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 ;; Maintainers: D. Goel <deego@gnufans.org>
7 ;; Colin Walters <walters@debian.org>
9 ;; This file is part of GNU Emacs.
11 ;; GNU Emacs is distributed in the hope that it will be useful,
12 ;; but WITHOUT ANY WARRANTY. No author or distributor
13 ;; accepts responsibility to anyone for the consequences of using it
14 ;; or for whether it serves any particular purpose or works at all,
15 ;; unless he says so in writing. Refer to the GNU Emacs General Public
16 ;; License for full details.
18 ;; Everyone is granted permission to copy, modify and redistribute
19 ;; GNU Emacs, but only under the conditions described in the
20 ;; GNU Emacs General Public License. A copy of this license is
21 ;; supposed to have been given to you along with GNU Emacs so you
22 ;; can know your rights and responsibilities. It should be in a
23 ;; file named COPYING. Among other things, the copyright notice
24 ;; and this notice must be preserved on all copies.
30 ;; This file is autoloaded from calc-ext.el.
35 (defun calc-Need-calc-arith () nil
)
43 (calc-binary-op "min" 'calcFunc-min arg
'(var inf var-inf
))))
48 (calc-binary-op "max" 'calcFunc-max arg
'(neg (var inf var-inf
)))))
53 (calc-unary-op "abs" 'calcFunc-abs arg
)))
56 (defun calc-idiv (arg)
59 (calc-binary-op "\\" 'calcFunc-idiv arg
1)))
62 (defun calc-floor (arg)
66 (if (calc-is-hyperbolic)
67 (calc-unary-op "ceil" 'calcFunc-fceil arg
)
68 (calc-unary-op "ceil" 'calcFunc-ceil arg
))
69 (if (calc-is-hyperbolic)
70 (calc-unary-op "flor" 'calcFunc-ffloor arg
)
71 (calc-unary-op "flor" 'calcFunc-floor arg
)))))
73 (defun calc-ceiling (arg)
78 (defun calc-round (arg)
82 (if (calc-is-hyperbolic)
83 (calc-unary-op "trnc" 'calcFunc-ftrunc arg
)
84 (calc-unary-op "trnc" 'calcFunc-trunc arg
))
85 (if (calc-is-hyperbolic)
86 (calc-unary-op "rond" 'calcFunc-fround arg
)
87 (calc-unary-op "rond" 'calcFunc-round arg
)))))
89 (defun calc-trunc (arg)
94 (defun calc-mant-part (arg)
97 (calc-unary-op "mant" 'calcFunc-mant arg
)))
99 (defun calc-xpon-part (arg)
102 (calc-unary-op "xpon" 'calcFunc-xpon arg
)))
104 (defun calc-scale-float (arg)
107 (calc-binary-op "scal" 'calcFunc-scf arg
)))
109 (defun calc-abssqr (arg)
112 (calc-unary-op "absq" 'calcFunc-abssqr arg
)))
114 (defun calc-sign (arg)
117 (calc-unary-op "sign" 'calcFunc-sign arg
)))
119 (defun calc-increment (arg)
122 (calc-enter-result 1 "incr" (list 'calcFunc-incr
(calc-top-n 1) arg
))))
124 (defun calc-decrement (arg)
127 (calc-enter-result 1 "decr" (list 'calcFunc-decr
(calc-top-n 1) arg
))))
130 (defun math-abs-approx (a)
136 (math-add (math-abs (nth 1 a
)) (math-abs (nth 2 a
))))
140 (math-abs-approx (nth 1 a
)))
142 (math-max (math-abs (nth 2 a
)) (math-abs (nth 3 a
))))
146 (math-reduce-vec 'math-add-abs-approx a
))
147 ((eq (car a
) 'calcFunc-abs
)
151 (defun math-add-abs-approx (a b
)
152 (math-add (math-abs-approx a
) (math-abs-approx b
)))
157 (defvar math-decls-cache-tag nil
)
158 (defvar math-decls-cache nil
)
159 (defvar math-decls-all nil
)
161 ;;; Math-decls-cache is an a-list where each entry is a list of the form:
162 ;;; (VAR TYPES RANGE)
163 ;;; where VAR is a variable name (with var- prefix) or function name;
164 ;;; TYPES is a list of type symbols (any, int, frac, ...)
165 ;;; RANGE is a sorted vector of intervals describing the range.
167 (defun math-setup-declarations ()
168 (or (eq math-decls-cache-tag
(calc-var-value 'var-Decls
))
169 (let ((p (calc-var-value 'var-Decls
))
171 (setq math-decls-cache-tag p
172 math-decls-cache nil
)
173 (and (eq (car-safe p
) 'vec
)
174 (while (setq p
(cdr p
))
175 (and (eq (car-safe (car p
)) 'vec
)
176 (setq vec
(nth 2 (car p
)))
178 (let ((v (nth 1 (car p
))))
179 (setq type nil range nil
)
180 (or (eq (car-safe vec
) 'vec
)
181 (setq vec
(list 'vec vec
)))
182 (while (and (setq vec
(cdr vec
))
183 (not (Math-objectp (car vec
))))
184 (and (eq (car-safe (car vec
)) 'var
)
185 (let ((st (assq (nth 1 (car vec
))
187 (cond (st (setq type
(append type st
)))
188 ((eq (nth 1 (car vec
)) 'pos
)
189 (setq type
(append type
192 '(intv 1 0 (var inf var-inf
))))
193 ((eq (nth 1 (car vec
)) 'nonneg
)
194 (setq type
(append type
198 (var inf var-inf
))))))))
200 (setq type
(append type
'(real number
))
201 range
(math-prepare-set (cons 'vec vec
))))
202 (setq type
(list type range
))
203 (or (eq (car-safe v
) 'vec
)
204 (setq v
(list 'vec v
)))
205 (while (setq v
(cdr v
))
206 (if (or (eq (car-safe (car v
)) 'var
)
207 (not (Math-primp (car v
))))
208 (setq math-decls-cache
209 (cons (cons (if (eq (car (car v
)) 'var
)
213 math-decls-cache
)))))
215 (setq math-decls-all
(assq 'var-All math-decls-cache
)))))
217 (defvar math-super-types
218 '((int numint rat real number
)
220 (frac rat real number
)
230 (defun math-known-scalarp (a &optional assume-scalar
)
231 (math-setup-declarations)
232 (if (if calc-matrix-mode
233 (eq calc-matrix-mode
'scalar
)
235 (not (math-check-known-matrixp a
))
236 (math-check-known-scalarp a
)))
238 (defun math-known-matrixp (a)
239 (and (not (Math-scalarp a
))
240 (not (math-known-scalarp a t
))))
242 ;;; Try to prove that A is a scalar (i.e., a non-vector).
243 (defun math-check-known-scalarp (a)
244 (cond ((Math-objectp a
) t
)
245 ((memq (car a
) math-scalar-functions
)
247 ((memq (car a
) math-real-scalar-functions
)
249 ((memq (car a
) math-scalar-if-args-functions
)
250 (while (and (setq a
(cdr a
))
251 (math-check-known-scalarp (car a
))))
254 (math-check-known-scalarp (nth 1 a
)))
255 ((math-const-var a
) t
)
257 (let ((decl (if (eq (car a
) 'var
)
258 (or (assq (nth 2 a
) math-decls-cache
)
260 (assq (car a
) math-decls-cache
))))
261 (memq 'scalar
(nth 1 decl
))))))
263 ;;; Try to prove that A is *not* a scalar.
264 (defun math-check-known-matrixp (a)
265 (cond ((Math-objectp a
) nil
)
266 ((memq (car a
) math-nonscalar-functions
)
268 ((memq (car a
) math-scalar-if-args-functions
)
269 (while (and (setq a
(cdr a
))
270 (not (math-check-known-matrixp (car a
)))))
273 (math-check-known-matrixp (nth 1 a
)))
274 ((math-const-var a
) nil
)
276 (let ((decl (if (eq (car a
) 'var
)
277 (or (assq (nth 2 a
) math-decls-cache
)
279 (assq (car a
) math-decls-cache
))))
280 (memq 'vector
(nth 1 decl
))))))
283 ;;; Try to prove that A is a real (i.e., not complex).
284 (defun math-known-realp (a)
285 (< (math-possible-signs a
) 8))
287 ;;; Try to prove that A is real and positive.
288 (defun math-known-posp (a)
289 (eq (math-possible-signs a
) 4))
291 ;;; Try to prove that A is real and negative.
292 (defun math-known-negp (a)
293 (eq (math-possible-signs a
) 1))
295 ;;; Try to prove that A is real and nonnegative.
296 (defun math-known-nonnegp (a)
297 (memq (math-possible-signs a
) '(2 4 6)))
299 ;;; Try to prove that A is real and nonpositive.
300 (defun math-known-nonposp (a)
301 (memq (math-possible-signs a
) '(1 2 3)))
303 ;;; Try to prove that A is nonzero.
304 (defun math-known-nonzerop (a)
305 (memq (math-possible-signs a
) '(1 4 5 8 9 12 13)))
307 ;;; Return true if A is negative, or looks negative but we don't know.
308 (defun math-guess-if-neg (a)
309 (let ((sgn (math-possible-signs a
)))
310 (if (memq sgn
'(1 3))
312 (if (memq sgn
'(2 4 6))
314 (math-looks-negp a
)))))
316 ;;; Find the possible signs of A, assuming A is a number of some kind.
317 ;;; Returns an integer with bits: 1 may be negative,
319 ;;; 4 may be positive,
320 ;;; 8 may be nonreal.
322 (defun math-possible-signs (a &optional origin
)
323 (cond ((Math-objectp a
)
324 (if origin
(setq a
(math-sub a origin
)))
325 (cond ((Math-posp a
) 4)
329 (cond ((Math-zerop (nth 2 a
)) 6)
330 ((Math-zerop (nth 3 a
)) 3)
333 (if (math-known-realp (nth 1 a
)) 7 15))
335 ((memq (car a
) '(+ -
))
336 (cond ((Math-realp (nth 1 a
))
339 (math-possible-signs (nth 2 a
)
341 (math-add origin
(nth 1 a
))
343 (math-possible-signs (nth 2 a
)
345 (math-sub origin
(nth 1 a
))
346 (math-neg (nth 1 a
))))))
347 ((Math-realp (nth 2 a
))
348 (let ((org (if (eq (car a
) '-
)
350 (math-neg (nth 2 a
)))))
351 (math-possible-signs (nth 1 a
)
353 (math-add origin org
)
356 (let ((s1 (math-possible-signs (nth 1 a
) origin
))
357 (s2 (math-possible-signs (nth 2 a
))))
358 (if (eq (car a
) '-
) (setq s2
(math-neg-signs s2
)))
359 (cond ((eq s1 s2
) s1
)
364 ((and (eq s1
4) (eq s2
6)) 4)
365 ((and (eq s2
4) (eq s1
6)) 4)
366 ((and (eq s1
1) (eq s2
3)) 1)
367 ((and (eq s2
1) (eq s1
3)) 1)
370 (math-neg-signs (math-possible-signs
372 (and origin
(math-neg origin
)))))
373 ((and origin
(Math-zerop origin
) (setq origin nil
)
375 ((and (or (eq (car a
) '*)
376 (and (eq (car a
) '/) origin
))
377 (Math-realp (nth 1 a
)))
378 (let ((s (if (eq (car a
) '*)
379 (if (Math-zerop (nth 1 a
))
380 (math-possible-signs 0 origin
)
381 (math-possible-signs (nth 2 a
)
382 (math-div (or origin
0)
385 (math-possible-signs (nth 2 a
)
388 (if (Math-negp (nth 1 a
)) (math-neg-signs s
) s
)))
389 ((and (memq (car a
) '(* /)) (Math-realp (nth 2 a
)))
390 (let ((s (math-possible-signs (nth 1 a
)
392 (math-mul (or origin
0) (nth 2 a
))
393 (math-div (or origin
0) (nth 2 a
))))))
394 (if (Math-negp (nth 2 a
)) (math-neg-signs s
) s
)))
397 (while (and (setq a
(cdr a
)) (< signs
15))
398 (setq signs
(logior signs
(math-possible-signs
403 ((memq (car a
) '(* /))
404 (let ((s1 (math-possible-signs (nth 1 a
)))
405 (s2 (math-possible-signs (nth 2 a
))))
408 ((and (eq (car a
) '/) (memq s2
'(2 3 6 7))) 15)
410 (logior (if (memq s1
'(4 5 6 7)) s2
0)
411 (if (memq s1
'(2 3 6 7)) 2 0)
412 (if (memq s1
'(1 3 5 7))
413 (math-neg-signs s2
) 0))))))
415 (let ((s1 (math-possible-signs (nth 1 a
)))
416 (s2 (math-possible-signs (nth 2 a
))))
420 ((eq s1
2) (if (eq s2
4) 2 15))
421 ((eq s2
2) (if (memq s1
'(1 5)) 2 15))
422 ((Math-integerp (nth 2 a
))
423 (if (math-evenp (nth 2 a
))
424 (if (memq s1
'(3 6 7)) 6 4)
426 ((eq s1
6) (if (eq s2
4) 6 15))
429 (let ((s2 (math-possible-signs (nth 2 a
))))
435 ((and (memq (car a
) '(calcFunc-abs calcFunc-abssqr
))
437 (let ((s1 (math-possible-signs (nth 1 a
))))
439 ((memq s1
'(1 4 5)) 4)
441 ((and (eq (car a
) 'calcFunc-exp
) (= (length a
) 2))
442 (let ((s1 (math-possible-signs (nth 1 a
))))
445 (if (or (not origin
) (math-negp origin
))
447 (setq origin
(math-sub (or origin
0) 1))
448 (if (Math-zerop origin
) (setq origin nil
))
450 ((or (and (memq (car a
) '(calcFunc-ln calcFunc-log10
))
452 (and (eq (car a
) 'calcFunc-log
)
454 (math-known-posp (nth 2 a
))))
455 (if (math-known-nonnegp (nth 1 a
))
456 (math-possible-signs (nth 1 a
) 1)
458 ((and (eq (car a
) 'calcFunc-sqrt
) (= (length a
) 2))
459 (let ((s1 (math-possible-signs (nth 1 a
))))
460 (if (memq s1
'(2 4 6)) s1
15)))
461 ((memq (car a
) math-nonnegative-functions
) 6)
462 ((memq (car a
) math-positive-functions
) 4)
463 ((memq (car a
) math-real-functions
) 7)
464 ((memq (car a
) math-real-scalar-functions
) 7)
465 ((and (memq (car a
) math-real-if-arg-functions
)
467 (if (math-known-realp (nth 1 a
)) 7 15)))))
471 (if (Math-posp origin
)
472 (if (memq sign
'(1 2 3 8 9 10 11)) 1 7)
473 (if (memq sign
'(2 4 6 8 10 12 14)) 4 7)))
476 (cond ((eq (nth 2 a
) 'var-pi
)
478 (math-possible-signs (math-pi) origin
)
480 ((eq (nth 2 a
) 'var-e
)
482 (math-possible-signs (math-e) origin
)
484 ((eq (nth 2 a
) 'var-inf
) 4)
485 ((eq (nth 2 a
) 'var-uinf
) 13)
486 ((eq (nth 2 a
) 'var-i
) 8)
489 (math-setup-declarations)
490 (let ((decl (if (eq (car a
) 'var
)
491 (or (assq (nth 2 a
) math-decls-cache
)
493 (assq (car a
) math-decls-cache
))))
495 (memq 'int
(nth 1 decl
))
496 (not (Math-num-integerp origin
)))
499 (math-possible-signs (nth 2 decl
) origin
)
500 (if (memq 'real
(nth 1 decl
))
504 (defun math-neg-signs (s1)
506 (+ 8 (math-neg-signs (- s1
8)))
507 (+ (if (memq s1
'(1 3 5 7)) 4 0)
508 (if (memq s1
'(2 3 6 7)) 2 0)
509 (if (memq s1
'(4 5 6 7)) 1 0))))
512 ;;; Try to prove that A is an integer.
513 (defun math-known-integerp (a)
514 (eq (math-possible-types a
) 1))
516 (defun math-known-num-integerp (a)
517 (<= (math-possible-types a t
) 3))
519 (defun math-known-imagp (a)
520 (= (math-possible-types a
) 16))
523 ;;; Find the possible types of A.
524 ;;; Returns an integer with bits: 1 may be integer.
525 ;;; 2 may be integer-valued float.
526 ;;; 4 may be fraction.
527 ;;; 8 may be non-integer-valued float.
528 ;;; 16 may be imaginary.
529 ;;; 32 may be non-real, non-imaginary.
530 ;;; Real infinities count as integers for the purposes of this function.
531 (defun math-possible-types (a &optional num
)
532 (cond ((Math-objectp a
)
533 (cond ((Math-integerp a
) (if num
3 1))
534 ((Math-messy-integerp a
) (if num
3 2))
535 ((eq (car a
) 'frac
) (if num
12 4))
536 ((eq (car a
) 'float
) (if num
12 8))
538 (if (equal (nth 2 a
) (nth 3 a
))
539 (math-possible-types (nth 2 a
))
542 (if (math-known-realp (nth 1 a
)) 15 63))
544 (if (math-zerop (nth 1 a
)) 16 32))
546 (if (or (Math-equal (nth 2 a
) (math-quarter-circle nil
))
547 (Math-equal (nth 2 a
)
548 (math-neg (math-quarter-circle nil
))))
552 (let* ((t1 (math-possible-types (nth 1 a
) num
))
553 (t2 (math-possible-types (nth 2 a
) num
))
554 (t12 (logior t1 t2
)))
556 (if (> (logand t12
10) 0)
558 (if (or (= t1
4) (= t2
4) calc-prefer-frac
)
564 (if (< t2
16) 16 31))
569 ((memq (car a
) '(+ -
* %
))
570 (let* ((t1 (math-possible-types (nth 1 a
) num
))
571 (t2 (math-possible-types (nth 2 a
) num
))
572 (t12 (logior t1 t2
)))
574 (setq t1
(logand t1
15) t2
(logand t2
15) t12
(logand t12
15)))
576 (let ((mask (if (<= t12
3)
578 (if (and (or (and (<= t1
3) (= (logand t2
3) 0))
579 (and (<= t2
3) (= (logand t1
3) 0)))
580 (memq (car a
) '(+ -
)))
585 (logior (if (and (> (logand t1
5) 0) (> (logand t2
5) 0))
587 (if (> (logand t12
10) 0)
593 (if (< t2
16) 16 31))
598 (if (or (and (= t1
16) (< t2
16))
599 (and (= t2
16) (< t1
16))) 32 63)))
602 (math-possible-types (nth 1 a
)))
604 (let* ((t1 (math-possible-types (nth 1 a
) num
))
605 (t2 (math-possible-types (nth 2 a
) num
))
606 (t12 (logior t1 t2
)))
607 (if (and (<= t2
3) (math-known-nonnegp (nth 2 a
)) (< t1
16))
608 (let ((mask (logior (if (> (logand t1
3) 0) 1 0)
610 (if (> (logand t1
12) 0) 5 0))))
613 (logior (if (and (> (logand t1
5) 0) (> (logand t2
5) 0))
615 (if (> (logand t12
10) 0)
617 (if (and (math-known-nonnegp (nth 1 a
))
618 (math-known-posp (nth 2 a
)))
621 ((eq (car a
) 'calcFunc-sqrt
)
622 (let ((t1 (math-possible-signs (nth 1 a
))))
623 (logior (if (> (logand t1
2) 0) 3 0)
624 (if (> (logand t1
1) 0) 16 0)
625 (if (> (logand t1
4) 0) 15 0)
626 (if (> (logand t1
8) 0) 32 0))))
629 (while (and (setq a
(cdr a
)) (< types
63))
630 (setq types
(logior types
(math-possible-types (car a
) t
))))
632 ((or (memq (car a
) math-integer-functions
)
633 (and (memq (car a
) math-rounding-functions
)
634 (math-known-nonnegp (or (nth 2 a
) 0))))
636 ((or (memq (car a
) math-num-integer-functions
)
637 (and (memq (car a
) math-float-rounding-functions
)
638 (math-known-nonnegp (or (nth 2 a
) 0))))
640 ((eq (car a
) 'calcFunc-frac
)
642 ((and (eq (car a
) 'calcFunc-float
) (= (length a
) 2))
643 (let ((t1 (math-possible-types (nth 1 a
))))
644 (logior (if (> (logand t1
3) 0) 2 0)
645 (if (> (logand t1
12) 0) 8 0)
647 ((and (memq (car a
) '(calcFunc-abs calcFunc-abssqr
))
649 (let ((t1 (math-possible-types (nth 1 a
))))
654 (cond ((memq (nth 2 a
) '(var-e var-pi var-phi var-gamma
)) 8)
655 ((eq (nth 2 a
) 'var-inf
) 1)
656 ((eq (nth 2 a
) 'var-i
) 16)
659 (math-setup-declarations)
660 (let ((decl (if (eq (car a
) 'var
)
661 (or (assq (nth 2 a
) math-decls-cache
)
663 (assq (car a
) math-decls-cache
))))
664 (cond ((memq 'int
(nth 1 decl
))
666 ((memq 'numint
(nth 1 decl
))
668 ((memq 'frac
(nth 1 decl
))
670 ((memq 'rat
(nth 1 decl
))
672 ((memq 'float
(nth 1 decl
))
675 (math-possible-types (nth 2 decl
)))
676 ((memq 'real
(nth 1 decl
))
680 (defun math-known-evenp (a)
681 (cond ((Math-integerp a
)
683 ((Math-messy-integerp a
)
685 (math-evenp (math-trunc a
))))
687 (if (math-known-evenp (nth 1 a
))
688 (math-known-num-integerp (nth 2 a
))
689 (if (math-known-num-integerp (nth 1 a
))
690 (math-known-evenp (nth 2 a
)))))
691 ((memq (car a
) '(+ -
))
692 (or (and (math-known-evenp (nth 1 a
))
693 (math-known-evenp (nth 2 a
)))
694 (and (math-known-oddp (nth 1 a
))
695 (math-known-oddp (nth 2 a
)))))
697 (math-known-evenp (nth 1 a
)))))
699 (defun math-known-oddp (a)
700 (cond ((Math-integerp a
)
702 ((Math-messy-integerp a
)
703 (and (<= (nth 2 a
) 0)
704 (math-oddp (math-trunc a
))))
705 ((memq (car a
) '(+ -
))
706 (or (and (math-known-evenp (nth 1 a
))
707 (math-known-oddp (nth 2 a
)))
708 (and (math-known-oddp (nth 1 a
))
709 (math-known-evenp (nth 2 a
)))))
711 (math-known-oddp (nth 1 a
)))))
714 (defun calcFunc-dreal (expr)
715 (let ((types (math-possible-types expr
)))
717 (if (= (logand types
15) 0) 0
718 (math-reject-arg expr
'realp
'quiet
)))))
720 (defun calcFunc-dimag (expr)
721 (let ((types (math-possible-types expr
)))
723 (if (= (logand types
16) 0) 0
724 (math-reject-arg expr
"Expected an imaginary number")))))
726 (defun calcFunc-dpos (expr)
727 (let ((signs (math-possible-signs expr
)))
729 (if (memq signs
'(1 2 3)) 0
730 (math-reject-arg expr
'posp
'quiet
)))))
732 (defun calcFunc-dneg (expr)
733 (let ((signs (math-possible-signs expr
)))
735 (if (memq signs
'(2 4 6)) 0
736 (math-reject-arg expr
'negp
'quiet
)))))
738 (defun calcFunc-dnonneg (expr)
739 (let ((signs (math-possible-signs expr
)))
740 (if (memq signs
'(2 4 6)) 1
742 (math-reject-arg expr
'posp
'quiet
)))))
744 (defun calcFunc-dnonzero (expr)
745 (let ((signs (math-possible-signs expr
)))
746 (if (memq signs
'(1 4 5 8 9 12 13)) 1
748 (math-reject-arg expr
'nonzerop
'quiet
)))))
750 (defun calcFunc-dint (expr)
751 (let ((types (math-possible-types expr
)))
753 (if (= (logand types
1) 0) 0
754 (math-reject-arg expr
'integerp
'quiet
)))))
756 (defun calcFunc-dnumint (expr)
757 (let ((types (math-possible-types expr t
)))
759 (if (= (logand types
3) 0) 0
760 (math-reject-arg expr
'integerp
'quiet
)))))
762 (defun calcFunc-dnatnum (expr)
763 (let ((res (calcFunc-dint expr
)))
765 (calcFunc-dnonneg expr
)
768 (defun calcFunc-deven (expr)
769 (if (math-known-evenp expr
)
771 (if (or (math-known-oddp expr
)
772 (= (logand (math-possible-types expr
) 3) 0))
774 (math-reject-arg expr
"Can't tell if expression is odd or even"))))
776 (defun calcFunc-dodd (expr)
777 (if (math-known-oddp expr
)
779 (if (or (math-known-evenp expr
)
780 (= (logand (math-possible-types expr
) 3) 0))
782 (math-reject-arg expr
"Can't tell if expression is odd or even"))))
784 (defun calcFunc-drat (expr)
785 (let ((types (math-possible-types expr
)))
786 (if (memq types
'(1 4 5)) 1
787 (if (= (logand types
5) 0) 0
788 (math-reject-arg expr
"Rational number expected")))))
790 (defun calcFunc-drange (expr)
791 (math-setup-declarations)
793 (if (Math-realp expr
)
795 (if (eq (car-safe expr
) 'intv
)
797 (if (eq (car-safe expr
) 'var
)
798 (setq range
(nth 2 (or (assq (nth 2 expr
) math-decls-cache
)
800 (setq range
(nth 2 (assq (car-safe expr
) math-decls-cache
))))
802 (math-clean-set (copy-sequence range
))
803 (setq range
(math-possible-signs expr
))
806 (intv 2 (neg (var inf var-inf
)) 0)
808 (intv 3 (neg (var inf var-inf
)) 0)
809 (intv 1 0 (var inf var-inf
))
810 (vec (intv 2 (neg (var inf var-inf
)) 0)
811 (intv 1 0 (var inf var-inf
)))
812 (intv 3 0 (var inf var-inf
))
813 (intv 3 (neg (var inf var-inf
)) (var inf var-inf
))] range
)
814 (math-reject-arg expr
'realp
'quiet
)))))))
816 (defun calcFunc-dscalar (a)
817 (if (math-known-scalarp a
) 1
818 (if (math-known-matrixp a
) 0
819 (math-reject-arg a
'objectp
'quiet
))))
822 ;;; The following lists are not exhaustive.
823 (defvar math-scalar-functions
'(calcFunc-det
824 calcFunc-cnorm calcFunc-rnorm
825 calcFunc-vlen calcFunc-vcount
826 calcFunc-vsum calcFunc-vprod
827 calcFunc-vmin calcFunc-vmax
))
829 (defvar math-nonscalar-functions
'(vec calcFunc-idn calcFunc-diag
830 calcFunc-cvec calcFunc-index
833 calcFunc-cons calcFunc-rcons
834 calcFunc-tail calcFunc-rhead
))
836 (defvar math-scalar-if-args-functions
'(+ -
* / neg
))
838 (defvar math-real-functions
'(calcFunc-arg
839 calcFunc-re calcFunc-im
840 calcFunc-floor calcFunc-ceil
841 calcFunc-trunc calcFunc-round
842 calcFunc-rounde calcFunc-roundu
843 calcFunc-ffloor calcFunc-fceil
844 calcFunc-ftrunc calcFunc-fround
845 calcFunc-frounde calcFunc-froundu
))
847 (defvar math-positive-functions
'())
849 (defvar math-nonnegative-functions
'(calcFunc-cnorm calcFunc-rnorm
850 calcFunc-vlen calcFunc-vcount
))
852 (defvar math-real-scalar-functions
'(% calcFunc-idiv calcFunc-abs
853 calcFunc-choose calcFunc-perm
854 calcFunc-eq calcFunc-neq
855 calcFunc-lt calcFunc-gt
856 calcFunc-leq calcFunc-geq
858 calcFunc-max calcFunc-min
))
860 (defvar math-real-if-arg-functions
'(calcFunc-sin calcFunc-cos
861 calcFunc-tan calcFunc-arctan
862 calcFunc-sinh calcFunc-cosh
863 calcFunc-tanh calcFunc-exp
864 calcFunc-gamma calcFunc-fact
))
866 (defvar math-integer-functions
'(calcFunc-idiv
867 calcFunc-isqrt calcFunc-ilog
868 calcFunc-vlen calcFunc-vcount
))
870 (defvar math-num-integer-functions
'())
872 (defvar math-rounding-functions
'(calcFunc-floor
874 calcFunc-round calcFunc-trunc
875 calcFunc-rounde calcFunc-roundu
))
877 (defvar math-float-rounding-functions
'(calcFunc-ffloor
879 calcFunc-fround calcFunc-ftrunc
880 calcFunc-frounde calcFunc-froundu
))
882 (defvar math-integer-if-args-functions
'(+ -
* % neg calcFunc-abs
883 calcFunc-min calcFunc-max
884 calcFunc-choose calcFunc-perm
))
889 (defsubst calcFunc-neg
(a)
890 (math-normalize (list 'neg a
)))
892 (defun math-neg-fancy (a)
893 (cond ((eq (car a
) 'polar
)
896 (if (math-posp (nth 2 a
))
897 (math-sub (nth 2 a
) (math-half-circle nil
))
898 (math-add (nth 2 a
) (math-half-circle nil
)))))
900 (if (math-zerop (nth 1 a
))
902 (list 'mod
(math-sub (nth 2 a
) (nth 1 a
)) (nth 2 a
))))
904 (list 'sdev
(math-neg (nth 1 a
)) (nth 2 a
)))
906 (math-make-intv (aref [0 2 1 3] (nth 1 a
))
908 (math-neg (nth 2 a
))))
909 ((and math-simplify-only
910 (not (equal a math-simplify-only
)))
913 (math-sub (math-neg (nth 1 a
)) (nth 2 a
)))
915 (math-sub (nth 2 a
) (nth 1 a
)))
916 ((and (memq (car a
) '(* /))
917 (math-okay-neg (nth 1 a
)))
918 (list (car a
) (math-neg (nth 1 a
)) (nth 2 a
)))
919 ((and (memq (car a
) '(* /))
920 (math-okay-neg (nth 2 a
)))
921 (list (car a
) (nth 1 a
) (math-neg (nth 2 a
))))
922 ((and (memq (car a
) '(* /))
923 (or (math-objectp (nth 1 a
))
924 (and (eq (car (nth 1 a
)) '*)
925 (math-objectp (nth 1 (nth 1 a
))))))
926 (list (car a
) (math-neg (nth 1 a
)) (nth 2 a
)))
927 ((and (eq (car a
) '/)
928 (or (math-objectp (nth 2 a
))
929 (and (eq (car (nth 2 a
)) '*)
930 (math-objectp (nth 1 (nth 2 a
))))))
931 (list (car a
) (nth 1 a
) (math-neg (nth 2 a
))))
932 ((and (eq (car a
) 'var
) (memq (nth 2 a
) '(var-uinf var-nan
)))
938 (defun math-okay-neg (a)
939 (or (math-looks-negp a
)
940 (eq (car-safe a
) '-
)))
942 (defun math-neg-float (a)
943 (list 'float
(Math-integer-neg (nth 1 a
)) (nth 2 a
)))
946 (defun calcFunc-add (&rest rest
)
948 (let ((a (car rest
)))
949 (while (setq rest
(cdr rest
))
950 (setq a
(list '+ a
(car rest
))))
954 (defun calcFunc-sub (&rest rest
)
956 (let ((a (car rest
)))
957 (while (setq rest
(cdr rest
))
958 (setq a
(list '- a
(car rest
))))
962 (defun math-add-objects-fancy (a b
)
963 (cond ((and (Math-numberp a
) (Math-numberp b
))
964 (let ((aa (math-complex a
))
965 (bb (math-complex b
)))
967 (let ((res (list 'cplx
968 (math-add (nth 1 aa
) (nth 1 bb
))
969 (math-add (nth 2 aa
) (nth 2 bb
)))))
970 (if (math-want-polar a b
)
973 ((or (Math-vectorp a
) (Math-vectorp b
))
974 (math-map-vec-2 'math-add a b
))
975 ((eq (car-safe a
) 'sdev
)
976 (if (eq (car-safe b
) 'sdev
)
977 (math-make-sdev (math-add (nth 1 a
) (nth 1 b
))
978 (math-hypot (nth 2 a
) (nth 2 b
)))
979 (and (or (Math-scalarp b
)
980 (not (Math-objvecp b
)))
981 (math-make-sdev (math-add (nth 1 a
) b
) (nth 2 a
)))))
982 ((and (eq (car-safe b
) 'sdev
)
984 (not (Math-objvecp a
))))
985 (math-make-sdev (math-add a
(nth 1 b
)) (nth 2 b
)))
986 ((eq (car-safe a
) 'intv
)
987 (if (eq (car-safe b
) 'intv
)
988 (math-make-intv (logior (logand (nth 1 a
) (nth 1 b
))
990 '(neg (var inf var-inf
)))
991 (logand (nth 1 a
) 2) 0)
993 '(neg (var inf var-inf
)))
994 (logand (nth 1 b
) 2) 0)
995 (if (equal (nth 3 a
) '(var inf var-inf
))
996 (logand (nth 1 a
) 1) 0)
997 (if (equal (nth 3 b
) '(var inf var-inf
))
998 (logand (nth 1 b
) 1) 0))
999 (math-add (nth 2 a
) (nth 2 b
))
1000 (math-add (nth 3 a
) (nth 3 b
)))
1001 (and (or (Math-anglep b
)
1003 (not (Math-objvecp b
)))
1004 (math-make-intv (nth 1 a
)
1005 (math-add (nth 2 a
) b
)
1006 (math-add (nth 3 a
) b
)))))
1007 ((and (eq (car-safe b
) 'intv
)
1010 (not (Math-objvecp a
))))
1011 (math-make-intv (nth 1 b
)
1012 (math-add a
(nth 2 b
))
1013 (math-add a
(nth 3 b
))))
1014 ((eq (car-safe a
) 'date
)
1015 (cond ((eq (car-safe b
) 'date
)
1016 (math-add (nth 1 a
) (nth 1 b
)))
1017 ((eq (car-safe b
) 'hms
)
1018 (let ((parts (math-date-parts (nth 1 a
))))
1020 (math-add (car parts
) ; this minimizes roundoff
1022 (math-add (nth 1 parts
)
1025 (math-mul (nth 1 b
) 3600)
1026 (math-add (math-mul (nth 2 b
) 60)
1030 (list 'date
(math-add (nth 1 a
) b
)))
1032 ((eq (car-safe b
) 'date
)
1033 (math-add-objects-fancy b a
))
1034 ((and (eq (car-safe a
) 'mod
)
1035 (eq (car-safe b
) 'mod
)
1036 (equal (nth 2 a
) (nth 2 b
)))
1037 (math-make-mod (math-add (nth 1 a
) (nth 1 b
)) (nth 2 a
)))
1038 ((and (eq (car-safe a
) 'mod
)
1040 (math-make-mod (math-add (nth 1 a
) b
) (nth 2 a
)))
1041 ((and (eq (car-safe b
) 'mod
)
1043 (math-make-mod (math-add a
(nth 1 b
)) (nth 2 b
)))
1044 ((and (or (eq (car-safe a
) 'hms
) (eq (car-safe b
) 'hms
))
1045 (and (Math-anglep a
) (Math-anglep b
)))
1046 (or (eq (car-safe a
) 'hms
) (setq a
(math-to-hms a
)))
1047 (or (eq (car-safe b
) 'hms
) (setq b
(math-to-hms b
)))
1050 (math-neg (math-add (math-neg a
) (math-neg b
)))
1052 (let* ((s (math-add (nth 3 a
) (nth 3 b
)))
1053 (m (math-add (nth 2 a
) (nth 2 b
)))
1054 (h (math-add (nth 1 a
) (nth 1 b
))))
1056 (setq s
(math-add s
60)
1059 (setq m
(math-add m
60)
1064 (let* ((s (math-add (nth 3 a
) (nth 3 b
)))
1065 (m (math-add (nth 2 a
) (nth 2 b
)))
1066 (h (math-add (nth 1 a
) (nth 1 b
))))
1067 (list 'hms h m s
))))))
1068 (t (calc-record-why "*Incompatible arguments for +" a b
))))
1070 (defun math-add-symb-fancy (a b
)
1071 (or (and math-simplify-only
1072 (not (equal a math-simplify-only
))
1074 (and (eq (car-safe b
) '+)
1075 (math-add (math-add a
(nth 1 b
))
1077 (and (eq (car-safe b
) '-
)
1078 (math-sub (math-add a
(nth 1 b
))
1080 (and (eq (car-safe b
) 'neg
)
1081 (eq (car-safe (nth 1 b
)) '+)
1082 (math-sub (math-sub a
(nth 1 (nth 1 b
)))
1084 (and (or (and (Math-vectorp a
) (math-known-scalarp b
))
1085 (and (Math-vectorp b
) (math-known-scalarp a
)))
1086 (math-map-vec-2 'math-add a b
))
1087 (let ((inf (math-infinitep a
)))
1090 (let ((inf2 (math-infinitep b
)))
1092 (if (or (memq (nth 2 inf
) '(var-uinf var-nan
))
1093 (memq (nth 2 inf2
) '(var-uinf var-nan
)))
1095 (let ((dir (math-infinite-dir a inf
))
1096 (dir2 (math-infinite-dir b inf2
)))
1097 (if (and (Math-objectp dir
) (Math-objectp dir2
))
1098 (if (Math-equal dir dir2
)
1100 '(var nan var-nan
)))))
1101 (if (and (equal a
'(var inf var-inf
))
1102 (eq (car-safe b
) 'intv
)
1103 (memq (nth 1 b
) '(2 3))
1104 (equal (nth 2 b
) '(neg (var inf var-inf
))))
1105 (list 'intv
3 (nth 2 b
) a
)
1106 (if (and (equal a
'(neg (var inf var-inf
)))
1107 (eq (car-safe b
) 'intv
)
1108 (memq (nth 1 b
) '(1 3))
1109 (equal (nth 3 b
) '(var inf var-inf
)))
1110 (list 'intv
3 a
(nth 3 b
))
1113 (if (eq (car-safe a
) 'intv
)
1116 ((eq (car-safe a
) '+)
1117 (let ((temp (math-combine-sum (nth 2 a
) b nil nil t
)))
1119 (math-add (nth 1 a
) temp
))))
1120 ((eq (car-safe a
) '-
)
1121 (let ((temp (math-combine-sum (nth 2 a
) b t nil t
)))
1123 (math-add (nth 1 a
) temp
))))
1124 ((and (Math-objectp a
) (Math-objectp b
))
1127 (math-combine-sum a b nil nil nil
))))
1128 (and (Math-looks-negp b
)
1129 (list '- a
(math-neg b
)))
1130 (and (Math-looks-negp a
)
1131 (list '- b
(math-neg a
)))
1132 (and (eq (car-safe a
) 'calcFunc-idn
)
1134 (or (and (eq (car-safe b
) 'calcFunc-idn
)
1136 (list 'calcFunc-idn
(math-add (nth 1 a
) (nth 1 b
))))
1137 (and (math-square-matrixp b
)
1138 (math-add (math-mimic-ident (nth 1 a
) b
) b
))
1139 (and (math-known-scalarp b
)
1140 (math-add (nth 1 a
) b
))))
1141 (and (eq (car-safe b
) 'calcFunc-idn
)
1143 (or (and (math-square-matrixp a
)
1144 (math-add a
(math-mimic-ident (nth 1 b
) a
)))
1145 (and (math-known-scalarp a
)
1146 (math-add a
(nth 1 b
)))))
1150 (defun calcFunc-mul (&rest rest
)
1152 (let ((a (car rest
)))
1153 (while (setq rest
(cdr rest
))
1154 (setq a
(list '* a
(car rest
))))
1158 (defun math-mul-objects-fancy (a b
)
1159 (cond ((and (Math-numberp a
) (Math-numberp b
))
1161 (if (math-want-polar a b
)
1162 (let ((a (math-polar a
))
1165 (math-mul (nth 1 a
) (nth 1 b
))
1166 (math-fix-circular (math-add (nth 2 a
) (nth 2 b
)))))
1167 (setq a
(math-complex a
)
1170 (math-sub (math-mul (nth 1 a
) (nth 1 b
))
1171 (math-mul (nth 2 a
) (nth 2 b
)))
1172 (math-add (math-mul (nth 1 a
) (nth 2 b
))
1173 (math-mul (nth 2 a
) (nth 1 b
)))))))
1175 (if (Math-vectorp b
)
1176 (if (math-matrixp a
)
1177 (if (math-matrixp b
)
1178 (if (= (length (nth 1 a
)) (length b
))
1180 (math-dimension-error))
1181 (if (= (length (nth 1 a
)) 2)
1182 (if (= (length a
) (length b
))
1183 (math-mul-mats a
(list 'vec b
))
1184 (math-dimension-error))
1185 (if (= (length (nth 1 a
)) (length b
))
1186 (math-mul-mat-vec a b
)
1187 (math-dimension-error))))
1188 (if (math-matrixp b
)
1189 (if (= (length a
) (length b
))
1190 (nth 1 (math-mul-mats (list 'vec a
) b
))
1191 (math-dimension-error))
1192 (if (= (length a
) (length b
))
1193 (math-dot-product a b
)
1194 (math-dimension-error))))
1195 (math-map-vec-2 'math-mul a b
)))
1197 (math-map-vec-2 'math-mul a b
))
1198 ((eq (car-safe a
) 'sdev
)
1199 (if (eq (car-safe b
) 'sdev
)
1200 (math-make-sdev (math-mul (nth 1 a
) (nth 1 b
))
1201 (math-hypot (math-mul (nth 2 a
) (nth 1 b
))
1202 (math-mul (nth 2 b
) (nth 1 a
))))
1203 (and (or (Math-scalarp b
)
1204 (not (Math-objvecp b
)))
1205 (math-make-sdev (math-mul (nth 1 a
) b
)
1206 (math-mul (nth 2 a
) b
)))))
1207 ((and (eq (car-safe b
) 'sdev
)
1208 (or (Math-scalarp a
)
1209 (not (Math-objvecp a
))))
1210 (math-make-sdev (math-mul a
(nth 1 b
)) (math-mul a
(nth 2 b
))))
1211 ((and (eq (car-safe a
) 'intv
) (Math-anglep b
))
1213 (math-neg (math-mul a
(math-neg b
)))
1214 (math-make-intv (nth 1 a
)
1215 (math-mul (nth 2 a
) b
)
1216 (math-mul (nth 3 a
) b
))))
1217 ((and (eq (car-safe b
) 'intv
) (Math-anglep a
))
1219 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
1220 (eq (car-safe b
) 'intv
) (math-intv-constp b
))
1221 (let ((lo (math-mul a
(nth 2 b
)))
1222 (hi (math-mul a
(nth 3 b
))))
1223 (or (eq (car-safe lo
) 'intv
)
1224 (setq lo
(list 'intv
(if (memq (nth 1 b
) '(2 3)) 3 0) lo lo
)))
1225 (or (eq (car-safe hi
) 'intv
)
1226 (setq hi
(list 'intv
(if (memq (nth 1 b
) '(1 3)) 3 0) hi hi
)))
1227 (math-combine-intervals
1228 (nth 2 lo
) (and (or (memq (nth 1 b
) '(2 3))
1229 (math-infinitep (nth 2 lo
)))
1230 (memq (nth 1 lo
) '(2 3)))
1231 (nth 3 lo
) (and (or (memq (nth 1 b
) '(2 3))
1232 (math-infinitep (nth 3 lo
)))
1233 (memq (nth 1 lo
) '(1 3)))
1234 (nth 2 hi
) (and (or (memq (nth 1 b
) '(1 3))
1235 (math-infinitep (nth 2 hi
)))
1236 (memq (nth 1 hi
) '(2 3)))
1237 (nth 3 hi
) (and (or (memq (nth 1 b
) '(1 3))
1238 (math-infinitep (nth 3 hi
)))
1239 (memq (nth 1 hi
) '(1 3))))))
1240 ((and (eq (car-safe a
) 'mod
)
1241 (eq (car-safe b
) 'mod
)
1242 (equal (nth 2 a
) (nth 2 b
)))
1243 (math-make-mod (math-mul (nth 1 a
) (nth 1 b
)) (nth 2 a
)))
1244 ((and (eq (car-safe a
) 'mod
)
1246 (math-make-mod (math-mul (nth 1 a
) b
) (nth 2 a
)))
1247 ((and (eq (car-safe b
) 'mod
)
1249 (math-make-mod (math-mul a
(nth 1 b
)) (nth 2 b
)))
1250 ((and (eq (car-safe a
) 'hms
) (Math-realp b
))
1251 (math-with-extra-prec 2
1252 (math-to-hms (math-mul (math-from-hms a
'deg
) b
) 'deg
)))
1253 ((and (eq (car-safe b
) 'hms
) (Math-realp a
))
1255 (t (calc-record-why "*Incompatible arguments for *" a b
))))
1257 ;;; Fast function to multiply floating-point numbers.
1258 (defun math-mul-float (a b
) ; [F F F]
1259 (math-make-float (math-mul (nth 1 a
) (nth 1 b
))
1260 (+ (nth 2 a
) (nth 2 b
))))
1262 (defun math-sqr-float (a) ; [F F]
1263 (math-make-float (math-mul (nth 1 a
) (nth 1 a
))
1264 (+ (nth 2 a
) (nth 2 a
))))
1266 (defun math-intv-constp (a &optional finite
)
1267 (and (or (Math-anglep (nth 2 a
))
1268 (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
1270 (memq (nth 1 a
) '(0 1)))))
1271 (or (Math-anglep (nth 3 a
))
1272 (and (equal (nth 3 a
) '(var inf var-inf
))
1274 (memq (nth 1 a
) '(0 2)))))))
1276 (defun math-mul-zero (a b
)
1277 (if (math-known-matrixp b
)
1278 (if (math-vectorp b
)
1279 (math-map-vec-2 'math-mul a b
)
1280 (math-mimic-ident 0 b
))
1281 (if (math-infinitep b
)
1283 (let ((aa nil
) (bb nil
))
1284 (if (and (eq (car-safe b
) 'intv
)
1286 (and (equal (nth 2 b
) '(neg (var inf var-inf
)))
1287 (memq (nth 1 b
) '(2 3))
1288 (setq aa
(nth 2 b
)))
1289 (and (equal (nth 3 b
) '(var inf var-inf
))
1290 (memq (nth 1 b
) '(1 3))
1291 (setq bb
(nth 3 b
)))
1293 (if (or (math-posp a
)
1295 (or (memq calc-infinite-mode
'(-1 1))
1296 (setq aa
'(neg (var inf var-inf
))
1297 bb
'(var inf var-inf
)))))
1298 (list 'intv
3 (or aa
0) (or bb
0))
1300 (math-neg (list 'intv
3 (or aa
0) (or bb
0)))
1301 '(var nan var-nan
)))
1302 (if (or (math-floatp a
) (math-floatp b
)) '(float 0 0) 0))))))
1305 (defun math-mul-symb-fancy (a b
)
1306 (or (and math-simplify-only
1307 (not (equal a math-simplify-only
))
1309 (and (Math-equal-int a
1)
1311 (and (Math-equal-int a -
1)
1313 (and (or (and (Math-vectorp a
) (math-known-scalarp b
))
1314 (and (Math-vectorp b
) (math-known-scalarp a
)))
1315 (math-map-vec-2 'math-mul a b
))
1316 (and (Math-objectp b
) (not (Math-objectp a
))
1318 (and (eq (car-safe a
) 'neg
)
1319 (math-neg (math-mul (nth 1 a
) b
)))
1320 (and (eq (car-safe b
) 'neg
)
1321 (math-neg (math-mul a
(nth 1 b
))))
1322 (and (eq (car-safe a
) '*)
1324 (math-mul (nth 2 a
) b
)))
1325 (and (eq (car-safe a
) '^
)
1326 (Math-looks-negp (nth 2 a
))
1327 (not (and (eq (car-safe b
) '^
) (Math-looks-negp (nth 2 b
))))
1328 (math-known-scalarp b t
)
1329 (math-div b
(math-normalize
1330 (list '^
(nth 1 a
) (math-neg (nth 2 a
))))))
1331 (and (eq (car-safe b
) '^
)
1332 (Math-looks-negp (nth 2 b
))
1333 (not (and (eq (car-safe a
) '^
) (Math-looks-negp (nth 2 a
))))
1334 (math-div a
(math-normalize
1335 (list '^
(nth 1 b
) (math-neg (nth 2 b
))))))
1336 (and (eq (car-safe a
) '/)
1337 (or (math-known-scalarp a t
) (math-known-scalarp b t
))
1338 (let ((temp (math-combine-prod (nth 2 a
) b t nil t
)))
1340 (math-mul (nth 1 a
) temp
)
1341 (math-div (math-mul (nth 1 a
) b
) (nth 2 a
)))))
1342 (and (eq (car-safe b
) '/)
1343 (math-div (math-mul a
(nth 1 b
)) (nth 2 b
)))
1344 (and (eq (car-safe b
) '+)
1346 (or (Math-numberp (nth 1 b
))
1347 (Math-numberp (nth 2 b
)))
1348 (math-add (math-mul a
(nth 1 b
))
1349 (math-mul a
(nth 2 b
))))
1350 (and (eq (car-safe b
) '-
)
1352 (or (Math-numberp (nth 1 b
))
1353 (Math-numberp (nth 2 b
)))
1354 (math-sub (math-mul a
(nth 1 b
))
1355 (math-mul a
(nth 2 b
))))
1356 (and (eq (car-safe b
) '*)
1357 (Math-numberp (nth 1 b
))
1358 (not (Math-numberp a
))
1359 (math-mul (nth 1 b
) (math-mul a
(nth 2 b
))))
1360 (and (eq (car-safe a
) 'calcFunc-idn
)
1362 (or (and (eq (car-safe b
) 'calcFunc-idn
)
1364 (list 'calcFunc-idn
(math-mul (nth 1 a
) (nth 1 b
))))
1365 (and (math-known-scalarp b
)
1366 (list 'calcFunc-idn
(math-mul (nth 1 a
) b
)))
1367 (and (math-known-matrixp b
)
1368 (math-mul (nth 1 a
) b
))))
1369 (and (eq (car-safe b
) 'calcFunc-idn
)
1371 (or (and (math-known-scalarp a
)
1372 (list 'calcFunc-idn
(math-mul a
(nth 1 b
))))
1373 (and (math-known-matrixp a
)
1374 (math-mul a
(nth 1 b
)))))
1375 (and (math-looks-negp b
)
1376 (math-mul (math-neg a
) (math-neg b
)))
1377 (and (eq (car-safe b
) '-
)
1379 (math-mul (math-neg a
) (math-neg b
)))
1381 ((eq (car-safe b
) '*)
1382 (let ((temp (math-combine-prod a
(nth 1 b
) nil nil t
)))
1384 (math-mul temp
(nth 2 b
)))))
1386 (math-combine-prod a b nil nil nil
)))
1387 (and (equal a
'(var nan var-nan
))
1389 (and (equal b
'(var nan var-nan
))
1391 (and (equal a
'(var uinf var-uinf
))
1393 (and (equal b
'(var uinf var-uinf
))
1395 (and (equal b
'(var inf var-inf
))
1396 (let ((s1 (math-possible-signs a
)))
1400 '(intv 3 0 (var inf var-inf
)))
1404 '(intv 3 (neg (var inf var-inf
)) 0))
1405 ((and (eq (car a
) 'intv
) (math-intv-constp a
))
1406 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
)))
1407 ((and (eq (car a
) 'cplx
)
1408 (math-zerop (nth 1 a
)))
1409 (list '* (list 'cplx
0 (calcFunc-sign (nth 2 a
))) b
))
1410 ((eq (car a
) 'polar
)
1411 (list '* (list 'polar
1 (nth 2 a
)) b
)))))
1412 (and (equal a
'(var inf var-inf
))
1417 (defun calcFunc-div (a &rest rest
)
1419 (setq a
(list '/ a
(car rest
))
1423 (defun math-div-objects-fancy (a b
)
1424 (cond ((and (Math-numberp a
) (Math-numberp b
))
1426 (cond ((math-want-polar a b
)
1427 (let ((a (math-polar a
))
1430 (math-div (nth 1 a
) (nth 1 b
))
1431 (math-fix-circular (math-sub (nth 2 a
)
1434 (setq a
(math-complex a
))
1435 (list 'cplx
(math-div (nth 1 a
) b
)
1436 (math-div (nth 2 a
) b
)))
1438 (setq a
(math-complex a
)
1442 (math-add (math-mul (nth 1 a
) (nth 1 b
))
1443 (math-mul (nth 2 a
) (nth 2 b
)))
1444 (math-sub (math-mul (nth 2 a
) (nth 1 b
))
1445 (math-mul (nth 1 a
) (nth 2 b
))))
1446 (math-add (math-sqr (nth 1 b
))
1447 (math-sqr (nth 2 b
))))))))
1449 (if (math-square-matrixp b
)
1450 (let ((n1 (length b
)))
1451 (if (Math-vectorp a
)
1452 (if (math-matrixp a
)
1453 (if (= (length a
) n1
)
1454 (math-lud-solve (math-matrix-lud b
) a b
)
1455 (if (= (length (nth 1 a
)) n1
)
1457 (math-lud-solve (math-matrix-lud
1459 (math-transpose a
) b
))
1460 (math-dimension-error)))
1461 (if (= (length a
) n1
)
1462 (math-mat-col (math-lud-solve (math-matrix-lud b
)
1463 (math-col-matrix a
) b
)
1465 (math-dimension-error)))
1466 (if (Math-equal-int a
1)
1468 (math-mul a
(calcFunc-inv b
)))))
1469 (math-reject-arg b
'square-matrixp
)))
1470 ((and (Math-vectorp a
) (Math-objectp b
))
1471 (math-map-vec-2 'math-div a b
))
1472 ((eq (car-safe a
) 'sdev
)
1473 (if (eq (car-safe b
) 'sdev
)
1474 (let ((x (math-div (nth 1 a
) (nth 1 b
))))
1476 (math-div (math-hypot (nth 2 a
)
1477 (math-mul (nth 2 b
) x
))
1479 (if (or (Math-scalarp b
)
1480 (not (Math-objvecp b
)))
1481 (math-make-sdev (math-div (nth 1 a
) b
) (math-div (nth 2 a
) b
))
1482 (math-reject-arg 'realp b
))))
1483 ((and (eq (car-safe b
) 'sdev
)
1484 (or (Math-scalarp a
)
1485 (not (Math-objvecp a
))))
1486 (let ((x (math-div a
(nth 1 b
))))
1488 (math-div (math-mul (nth 2 b
) x
) (nth 1 b
)))))
1489 ((and (eq (car-safe a
) 'intv
) (Math-anglep b
))
1491 (math-neg (math-div a
(math-neg b
)))
1492 (math-make-intv (nth 1 a
)
1493 (math-div (nth 2 a
) b
)
1494 (math-div (nth 3 a
) b
))))
1495 ((and (eq (car-safe b
) 'intv
) (Math-anglep a
))
1496 (if (or (Math-posp (nth 2 b
))
1497 (and (Math-zerop (nth 2 b
)) (or (memq (nth 1 b
) '(0 1))
1498 calc-infinite-mode
)))
1500 (math-neg (math-div (math-neg a
) b
))
1501 (let ((calc-infinite-mode 1))
1502 (math-make-intv (aref [0 2 1 3] (nth 1 b
))
1503 (math-div a
(nth 3 b
))
1504 (math-div a
(nth 2 b
)))))
1505 (if (or (Math-negp (nth 3 b
))
1506 (and (Math-zerop (nth 3 b
)) (or (memq (nth 1 b
) '(0 2))
1507 calc-infinite-mode
)))
1508 (math-neg (math-div a
(math-neg b
)))
1509 (if calc-infinite-mode
1510 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1511 (math-reject-arg b
"*Division by zero")))))
1512 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
1513 (eq (car-safe b
) 'intv
) (math-intv-constp b
))
1514 (if (or (Math-posp (nth 2 b
))
1515 (and (Math-zerop (nth 2 b
)) (or (memq (nth 1 b
) '(0 1))
1516 calc-infinite-mode
)))
1517 (let* ((calc-infinite-mode 1)
1518 (lo (math-div a
(nth 2 b
)))
1519 (hi (math-div a
(nth 3 b
))))
1520 (or (eq (car-safe lo
) 'intv
)
1521 (setq lo
(list 'intv
(if (memq (nth 1 b
) '(2 3)) 3 0)
1523 (or (eq (car-safe hi
) 'intv
)
1524 (setq hi
(list 'intv
(if (memq (nth 1 b
) '(1 3)) 3 0)
1526 (math-combine-intervals
1527 (nth 2 lo
) (and (or (memq (nth 1 b
) '(2 3))
1528 (and (math-infinitep (nth 2 lo
))
1529 (not (math-zerop (nth 2 b
)))))
1530 (memq (nth 1 lo
) '(2 3)))
1531 (nth 3 lo
) (and (or (memq (nth 1 b
) '(2 3))
1532 (and (math-infinitep (nth 3 lo
))
1533 (not (math-zerop (nth 2 b
)))))
1534 (memq (nth 1 lo
) '(1 3)))
1535 (nth 2 hi
) (and (or (memq (nth 1 b
) '(1 3))
1536 (and (math-infinitep (nth 2 hi
))
1537 (not (math-zerop (nth 3 b
)))))
1538 (memq (nth 1 hi
) '(2 3)))
1539 (nth 3 hi
) (and (or (memq (nth 1 b
) '(1 3))
1540 (and (math-infinitep (nth 3 hi
))
1541 (not (math-zerop (nth 3 b
)))))
1542 (memq (nth 1 hi
) '(1 3)))))
1543 (if (or (Math-negp (nth 3 b
))
1544 (and (Math-zerop (nth 3 b
)) (or (memq (nth 1 b
) '(0 2))
1545 calc-infinite-mode
)))
1546 (math-neg (math-div a
(math-neg b
)))
1547 (if calc-infinite-mode
1548 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1549 (math-reject-arg b
"*Division by zero")))))
1550 ((and (eq (car-safe a
) 'mod
)
1551 (eq (car-safe b
) 'mod
)
1552 (equal (nth 2 a
) (nth 2 b
)))
1553 (math-make-mod (math-div-mod (nth 1 a
) (nth 1 b
) (nth 2 a
))
1555 ((and (eq (car-safe a
) 'mod
)
1557 (math-make-mod (math-div-mod (nth 1 a
) b
(nth 2 a
)) (nth 2 a
)))
1558 ((and (eq (car-safe b
) 'mod
)
1560 (math-make-mod (math-div-mod a
(nth 1 b
) (nth 2 b
)) (nth 2 b
)))
1561 ((eq (car-safe a
) 'hms
)
1562 (if (eq (car-safe b
) 'hms
)
1563 (math-with-extra-prec 1
1564 (math-div (math-from-hms a
'deg
)
1565 (math-from-hms b
'deg
)))
1566 (math-with-extra-prec 2
1567 (math-to-hms (math-div (math-from-hms a
'deg
) b
) 'deg
))))
1568 (t (calc-record-why "*Incompatible arguments for /" a b
))))
1570 (defun math-div-by-zero (a b
)
1571 (if (math-infinitep a
)
1572 (if (or (equal a
'(var nan var-nan
))
1573 (equal b
'(var uinf var-uinf
))
1574 (memq calc-infinite-mode
'(-1 1)))
1576 '(var uinf var-uinf
))
1577 (if calc-infinite-mode
1580 (if (eq calc-infinite-mode
1)
1581 (math-mul a
'(var inf var-inf
))
1582 (if (eq calc-infinite-mode -
1)
1583 (math-mul a
'(neg (var inf var-inf
)))
1584 (if (eq (car-safe a
) 'intv
)
1585 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1586 '(var uinf var-uinf
)))))
1587 (math-reject-arg a
"*Division by zero"))))
1589 (defun math-div-zero (a b
)
1590 (if (math-known-matrixp b
)
1591 (if (math-vectorp b
)
1592 (math-map-vec-2 'math-div a b
)
1593 (math-mimic-ident 0 b
))
1594 (if (equal b
'(var nan var-nan
))
1596 (if (and (eq (car-safe b
) 'intv
) (math-intv-constp b
)
1597 (not (math-posp b
)) (not (math-negp b
)))
1598 (if calc-infinite-mode
1600 (if (and (math-zerop (nth 2 b
))
1601 (memq calc-infinite-mode
'(1 -
1)))
1602 (nth 2 b
) '(neg (var inf var-inf
)))
1603 (if (and (math-zerop (nth 3 b
))
1604 (memq calc-infinite-mode
'(1 -
1)))
1605 (nth 3 b
) '(var inf var-inf
)))
1606 (math-reject-arg b
"*Division by zero"))
1609 (defun math-div-symb-fancy (a b
)
1610 (or (and math-simplify-only
1611 (not (equal a math-simplify-only
))
1613 (and (Math-equal-int b
1) a
)
1614 (and (Math-equal-int b -
1) (math-neg a
))
1615 (and (Math-vectorp a
) (math-known-scalarp b
)
1616 (math-map-vec-2 'math-div a b
))
1617 (and (eq (car-safe b
) '^
)
1618 (or (Math-looks-negp (nth 2 b
)) (Math-equal-int a
1))
1619 (math-mul a
(math-normalize
1620 (list '^
(nth 1 b
) (math-neg (nth 2 b
))))))
1621 (and (eq (car-safe a
) 'neg
)
1622 (math-neg (math-div (nth 1 a
) b
)))
1623 (and (eq (car-safe b
) 'neg
)
1624 (math-neg (math-div a
(nth 1 b
))))
1625 (and (eq (car-safe a
) '/)
1626 (math-div (nth 1 a
) (math-mul (nth 2 a
) b
)))
1627 (and (eq (car-safe b
) '/)
1628 (or (math-known-scalarp (nth 1 b
) t
)
1629 (math-known-scalarp (nth 2 b
) t
))
1630 (math-div (math-mul a
(nth 2 b
)) (nth 1 b
)))
1631 (and (eq (car-safe b
) 'frac
)
1632 (math-mul (math-make-frac (nth 2 b
) (nth 1 b
)) a
))
1633 (and (eq (car-safe a
) '+)
1634 (or (Math-numberp (nth 1 a
))
1635 (Math-numberp (nth 2 a
)))
1637 (math-add (math-div (nth 1 a
) b
)
1638 (math-div (nth 2 a
) b
)))
1639 (and (eq (car-safe a
) '-
)
1640 (or (Math-numberp (nth 1 a
))
1641 (Math-numberp (nth 2 a
)))
1643 (math-sub (math-div (nth 1 a
) b
)
1644 (math-div (nth 2 a
) b
)))
1645 (and (or (eq (car-safe a
) '-
)
1646 (math-looks-negp a
))
1648 (math-div (math-neg a
) (math-neg b
)))
1649 (and (eq (car-safe b
) '-
)
1651 (math-div (math-neg a
) (math-neg b
)))
1652 (and (eq (car-safe a
) 'calcFunc-idn
)
1654 (or (and (eq (car-safe b
) 'calcFunc-idn
)
1656 (list 'calcFunc-idn
(math-div (nth 1 a
) (nth 1 b
))))
1657 (and (math-known-scalarp b
)
1658 (list 'calcFunc-idn
(math-div (nth 1 a
) b
)))
1659 (and (math-known-matrixp b
)
1660 (math-div (nth 1 a
) b
))))
1661 (and (eq (car-safe b
) 'calcFunc-idn
)
1663 (or (and (math-known-scalarp a
)
1664 (list 'calcFunc-idn
(math-div a
(nth 1 b
))))
1665 (and (math-known-matrixp a
)
1666 (math-div a
(nth 1 b
)))))
1667 (if (and calc-matrix-mode
1668 (or (math-known-matrixp a
) (math-known-matrixp b
)))
1669 (math-combine-prod a b nil t nil
)
1670 (if (eq (car-safe a
) '*)
1671 (if (eq (car-safe b
) '*)
1672 (let ((c (math-combine-prod (nth 1 a
) (nth 1 b
) nil t t
)))
1674 (math-div (math-mul c
(nth 2 a
)) (nth 2 b
))))
1675 (let ((c (math-combine-prod (nth 1 a
) b nil t t
)))
1677 (math-mul c
(nth 2 a
)))))
1678 (if (eq (car-safe b
) '*)
1679 (let ((c (math-combine-prod a
(nth 1 b
) nil t t
)))
1681 (math-div c
(nth 2 b
))))
1682 (math-combine-prod a b nil t nil
))))
1683 (and (math-infinitep a
)
1684 (if (math-infinitep b
)
1686 (if (or (equal a
'(var nan var-nan
))
1687 (equal a
'(var uinf var-uinf
)))
1689 (if (equal a
'(var inf var-inf
))
1690 (if (or (math-posp b
)
1691 (and (eq (car-safe b
) 'intv
)
1692 (math-zerop (nth 2 b
))))
1693 (if (and (eq (car-safe b
) 'intv
)
1694 (not (math-intv-constp b t
)))
1695 '(intv 3 0 (var inf var-inf
))
1697 (if (or (math-negp b
)
1698 (and (eq (car-safe b
) 'intv
)
1699 (math-zerop (nth 3 b
))))
1700 (if (and (eq (car-safe b
) 'intv
)
1701 (not (math-intv-constp b t
)))
1702 '(intv 3 (neg (var inf var-inf
)) 0)
1704 (if (and (eq (car-safe b
) 'intv
)
1705 (math-negp (nth 2 b
)) (math-posp (nth 3 b
)))
1706 '(intv 3 (neg (var inf var-inf
))
1707 (var inf var-inf
)))))))))
1708 (and (math-infinitep b
)
1709 (if (equal b
'(var nan var-nan
))
1711 (let ((calc-infinite-mode 1))
1712 (math-mul-zero b a
))))
1716 (defun calcFunc-mod (a b
)
1717 (math-normalize (list '% a b
)))
1719 (defun math-mod-fancy (a b
)
1720 (cond ((equal b
'(var inf var-inf
))
1721 (if (or (math-posp a
) (math-zerop a
))
1725 (if (eq (car-safe a
) 'intv
)
1726 (if (math-negp (nth 2 a
))
1727 '(intv 3 0 (var inf var-inf
))
1730 ((and (eq (car-safe a
) 'mod
) (Math-realp b
) (math-posp b
))
1731 (math-make-mod (nth 1 a
) b
))
1732 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a t
) (math-posp b
))
1733 (math-mod-intv a b
))
1736 (calc-record-why 'anglep b
)
1737 (calc-record-why 'anglep a
))
1741 (defun calcFunc-pow (a b
)
1742 (math-normalize (list '^ a b
)))
1744 (defun math-pow-of-zero (a b
)
1746 (if calc-infinite-mode
1748 (math-reject-arg (list '^ a b
) "*Indeterminate form"))
1749 (if (math-floatp b
) (setq a
(math-float a
)))
1754 (if (math-infinitep b
)
1756 (if (and (eq (car b
) 'intv
) (math-intv-constp b
)
1758 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1759 (if (math-objectp b
)
1763 (defun math-pow-zero (a b
)
1764 (if (eq (car-safe a
) 'mod
)
1765 (math-make-mod 1 (nth 2 a
))
1766 (if (math-known-matrixp a
)
1767 (math-mimic-ident 1 a
)
1768 (if (math-infinitep a
)
1770 (if (and (eq (car a
) 'intv
) (math-intv-constp a
)
1771 (or (and (not (math-posp a
)) (not (math-negp a
)))
1772 (not (math-intv-constp a t
))))
1773 '(intv 3 (neg (var inf var-inf
)) (var inf var-inf
))
1774 (if (or (math-floatp a
) (math-floatp b
))
1775 '(float 1 0) 1))))))
1777 (defun math-pow-fancy (a b
)
1778 (cond ((and (Math-numberp a
) (Math-numberp b
))
1779 (or (if (memq (math-quarter-integer b
) '(1 2 3))
1780 (let ((sqrt (math-sqrt (if (math-floatp b
)
1781 (math-float a
) a
))))
1782 (and (Math-numberp sqrt
)
1783 (math-pow sqrt
(math-mul 2 b
))))
1784 (and (eq (car b
) 'frac
)
1785 (integerp (nth 2 b
))
1787 (let ((root (math-nth-root a
(nth 2 b
))))
1788 (and root
(math-ipow root
(nth 1 b
))))))
1789 (and (or (eq a
10) (equal a
'(float 1 1)))
1790 (math-num-integerp b
)
1791 (calcFunc-scf '(float 1 0) b
))
1792 (and calc-symbolic-mode
1794 (math-with-extra-prec 2
1796 (math-float (math-mul b
(math-ln-raw (math-float a
))))))))
1797 ((or (not (Math-objvecp a
))
1798 (not (Math-objectp b
)))
1800 (cond ((and math-simplify-only
1801 (not (equal a math-simplify-only
)))
1803 ((and (eq (car-safe a
) '*)
1804 (or (math-known-num-integerp b
)
1805 (math-known-nonnegp (nth 1 a
))
1806 (math-known-nonnegp (nth 2 a
))))
1807 (math-mul (math-pow (nth 1 a
) b
)
1808 (math-pow (nth 2 a
) b
)))
1809 ((and (eq (car-safe a
) '/)
1810 (or (math-known-num-integerp b
)
1811 (math-known-nonnegp (nth 2 a
))))
1812 (math-div (math-pow (nth 1 a
) b
)
1813 (math-pow (nth 2 a
) b
)))
1814 ((and (eq (car-safe a
) '/)
1815 (math-known-nonnegp (nth 1 a
))
1816 (not (math-equal-int (nth 1 a
) 1)))
1817 (math-mul (math-pow (nth 1 a
) b
)
1818 (math-pow (math-div 1 (nth 2 a
)) b
)))
1819 ((and (eq (car-safe a
) '^
)
1820 (or (math-known-num-integerp b
)
1821 (math-known-nonnegp (nth 1 a
))))
1822 (math-pow (nth 1 a
) (math-mul (nth 2 a
) b
)))
1823 ((and (eq (car-safe a
) 'calcFunc-sqrt
)
1824 (or (math-known-num-integerp b
)
1825 (math-known-nonnegp (nth 1 a
))))
1826 (math-pow (nth 1 a
) (math-div b
2)))
1827 ((and (eq (car-safe a
) '^
)
1828 (math-known-evenp (nth 2 a
))
1829 (memq (math-quarter-integer b
) '(1 2 3))
1830 (math-known-realp (nth 1 a
)))
1831 (math-abs (math-pow (nth 1 a
) (math-mul (nth 2 a
) b
))))
1832 ((and (math-looks-negp a
)
1833 (math-known-integerp b
)
1834 (setq temp
(or (and (math-known-evenp b
)
1835 (math-pow (math-neg a
) b
))
1836 (and (math-known-oddp b
)
1837 (math-neg (math-pow (math-neg a
)
1840 ((and (eq (car-safe a
) 'calcFunc-abs
)
1841 (math-known-realp (nth 1 a
))
1842 (math-known-evenp b
))
1843 (math-pow (nth 1 a
) b
))
1845 (cond ((equal a
'(var nan var-nan
))
1848 (math-mul (math-pow -
1 b
) (math-pow (nth 1 a
) b
)))
1852 (if (math-floatp b
) '(float 0 0) 0))
1853 ((and (eq (car-safe b
) 'intv
)
1854 (math-intv-constp b
))
1855 '(intv 3 0 (var inf var-inf
)))
1857 '(var nan var-nan
))))
1860 (cond ((math-negp b
)
1861 (math-pow (math-div 1 a
) (math-neg b
)))
1862 ((not (math-posp b
))
1864 ((math-equal-int (setq scale
(calcFunc-abssqr a
)) 1)
1866 ((Math-lessp scale
1)
1867 (if (math-floatp a
) '(float 0 0) 0))
1871 '(var uinf var-uinf
))
1872 ((and (eq (car a
) 'intv
)
1873 (math-intv-constp a
))
1874 (if (Math-lessp -
1 a
)
1875 (if (math-equal-int (nth 3 a
) 1)
1877 '(intv 3 0 (var inf var-inf
)))
1878 '(intv 3 (neg (var inf var-inf
))
1879 (var inf var-inf
))))
1880 (t (list '^ a b
)))))
1881 ((and (eq (car-safe a
) 'calcFunc-idn
)
1883 (math-known-num-integerp b
))
1884 (list 'calcFunc-idn
(math-pow (nth 1 a
) b
)))
1885 (t (if (Math-objectp a
)
1886 (calc-record-why 'objectp b
)
1887 (calc-record-why 'objectp a
))
1889 ((and (eq (car-safe a
) 'sdev
) (eq (car-safe b
) 'sdev
))
1890 (if (and (math-constp a
) (math-constp b
))
1891 (math-with-extra-prec 2
1892 (let* ((ln (math-ln-raw (math-float (nth 1 a
))))
1894 (math-float (math-mul (nth 1 b
) ln
)))))
1899 (math-hypot (math-mul (nth 2 a
)
1900 (math-div (nth 1 b
) (nth 1 a
)))
1901 (math-mul (nth 2 b
) ln
))))))
1902 (let ((pow (math-pow (nth 1 a
) (nth 1 b
))))
1906 (math-hypot (math-mul (nth 2 a
)
1907 (math-div (nth 1 b
) (nth 1 a
)))
1908 (math-mul (nth 2 b
) (calcFunc-ln
1910 ((and (eq (car-safe a
) 'sdev
) (Math-numberp b
))
1912 (math-with-extra-prec 2
1913 (let ((pow (math-pow (nth 1 a
) (math-sub b
1))))
1914 (math-make-sdev (math-mul pow
(nth 1 a
))
1915 (math-mul pow
(math-mul (nth 2 a
) b
)))))
1916 (math-make-sdev (math-pow (nth 1 a
) b
)
1917 (math-mul (math-pow (nth 1 a
) (math-add b -
1))
1918 (math-mul (nth 2 a
) b
)))))
1919 ((and (eq (car-safe b
) 'sdev
) (Math-numberp a
))
1920 (math-with-extra-prec 2
1921 (let* ((ln (math-ln-raw (math-float a
)))
1922 (pow (calcFunc-exp (math-mul (nth 1 b
) ln
))))
1923 (math-make-sdev pow
(math-mul pow
(math-mul (nth 2 b
) ln
))))))
1924 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
1926 (or (Math-natnump b
)
1927 (Math-posp (nth 2 a
))
1928 (and (math-zerop (nth 2 a
))
1930 (and (Math-integerp b
) calc-infinite-mode
)))
1931 (Math-negp (nth 3 a
))
1932 (and (math-zerop (nth 3 a
))
1934 (and (Math-integerp b
) calc-infinite-mode
)))))
1936 (setq a
(math-abs a
)))
1937 (let ((calc-infinite-mode (if (math-zerop (nth 3 a
)) -
1 1)))
1938 (math-sort-intv (nth 1 a
)
1939 (math-pow (nth 2 a
) b
)
1940 (math-pow (nth 3 a
) b
))))
1941 ((and (eq (car-safe b
) 'intv
) (math-intv-constp b
)
1942 (Math-realp a
) (Math-posp a
))
1943 (math-sort-intv (nth 1 b
)
1944 (math-pow a
(nth 2 b
))
1945 (math-pow a
(nth 3 b
))))
1946 ((and (eq (car-safe a
) 'intv
) (math-intv-constp a
)
1947 (eq (car-safe b
) 'intv
) (math-intv-constp b
)
1948 (or (and (not (Math-negp (nth 2 a
)))
1949 (not (Math-negp (nth 2 b
))))
1950 (and (Math-posp (nth 2 a
))
1951 (not (Math-posp (nth 3 b
))))))
1952 (let ((lo (math-pow a
(nth 2 b
)))
1953 (hi (math-pow a
(nth 3 b
))))
1954 (or (eq (car-safe lo
) 'intv
)
1955 (setq lo
(list 'intv
(if (memq (nth 1 b
) '(2 3)) 3 0) lo lo
)))
1956 (or (eq (car-safe hi
) 'intv
)
1957 (setq hi
(list 'intv
(if (memq (nth 1 b
) '(1 3)) 3 0) hi hi
)))
1958 (math-combine-intervals
1959 (nth 2 lo
) (and (or (memq (nth 1 b
) '(2 3))
1960 (math-infinitep (nth 2 lo
)))
1961 (memq (nth 1 lo
) '(2 3)))
1962 (nth 3 lo
) (and (or (memq (nth 1 b
) '(2 3))
1963 (math-infinitep (nth 3 lo
)))
1964 (memq (nth 1 lo
) '(1 3)))
1965 (nth 2 hi
) (and (or (memq (nth 1 b
) '(1 3))
1966 (math-infinitep (nth 2 hi
)))
1967 (memq (nth 1 hi
) '(2 3)))
1968 (nth 3 hi
) (and (or (memq (nth 1 b
) '(1 3))
1969 (math-infinitep (nth 3 hi
)))
1970 (memq (nth 1 hi
) '(1 3))))))
1971 ((and (eq (car-safe a
) 'mod
) (eq (car-safe b
) 'mod
)
1972 (equal (nth 2 a
) (nth 2 b
)))
1973 (math-make-mod (math-pow-mod (nth 1 a
) (nth 1 b
) (nth 2 a
))
1975 ((and (eq (car-safe a
) 'mod
) (Math-anglep b
))
1976 (math-make-mod (math-pow-mod (nth 1 a
) b
(nth 2 a
)) (nth 2 a
)))
1977 ((and (eq (car-safe b
) 'mod
) (Math-anglep a
))
1978 (math-make-mod (math-pow-mod a
(nth 1 b
) (nth 2 b
)) (nth 2 b
)))
1979 ((not (Math-numberp a
))
1980 (math-reject-arg a
'numberp
))
1982 (math-reject-arg b
'numberp
))))
1984 (defun math-quarter-integer (x)
1985 (if (Math-integerp x
)
1989 (setq x
(math-quarter-integer (math-neg x
)))
1991 (if (eq (car x
) 'frac
)
1992 (if (eq (nth 2 x
) 2)
1994 (and (eq (nth 2 x
) 4)
1997 (%
(if (consp x
) (nth 1 x
) x
) 4))))
1998 (if (eq (car x
) 'float
)
1999 (if (>= (nth 2 x
) 0)
2001 (if (= (nth 2 x
) -
1)
2004 (and (= (%
(if (consp x
) (nth 1 x
) x
) 10) 5) 2))
2005 (if (= (nth 2 x
) -
2)
2008 x
(%
(if (consp x
) (nth 1 x
) x
) 100))
2010 (if (= x
75) 3)))))))))))
2012 ;;; This assumes A < M and M > 0.
2013 (defun math-pow-mod (a b m
) ; [R R R R]
2014 (if (and (Math-integerp a
) (Math-integerp b
) (Math-integerp m
))
2016 (math-div-mod 1 (math-pow-mod a
(Math-integer-neg b
) m
) m
)
2019 (math-pow-mod-step a b m
)))
2020 (math-mod (math-pow a b
) m
)))
2022 (defun math-pow-mod-step (a n m
) ; [I I I I]
2023 (math-working "pow" a
)
2028 (let ((rest (math-pow-mod-step
2029 (math-imod (math-mul a a
) m
)
2034 (math-mod (math-mul a rest
) m
)))))))
2035 (math-working "pow" val
)
2039 ;;; Compute the minimum of two real numbers. [R R R] [Public]
2040 (defun math-min (a b
)
2041 (if (and (consp a
) (eq (car a
) 'intv
))
2042 (if (and (consp b
) (eq (car b
) 'intv
))
2043 (let ((lo (nth 2 a
))
2044 (lom (memq (nth 1 a
) '(2 3)))
2046 (him (memq (nth 1 a
) '(1 3)))
2048 (if (= (setq res
(math-compare (nth 2 b
) lo
)) -
1)
2049 (setq lo
(nth 2 b
) lom
(memq (nth 1 b
) '(2 3)))
2051 (setq lom
(or lom
(memq (nth 1 b
) '(2 3))))))
2052 (if (= (setq res
(math-compare (nth 3 b
) hi
)) -
1)
2053 (setq hi
(nth 3 b
) him
(memq (nth 1 b
) '(1 3)))
2055 (setq him
(or him
(memq (nth 1 b
) '(1 3))))))
2056 (math-make-intv (+ (if lom
2 0) (if him
1 0)) lo hi
))
2057 (math-min a
(list 'intv
3 b b
)))
2058 (if (and (consp b
) (eq (car b
) 'intv
))
2059 (math-min (list 'intv
3 a a
) b
)
2060 (let ((res (math-compare a b
)))
2067 (defun calcFunc-min (&optional a
&rest b
)
2070 (if (not (or (Math-anglep a
) (eq (car a
) 'date
)
2071 (and (eq (car a
) 'intv
) (math-intv-constp a
))
2072 (math-infinitep a
)))
2073 (math-reject-arg a
'anglep
))
2074 (math-min-list a b
)))
2076 (defun math-min-list (a b
)
2078 (if (or (Math-anglep (car b
)) (eq (car b
) 'date
)
2079 (and (eq (car (car b
)) 'intv
) (math-intv-constp (car b
)))
2080 (math-infinitep (car b
)))
2081 (math-min-list (math-min a
(car b
)) (cdr b
))
2082 (math-reject-arg (car b
) 'anglep
))
2085 ;;; Compute the maximum of two real numbers. [R R R] [Public]
2086 (defun math-max (a b
)
2087 (if (or (and (consp a
) (eq (car a
) 'intv
))
2088 (and (consp b
) (eq (car b
) 'intv
)))
2089 (math-neg (math-min (math-neg a
) (math-neg b
)))
2090 (let ((res (math-compare a b
)))
2097 (defun calcFunc-max (&optional a
&rest b
)
2099 '(neg (var inf var-inf
))
2100 (if (not (or (Math-anglep a
) (eq (car a
) 'date
)
2101 (and (eq (car a
) 'intv
) (math-intv-constp a
))
2102 (math-infinitep a
)))
2103 (math-reject-arg a
'anglep
))
2104 (math-max-list a b
)))
2106 (defun math-max-list (a b
)
2108 (if (or (Math-anglep (car b
)) (eq (car b
) 'date
)
2109 (and (eq (car (car b
)) 'intv
) (math-intv-constp (car b
)))
2110 (math-infinitep (car b
)))
2111 (math-max-list (math-max a
(car b
)) (cdr b
))
2112 (math-reject-arg (car b
) 'anglep
))
2116 ;;; Compute the absolute value of A. [O O; r r] [Public]
2118 (cond ((Math-negp a
)
2123 (math-hypot (nth 1 a
) (nth 2 a
)))
2124 ((eq (car a
) 'polar
)
2127 (if (cdr (cdr (cdr a
)))
2128 (math-sqrt (calcFunc-abssqr a
))
2130 (math-hypot (nth 1 a
) (nth 2 a
))
2132 (math-abs (nth 1 a
))
2135 (list 'sdev
(math-abs (nth 1 a
)) (nth 2 a
)))
2136 ((and (eq (car a
) 'intv
) (math-intv-constp a
))
2139 (let* ((nlo (math-neg (nth 2 a
)))
2140 (res (math-compare nlo
(nth 3 a
))))
2142 (math-make-intv (if (memq (nth 1 a
) '(0 1)) 2 3) 0 nlo
))
2144 (math-make-intv (if (eq (nth 1 a
) 0) 2 3) 0 nlo
))
2146 (math-make-intv (if (memq (nth 1 a
) '(0 2)) 2 3)
2148 ((math-looks-negp a
)
2149 (list 'calcFunc-abs
(math-neg a
)))
2150 ((let ((signs (math-possible-signs a
)))
2151 (or (and (memq signs
'(2 4 6)) a
)
2152 (and (memq signs
'(1 3)) (math-neg a
)))))
2153 ((let ((inf (math-infinitep a
)))
2155 (if (equal inf
'(var nan var-nan
))
2157 '(var inf var-inf
)))))
2158 (t (calc-record-why 'numvecp a
)
2159 (list 'calcFunc-abs a
))))
2161 (defalias 'calcFunc-abs
'math-abs
)
2163 (defun math-float-fancy (a)
2164 (cond ((eq (car a
) 'intv
)
2165 (cons (car a
) (cons (nth 1 a
) (mapcar 'math-float
(nthcdr 2 a
)))))
2166 ((and (memq (car a
) '(* /))
2167 (math-numberp (nth 1 a
)))
2168 (list (car a
) (math-float (nth 1 a
))
2169 (list 'calcFunc-float
(nth 2 a
))))
2170 ((and (eq (car a
) '/)
2171 (eq (car (nth 1 a
)) '*)
2172 (math-numberp (nth 1 (nth 1 a
))))
2173 (list '* (math-float (nth 1 (nth 1 a
)))
2174 (list 'calcFunc-float
(list '/ (nth 2 (nth 1 a
)) (nth 2 a
)))))
2175 ((math-infinitep a
) a
)
2176 ((eq (car a
) 'calcFunc-float
) a
)
2177 ((let ((func (assq (car a
) '((calcFunc-floor . calcFunc-ffloor
)
2178 (calcFunc-ceil . calcFunc-fceil
)
2179 (calcFunc-trunc . calcFunc-ftrunc
)
2180 (calcFunc-round . calcFunc-fround
)
2181 (calcFunc-rounde . calcFunc-frounde
)
2182 (calcFunc-roundu . calcFunc-froundu
)))))
2183 (and func
(cons (cdr func
) (cdr a
)))))
2184 (t (math-reject-arg a
'objectp
))))
2186 (defalias 'calcFunc-float
'math-float
)
2188 (defun math-trunc-fancy (a)
2189 (cond ((eq (car a
) 'frac
) (math-quotient (nth 1 a
) (nth 2 a
)))
2190 ((eq (car a
) 'cplx
) (math-trunc (nth 1 a
)))
2191 ((eq (car a
) 'polar
) (math-trunc (math-complex a
)))
2192 ((eq (car a
) 'hms
) (list 'hms
(nth 1 a
) 0 0))
2193 ((eq (car a
) 'date
) (list 'date
(math-trunc (nth 1 a
))))
2195 (if (math-messy-integerp (nth 2 a
))
2196 (math-trunc (math-make-mod (nth 1 a
) (math-trunc (nth 2 a
))))
2197 (math-make-mod (math-trunc (nth 1 a
)) (nth 2 a
))))
2199 (math-make-intv (+ (if (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
2200 (memq (nth 1 a
) '(0 1)))
2202 (if (and (equal (nth 3 a
) '(var inf var-inf
))
2203 (memq (nth 1 a
) '(0 2)))
2205 (if (and (Math-negp (nth 2 a
))
2206 (Math-num-integerp (nth 2 a
))
2207 (memq (nth 1 a
) '(0 1)))
2208 (math-add (math-trunc (nth 2 a
)) 1)
2209 (math-trunc (nth 2 a
)))
2210 (if (and (Math-posp (nth 3 a
))
2211 (Math-num-integerp (nth 3 a
))
2212 (memq (nth 1 a
) '(0 2)))
2213 (math-add (math-trunc (nth 3 a
)) -
1)
2214 (math-trunc (nth 3 a
)))))
2215 ((math-provably-integerp a
) a
)
2217 (math-map-vec (function (lambda (x) (math-trunc x prec
))) a
))
2219 (if (or (math-posp a
) (math-negp a
))
2221 '(var nan var-nan
)))
2222 ((math-to-integer a
))
2223 (t (math-reject-arg a
'numberp
))))
2225 (defun math-trunc-special (a prec
)
2226 (if (Math-messy-integerp prec
)
2227 (setq prec
(math-trunc prec
)))
2229 (math-reject-arg prec
'fixnump
))
2230 (if (and (<= prec
0)
2231 (math-provably-integerp a
))
2233 (calcFunc-scf (math-trunc (let ((calc-prefer-frac t
))
2234 (calcFunc-scf a prec
)))
2237 (defun math-to-integer (a)
2238 (let ((func (assq (car-safe a
) '((calcFunc-ffloor . calcFunc-floor
)
2239 (calcFunc-fceil . calcFunc-ceil
)
2240 (calcFunc-ftrunc . calcFunc-trunc
)
2241 (calcFunc-fround . calcFunc-round
)
2242 (calcFunc-frounde . calcFunc-rounde
)
2243 (calcFunc-froundu . calcFunc-roundu
)))))
2244 (and func
(= (length a
) 2)
2245 (cons (cdr func
) (cdr a
)))))
2247 (defun calcFunc-ftrunc (a &optional prec
)
2248 (if (and (Math-messy-integerp a
)
2249 (or (not prec
) (and (integerp prec
)
2252 (math-float (math-trunc a prec
))))
2254 (defun math-floor-fancy (a)
2255 (cond ((math-provably-integerp a
) a
)
2257 (if (or (math-posp a
)
2258 (and (math-zerop (nth 2 a
))
2259 (math-zerop (nth 3 a
))))
2261 (math-add (math-trunc a
) -
1)))
2262 ((eq (car a
) 'date
) (list 'date
(math-floor (nth 1 a
))))
2264 (math-make-intv (+ (if (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
2265 (memq (nth 1 a
) '(0 1)))
2267 (if (and (equal (nth 3 a
) '(var inf var-inf
))
2268 (memq (nth 1 a
) '(0 2)))
2270 (math-floor (nth 2 a
))
2271 (if (and (Math-num-integerp (nth 3 a
))
2272 (memq (nth 1 a
) '(0 2)))
2273 (math-add (math-floor (nth 3 a
)) -
1)
2274 (math-floor (nth 3 a
)))))
2276 (math-map-vec (function (lambda (x) (math-floor x prec
))) a
))
2278 (if (or (math-posp a
) (math-negp a
))
2280 '(var nan var-nan
)))
2281 ((math-to-integer a
))
2282 (t (math-reject-arg a
'anglep
))))
2284 (defun math-floor-special (a prec
)
2285 (if (Math-messy-integerp prec
)
2286 (setq prec
(math-trunc prec
)))
2288 (math-reject-arg prec
'fixnump
))
2289 (if (and (<= prec
0)
2290 (math-provably-integerp a
))
2292 (calcFunc-scf (math-floor (let ((calc-prefer-frac t
))
2293 (calcFunc-scf a prec
)))
2296 (defun calcFunc-ffloor (a &optional prec
)
2297 (if (and (Math-messy-integerp a
)
2298 (or (not prec
) (and (integerp prec
)
2301 (math-float (math-floor a prec
))))
2303 ;;; Coerce A to be an integer (by truncation toward plus infinity). [I N]
2304 (defun math-ceiling (a &optional prec
) ; [Public]
2306 (if (Math-messy-integerp prec
)
2307 (setq prec
(math-trunc prec
)))
2309 (math-reject-arg prec
'fixnump
))
2310 (if (and (<= prec
0)
2311 (math-provably-integerp a
))
2313 (calcFunc-scf (math-ceiling (let ((calc-prefer-frac t
))
2314 (calcFunc-scf a prec
)))
2316 ((Math-integerp a
) a
)
2317 ((Math-messy-integerp a
) (math-trunc a
))
2320 (math-add (math-trunc a
) 1)
2322 ((math-provably-integerp a
) a
)
2324 (if (or (math-negp a
)
2325 (and (math-zerop (nth 2 a
))
2326 (math-zerop (nth 3 a
))))
2328 (math-add (math-trunc a
) 1)))
2329 ((eq (car a
) 'date
) (list 'date
(math-ceiling (nth 1 a
))))
2331 (math-make-intv (+ (if (and (equal (nth 2 a
) '(neg (var inf var-inf
)))
2332 (memq (nth 1 a
) '(0 1)))
2334 (if (and (equal (nth 3 a
) '(var inf var-inf
))
2335 (memq (nth 1 a
) '(0 2)))
2337 (if (and (Math-num-integerp (nth 2 a
))
2338 (memq (nth 1 a
) '(0 1)))
2339 (math-add (math-floor (nth 2 a
)) 1)
2340 (math-ceiling (nth 2 a
)))
2341 (math-ceiling (nth 3 a
))))
2343 (math-map-vec (function (lambda (x) (math-ceiling x prec
))) a
))
2345 (if (or (math-posp a
) (math-negp a
))
2347 '(var nan var-nan
)))
2348 ((math-to-integer a
))
2349 (t (math-reject-arg a
'anglep
))))
2351 (defalias 'calcFunc-ceil
'math-ceiling
)
2353 (defun calcFunc-fceil (a &optional prec
)
2354 (if (and (Math-messy-integerp a
)
2355 (or (not prec
) (and (integerp prec
)
2358 (math-float (math-ceiling a prec
))))
2360 (defvar math-rounding-mode nil
)
2362 ;;; Coerce A to be an integer (by rounding to nearest integer). [I N] [Public]
2363 (defun math-round (a &optional prec
)
2365 (if (Math-messy-integerp prec
)
2366 (setq prec
(math-trunc prec
)))
2368 (math-reject-arg prec
'fixnump
))
2369 (if (and (<= prec
0)
2370 (math-provably-integerp a
))
2372 (calcFunc-scf (math-round (let ((calc-prefer-frac t
))
2373 (calcFunc-scf a prec
)))
2376 (if (Math-num-integerp a
)
2378 (if (and (Math-negp a
) (not (eq math-rounding-mode
'up
)))
2379 (math-neg (math-round (math-neg a
)))
2380 (setq a
(let ((calc-angle-mode 'deg
)) ; in case of HMS forms
2381 (math-add a
(if (Math-ratp a
)
2384 (if (and (Math-num-integerp a
) (eq math-rounding-mode
'even
))
2386 (setq a
(math-floor a
))
2388 (setq a
(math-sub a
1)))
2391 ((math-provably-integerp a
) a
)
2392 ((eq (car a
) 'date
) (list 'date
(math-round (nth 1 a
))))
2394 (math-floor (math-add a
'(frac 1 2))))
2396 (math-map-vec (function (lambda (x) (math-round x prec
))) a
))
2398 (if (or (math-posp a
) (math-negp a
))
2400 '(var nan var-nan
)))
2401 ((math-to-integer a
))
2402 (t (math-reject-arg a
'anglep
))))
2404 (defalias 'calcFunc-round
'math-round
)
2406 (defsubst calcFunc-rounde
(a &optional prec
)
2407 (let ((math-rounding-mode 'even
))
2408 (math-round a prec
)))
2410 (defsubst calcFunc-roundu
(a &optional prec
)
2411 (let ((math-rounding-mode 'up
))
2412 (math-round a prec
)))
2414 (defun calcFunc-fround (a &optional prec
)
2415 (if (and (Math-messy-integerp a
)
2416 (or (not prec
) (and (integerp prec
)
2419 (math-float (math-round a prec
))))
2421 (defsubst calcFunc-frounde
(a &optional prec
)
2422 (let ((math-rounding-mode 'even
))
2423 (calcFunc-fround a prec
)))
2425 (defsubst calcFunc-froundu
(a &optional prec
)
2426 (let ((math-rounding-mode 'up
))
2427 (calcFunc-fround a prec
)))
2429 ;;; Pull floating-point values apart into mantissa and exponent.
2430 (defun calcFunc-mant (x)
2432 (if (or (Math-ratp x
)
2435 (list 'float
(nth 1 x
) (- 1 (math-numdigs (nth 1 x
)))))
2436 (calc-record-why 'realp x
)
2437 (list 'calcFunc-mant x
)))
2439 (defun calcFunc-xpon (x)
2441 (if (or (Math-ratp x
)
2444 (math-normalize (+ (nth 2 x
) (1- (math-numdigs (nth 1 x
))))))
2445 (calc-record-why 'realp x
)
2446 (list 'calcFunc-xpon x
)))
2448 (defun calcFunc-scf (x n
)
2454 (math-scale-int x n
)
2455 (math-div x
(math-scale-int 1 (- n
)))))
2458 (math-make-frac (math-scale-int (nth 1 x
) n
) (nth 2 x
))
2459 (math-make-frac (nth 1 x
) (math-scale-int (nth 2 x
) (- n
)))))
2460 ((eq (car x
) 'float
)
2461 (math-make-float (nth 1 x
) (+ (nth 2 x
) n
)))
2462 ((memq (car x
) '(cplx sdev
))
2465 (calcFunc-scf (nth 1 x
) n
)
2466 (calcFunc-scf (nth 2 x
) n
))))
2467 ((memq (car x
) '(polar mod
))
2470 (calcFunc-scf (nth 1 x
) n
)
2476 (calcFunc-scf (nth 2 x
) n
)
2477 (calcFunc-scf (nth 3 x
) n
))))
2479 (math-map-vec (function (lambda (x) (calcFunc-scf x n
))) x
))
2483 (calc-record-why 'realp x
)
2484 (list 'calcFunc-scf x n
)))
2485 (if (math-messy-integerp n
)
2486 (if (< (nth 2 n
) 10)
2487 (calcFunc-scf x
(math-trunc n
))
2489 (if (math-integerp n
)
2491 (calc-record-why 'integerp n
)
2492 (list 'calcFunc-scf x n
)))))
2495 (defun calcFunc-incr (x &optional step relative-to
)
2496 (or step
(setq step
1))
2497 (cond ((not (Math-integerp step
))
2498 (math-reject-arg step
'integerp
))
2501 ((eq (car x
) 'float
)
2502 (if (and (math-zerop x
)
2503 (eq (car-safe relative-to
) 'float
))
2505 (calcFunc-scf relative-to
(- 1 calc-internal-prec
)))
2506 (math-add-float x
(math-make-float
2509 (- (math-numdigs (nth 1 x
))
2510 calc-internal-prec
))))))
2512 (if (Math-integerp (nth 1 x
))
2514 (math-add x
(list 'hms
0 0 step
))))
2516 (math-reject-arg x
'realp
))))
2518 (defsubst calcFunc-decr
(x &optional step relative-to
)
2519 (calcFunc-incr x
(math-neg (or step
1)) relative-to
))
2521 (defun calcFunc-percent (x)
2522 (if (math-objectp x
)
2523 (let ((calc-prefer-frac nil
))
2525 (list 'calcFunc-percent x
)))
2527 (defun calcFunc-relch (x y
)
2528 (if (and (math-objectp x
) (math-objectp y
))
2529 (math-div (math-sub y x
) x
)
2530 (list 'calcFunc-relch x y
)))
2532 ;;; Compute the absolute value squared of A. [F N] [Public]
2533 (defun calcFunc-abssqr (a)
2534 (cond ((Math-realp a
)
2537 (math-add (math-sqr (nth 1 a
))
2538 (math-sqr (nth 2 a
))))
2539 ((eq (car a
) 'polar
)
2540 (math-sqr (nth 1 a
)))
2541 ((and (memq (car a
) '(sdev intv
)) (math-constp a
))
2542 (math-sqr (math-abs a
)))
2544 (math-reduce-vec 'math-add
(math-map-vec 'calcFunc-abssqr a
)))
2545 ((math-known-realp a
)
2547 ((let ((inf (math-infinitep a
)))
2549 (math-mul (calcFunc-abssqr (math-infinite-dir a inf
)) inf
))))
2550 (t (calc-record-why 'numvecp a
)
2551 (list 'calcFunc-abssqr a
))))
2553 (defsubst math-sqr
(a)
2558 (defun calcFunc-idiv (a b
) ; [I I I] [Public]
2559 (cond ((and (Math-natnump a
) (Math-natnump b
) (not (eq b
0)))
2560 (math-quotient a b
))
2563 (let ((calc-prefer-frac t
))
2564 (math-floor (math-div a b
)))
2565 (math-reject-arg b
'realp
)))
2566 ((eq (car-safe a
) 'hms
)
2567 (if (eq (car-safe b
) 'hms
)
2568 (let ((calc-prefer-frac t
))
2569 (math-floor (math-div a b
)))
2570 (math-reject-arg b
'hmsp
)))
2571 ((and (or (eq (car-safe a
) 'intv
) (Math-realp a
))
2572 (or (eq (car-safe b
) 'intv
) (Math-realp b
)))
2573 (math-floor (math-div a b
)))
2574 ((or (math-infinitep a
)
2577 (t (math-reject-arg a
'anglep
))))
2580 ;;; Combine two terms being added, if possible.
2581 (defun math-combine-sum (a b nega negb scalar-okay
)
2582 (if (and scalar-okay
(Math-objvecp a
) (Math-objvecp b
))
2583 (math-add-or-sub a b nega negb
)
2584 (let ((amult 1) (bmult 1))
2586 (cond ((and (eq (car a
) '*)
2587 (Math-objectp (nth 1 a
)))
2588 (setq amult
(nth 1 a
)
2590 ((and (eq (car a
) '/)
2591 (Math-objectp (nth 2 a
)))
2592 (setq amult
(if (Math-integerp (nth 2 a
))
2593 (list 'frac
1 (nth 2 a
))
2594 (math-div 1 (nth 2 a
)))
2600 (cond ((and (eq (car b
) '*)
2601 (Math-objectp (nth 1 b
)))
2602 (setq bmult
(nth 1 b
)
2604 ((and (eq (car b
) '/)
2605 (Math-objectp (nth 2 b
)))
2606 (setq bmult
(if (Math-integerp (nth 2 b
))
2607 (list 'frac
1 (nth 2 b
))
2608 (math-div 1 (nth 2 b
)))
2613 (and (if math-simplifying
2617 (if nega
(setq amult
(math-neg amult
)))
2618 (if negb
(setq bmult
(math-neg bmult
)))
2619 (setq amult
(math-add amult bmult
))
2620 (math-mul amult a
))))))
2622 (defun math-add-or-sub (a b aneg bneg
)
2623 (if aneg
(setq a
(math-neg a
)))
2624 (if bneg
(setq b
(math-neg b
)))
2625 (if (or (Math-vectorp a
) (Math-vectorp b
))
2626 (math-normalize (list '+ a b
))
2629 (defvar math-combine-prod-e
'(var e var-e
))
2631 ;;; The following is expanded out four ways for speed.
2632 (defun math-combine-prod (a b inva invb scalar-okay
)
2634 ((or (and inva
(Math-zerop a
))
2635 (and invb
(Math-zerop b
)))
2637 ((and scalar-okay
(Math-objvecp a
) (Math-objvecp b
))
2638 (setq a
(math-mul-or-div a b inva invb
))
2639 (and (Math-objvecp a
)
2641 ((and (eq (car-safe a
) '^
)
2643 (math-looks-negp (nth 2 a
)))
2644 (math-mul (math-pow (nth 1 a
) (math-neg (nth 2 a
))) b
))
2645 ((and (eq (car-safe b
) '^
)
2647 (math-looks-negp (nth 2 b
)))
2648 (math-mul a
(math-pow (nth 1 b
) (math-neg (nth 2 b
)))))
2649 (t (let ((apow 1) (bpow 1))
2651 (cond ((and (eq (car a
) '^
)
2652 (or math-simplifying
2653 (Math-numberp (nth 2 a
))))
2654 (setq apow
(nth 2 a
)
2656 ((eq (car a
) 'calcFunc-sqrt
)
2657 (setq apow
'(frac 1 2)
2659 ((and (eq (car a
) 'calcFunc-exp
)
2660 (or math-simplifying
2661 (Math-numberp (nth 1 a
))))
2662 (setq apow
(nth 1 a
)
2663 a math-combine-prod-e
))))
2664 (and (consp a
) (eq (car a
) 'frac
)
2665 (Math-lessp (nth 1 a
) (nth 2 a
))
2666 (setq a
(math-div 1 a
) apow
(math-neg apow
)))
2668 (cond ((and (eq (car b
) '^
)
2669 (or math-simplifying
2670 (Math-numberp (nth 2 b
))))
2671 (setq bpow
(nth 2 b
)
2673 ((eq (car b
) 'calcFunc-sqrt
)
2674 (setq bpow
'(frac 1 2)
2676 ((and (eq (car b
) 'calcFunc-exp
)
2677 (or math-simplifying
2678 (Math-numberp (nth 1 b
))))
2679 (setq bpow
(nth 1 b
)
2680 b math-combine-prod-e
))))
2681 (and (consp b
) (eq (car b
) 'frac
)
2682 (Math-lessp (nth 1 b
) (nth 2 b
))
2683 (setq b
(math-div 1 b
) bpow
(math-neg bpow
)))
2684 (if inva
(setq apow
(math-neg apow
)))
2685 (if invb
(setq bpow
(math-neg bpow
)))
2686 (or (and (if math-simplifying
2687 (math-commutative-equal a b
)
2689 (let ((sumpow (math-add apow bpow
)))
2690 (and (or (not (Math-integerp a
))
2692 (eq (eq (car-safe apow
) 'frac
)
2693 (eq (car-safe bpow
) 'frac
)))
2695 (and (math-looks-negp sumpow
)
2696 (Math-ratp a
) (Math-posp a
)
2697 (setq a
(math-div 1 a
)
2698 sumpow
(math-neg sumpow
)))
2699 (cond ((equal sumpow
'(frac 1 2))
2700 (list 'calcFunc-sqrt a
))
2701 ((equal sumpow
'(frac -
1 2))
2702 (math-div 1 (list 'calcFunc-sqrt a
)))
2703 ((and (eq a math-combine-prod-e
)
2705 (list 'calcFunc-exp sumpow
))
2709 (inexact-result (list '^ a sumpow
)))))))))
2710 (and math-simplifying-units
2711 math-combining-units
2712 (let* ((ua (math-check-unit-name a
))
2715 (eq ua
(setq ub
(math-check-unit-name b
)))
2717 (setq ua
(if (eq (nth 1 a
) (car ua
))
2719 (nth 1 (assq (aref (symbol-name (nth 1 a
))
2721 math-unit-prefixes
)))
2722 ub
(if (eq (nth 1 b
) (car ub
))
2724 (nth 1 (assq (aref (symbol-name (nth 1 b
))
2726 math-unit-prefixes
))))
2727 (if (Math-lessp ua ub
)
2729 (setq temp a a b b temp
2730 temp ua ua ub ub temp
2731 temp apow apow bpow bpow temp
)))
2732 (math-mul (math-pow (math-div ua ub
) apow
)
2733 (math-pow b
(math-add apow bpow
)))))))
2734 (and (equal apow bpow
)
2735 (Math-natnump a
) (Math-natnump b
)
2736 (cond ((equal apow
'(frac 1 2))
2737 (list 'calcFunc-sqrt
(math-mul a b
)))
2738 ((equal apow
'(frac -
1 2))
2739 (math-div 1 (list 'calcFunc-sqrt
(math-mul a b
))))
2741 (setq a
(math-mul a b
))
2744 (inexact-result (list '^ a apow
)))))))))))
2746 (defun math-mul-or-div (a b ainv binv
)
2747 (if (or (Math-vectorp a
) (Math-vectorp b
))
2751 (list '/ (math-div 1 a
) b
)
2758 (math-div (math-div 1 a
) b
)
2764 (defun math-commutative-equal (a b
)
2765 (if (memq (car-safe a
) '(+ -
))
2766 (and (memq (car-safe b
) '(+ -
))
2767 (let ((bterms nil
) aterms p
)
2768 (math-commutative-collect b nil
)
2769 (setq aterms bterms bterms nil
)
2770 (math-commutative-collect a nil
)
2771 (and (= (length aterms
) (length bterms
))
2776 (while (and p
(not (equal (car aterms
)
2780 (setq bterms
(delq (car p
) bterms
)
2781 aterms
(cdr aterms
)))
2785 (defun math-commutative-collect (b neg
)
2786 (if (eq (car-safe b
) '+)
2788 (math-commutative-collect (nth 1 b
) neg
)
2789 (math-commutative-collect (nth 2 b
) neg
))
2790 (if (eq (car-safe b
) '-
)
2792 (math-commutative-collect (nth 1 b
) neg
)
2793 (math-commutative-collect (nth 2 b
) (not neg
)))
2794 (setq bterms
(cons (if neg
(math-neg b
) b
) bterms
)))))
2796 ;;; calc-arith.el ends here
