2775 (eq (eq (car-safe apow
) 'frac
)
2776 (eq (car-safe bpow
) 'frac
)))
2778 (and (math-looks-negp sumpow
)
2779 (Math-ratp a
) (Math-posp a
)
2780 (setq a
(math-div 1 a
)
2781 sumpow
(math-neg sumpow
)))
2782 (cond ((equal sumpow
'(frac 1 2))
2783 (list 'calcFunc-sqrt a
))
2784 ((equal sumpow
'(frac -
1 2))
2785 (math-div 1 (list 'calcFunc-sqrt a
)))
2786 ((and (eq a math-combine-prod-e
)
2788 (list 'calcFunc-exp sumpow
))
2792 (inexact-result (list '^ a sumpow
)))))))))
2793 (and math-simplifying-units
2794 math-combining-units
2795 (let* ((ua (math-check-unit-name a
))
2798 (eq ua
(setq ub
(math-check-unit-name b
)))
2800 (setq ua
(if (eq (nth 1 a
) (car ua
))
2802 (nth 1 (assq (aref (symbol-name (nth 1 a
))
2804 math-unit-prefixes
)))
2805 ub
(if (eq (nth 1 b
) (car ub
))
2807 (nth 1 (assq (aref (symbol-name (nth 1 b
))
2809 math-unit-prefixes
))))
2810 (if (Math-lessp ua ub
)
2812 (setq temp a a b b temp
2813 temp ua ua ub ub temp
2814 temp apow apow bpow bpow temp
)))
2815 (math-mul (math-pow (math-div ua ub
) apow
)
2816 (math-pow b
(math-add apow bpow
)))))))
2817 (and (equal apow bpow
)
2818 (Math-natnump a
) (Math-natnump b
)
2819 (cond ((equal apow
'(frac 1 2))
2820 (list 'calcFunc-sqrt
(math-mul a b
)))
2821 ((equal apow
'(frac -
1 2))
2822 (math-div 1 (list 'calcFunc-sqrt
(math-mul a b
))))
2824 (setq a
(math-mul a b
))
2827 (inexact-result (list '^ a apow
)))))))))))
2829 (defun math-combine-prod-trig (a b
)
2831 ((and (eq (car-safe a
) 'calcFunc-sin
)
2832 (eq (car-safe b
) 'calcFunc-csc
)
2833 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2835 ((and (eq (car-safe a
) 'calcFunc-sin
)
2836 (eq (car-safe b
) 'calcFunc-sec
)
2837 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2838 (cons 'calcFunc-tan
(cdr a
)))
2839 ((and (eq (car-safe a
) 'calcFunc-sin
)
2840 (eq (car-safe b
) 'calcFunc-cot
)
2841 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2842 (cons 'calcFunc-cos
(cdr a
)))
2843 ((and (eq (car-safe a
) 'calcFunc-cos
)
2844 (eq (car-safe b
) 'calcFunc-sec
)
2845 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2847 ((and (eq (car-safe a
) 'calcFunc-cos
)
2848 (eq (car-safe b
) 'calcFunc-csc
)
2849 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2850 (cons 'calcFunc-cot
(cdr a
)))
2851 ((and (eq (car-safe a
) 'calcFunc-cos
)
2852 (eq (car-safe b
) 'calcFunc-tan
)
2853 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2854 (cons 'calcFunc-sin
(cdr a
)))
2855 ((and (eq (car-safe a
) 'calcFunc-tan
)
2856 (eq (car-safe b
) 'calcFunc-cot
)
2857 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2859 ((and (eq (car-safe a
) 'calcFunc-tan
)
2860 (eq (car-safe b
) 'calcFunc-csc
)
2861 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2862 (cons 'calcFunc-sec
(cdr a
)))
2863 ((and (eq (car-safe a
) 'calcFunc-sec
)
2864 (eq (car-safe b
) 'calcFunc-cot
)
2865 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2866 (cons 'calcFunc-csc
(cdr a
)))
2867 ((and (eq (car-safe a
) 'calcFunc-sinh
)
2868 (eq (car-safe b
) 'calcFunc-csch
)
2869 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2871 ((and (eq (car-safe a
) 'calcFunc-sinh
)
2872 (eq (car-safe b
) 'calcFunc-sech
)
2873 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2874 (cons 'calcFunc-tanh
(cdr a
)))
2875 ((and (eq (car-safe a
) 'calcFunc-sinh
)
2876 (eq (car-safe b
) 'calcFunc-coth
)
2877 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2878 (cons 'calcFunc-cosh
(cdr a
)))
2879 ((and (eq (car-safe a
) 'calcFunc-cosh
)
2880 (eq (car-safe b
) 'calcFunc-sech
)
2881 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2883 ((and (eq (car-safe a
) 'calcFunc-cosh
)
2884 (eq (car-safe b
) 'calcFunc-csch
)
2885 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2886 (cons 'calcFunc-coth
(cdr a
)))
2887 ((and (eq (car-safe a
) 'calcFunc-cosh
)
2888 (eq (car-safe b
) 'calcFunc-tanh
)
2889 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2890 (cons 'calcFunc-sinh
(cdr a
)))
2891 ((and (eq (car-safe a
) 'calcFunc-tanh
)
2892 (eq (car-safe b
) 'calcFunc-coth
)
2893 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2895 ((and (eq (car-safe a
) 'calcFunc-tanh
)
2896 (eq (car-safe b
) 'calcFunc-csch
)
2897 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2898 (cons 'calcFunc-sech
(cdr a
)))
2899 ((and (eq (car-safe a
) 'calcFunc-sech
)
2900 (eq (car-safe b
) 'calcFunc-coth
)
2901 (= 0 (math-simplify (math-sub (cdr a
) (cdr b
)))))
2902 (cons 'calcFunc-csch
(cdr a
)))
2906 (defun math-mul-or-div (a b ainv binv
)
2907 (if (or (Math-vectorp a
) (Math-vectorp b
))
2911 (list '/ (math-div 1 a
) b
)
2918 (math-div (math-div 1 a
) b
)
2924 ;; The variable math-com-bterms is local to math-commutative-equal,
2925 ;; but is used by math-commutative collect, which is called by
2926 ;; math-commutative-equal.
2927 (defvar math-com-bterms
)
2929 (defun math-commutative-equal (a b
)
2930 (if (memq (car-safe a
) '(+ -
))
2931 (and (memq (car-safe b
) '(+ -
))
2932 (let ((math-com-bterms nil
) aterms p
)
2933 (math-commutative-collect b nil
)
2934 (setq aterms math-com-bterms math-com-bterms nil
)
2935 (math-commutative-collect a nil
)
2936 (and (= (length aterms
) (length math-com-bterms
))
2940 (setq p math-com-bterms
)
2941 (while (and p
(not (equal (car aterms
)
2945 (setq math-com-bterms
(delq (car p
) math-com-bterms
)
2946 aterms
(cdr aterms
)))
2950 (defun math-commutative-collect (b neg
)
2951 (if (eq (car-safe b
) '+)
2953 (math-commutative-collect (nth 1 b
) neg
)
2954 (math-commutative-collect (nth 2 b
) neg
))
2955 (if (eq (car-safe b
) '-
)
2957 (math-commutative-collect (nth 1 b
) neg
)
2958 (math-commutative-collect (nth 2 b
) (not neg
)))
2959 (setq math-com-bterms
(cons (if neg
(math-neg b
) b
) math-com-bterms
)))))
2961 (provide 'calc-arith
)
2963 ;;; arch-tag: 6c396b5b-14c6-40ed-bb2a-7cc2e8111465
2964 ;;; calc-arith.el ends here
HCoop / bpt / emacs.git / blob