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3132f345 CW |
1 | ;;; calc-math.el --- mathematical functions for Calc |
2 | ||
58ba2f8f | 3 | ;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004, |
ae940284 | 4 | ;; 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc. |
3132f345 CW |
5 | |
6 | ;; Author: David Gillespie <daveg@synaptics.com> | |
e8fff8ed | 7 | ;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com> |
136211a9 EZ |
8 | |
9 | ;; This file is part of GNU Emacs. | |
10 | ||
662c9c64 | 11 | ;; GNU Emacs is free software: you can redistribute it and/or modify |
7c671b23 | 12 | ;; it under the terms of the GNU General Public License as published by |
662c9c64 GM |
13 | ;; the Free Software Foundation, either version 3 of the License, or |
14 | ;; (at your option) any later version. | |
7c671b23 | 15 | |
136211a9 | 16 | ;; GNU Emacs is distributed in the hope that it will be useful, |
7c671b23 GM |
17 | ;; but WITHOUT ANY WARRANTY; without even the implied warranty of |
18 | ;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
19 | ;; GNU General Public License for more details. | |
20 | ||
21 | ;; You should have received a copy of the GNU General Public License | |
662c9c64 | 22 | ;; along with GNU Emacs. If not, see <http://www.gnu.org/licenses/>. |
136211a9 | 23 | |
3132f345 | 24 | ;;; Commentary: |
136211a9 | 25 | |
3132f345 | 26 | ;;; Code: |
136211a9 EZ |
27 | |
28 | ;; This file is autoloaded from calc-ext.el. | |
136211a9 | 29 | |
95995a85 | 30 | (require 'calc-ext) |
136211a9 EZ |
31 | (require 'calc-macs) |
32 | ||
178b8baf JB |
33 | |
34 | ;;; Find out how many 9s in 9.9999... will give distinct Emacs floats, | |
35 | ;;; then back off by one. | |
36 | ||
37 | (defvar math-emacs-precision | |
38 | (let* ((n 1) | |
39 | (x 9) | |
40 | (xx (+ x (* 9 (expt 10 (- n)))))) | |
41 | (while (/= x xx) | |
42 | (progn | |
43 | (setq n (1+ n)) | |
44 | (setq x xx) | |
45 | (setq xx (+ x (* 9 (expt 10 (- n))))))) | |
46 | (1- n)) | |
47 | "The number of digits in an Emacs float.") | |
48 | ||
49 | ;;; Find the largest power of 10 which is an Emacs float, | |
50 | ;;; then back off by one so that any float d.dddd...eN | |
51 | ;;; is an Emacs float, for acceptable d.dddd.... | |
52 | ||
53 | (defvar math-largest-emacs-expt | |
86e405cf JB |
54 | (let ((x 1) |
55 | (pow 1e2)) | |
56 | ;; The following loop is for efficiency; it should stop when | |
57 | ;; 10^(2x) is too large. This could be indicated by a range | |
16635351 | 58 | ;; error when computing 10^(2x) or an infinite value for 10^(2x). |
86e405cf JB |
59 | (while (and |
60 | pow | |
16635351 | 61 | (< pow 1.0e+INF)) |
86e405cf JB |
62 | (setq x (* 2 x)) |
63 | (setq pow (condition-case nil | |
64 | (expt 10.0 (* 2 x)) | |
65 | (error nil)))) | |
66 | ;; The following loop should stop when 10^(x+1) is too large. | |
67 | (setq pow (condition-case nil | |
68 | (expt 10.0 (1+ x)) | |
69 | (error nil))) | |
70 | (while (and | |
71 | pow | |
16635351 | 72 | (< pow 1.0e+INF)) |
86e405cf JB |
73 | (setq x (1+ x)) |
74 | (setq pow (condition-case nil | |
75 | (expt 10.0 (1+ x)) | |
76 | (error nil)))) | |
77 | (1- x)) | |
178b8baf JB |
78 | "The largest exponent which Calc will convert to an Emacs float.") |
79 | ||
80 | (defvar math-smallest-emacs-expt | |
81 | (let ((x -1)) | |
82 | (while (condition-case nil | |
f1640784 | 83 | (> (expt 10.0 x) 0.0) |
178b8baf JB |
84 | (error nil)) |
85 | (setq x (* 2 x))) | |
86 | (setq x (/ x 2)) | |
87 | (while (condition-case nil | |
f1640784 | 88 | (> (expt 10.0 x) 0.0) |
178b8baf JB |
89 | (error nil)) |
90 | (setq x (1- x))) | |
91 | (+ x 2)) | |
92 | "The smallest exponent which Calc will convert to an Emacs float.") | |
93 | ||
94 | (defun math-use-emacs-fn (fn x) | |
95 | "Use the native Emacs function FN to evaluate the Calc number X. | |
96 | If this can't be done, return NIL." | |
97 | (and | |
98 | (<= calc-internal-prec math-emacs-precision) | |
99 | (math-realp x) | |
100 | (let* ((fx (math-float x)) | |
101 | (xpon (+ (nth 2 x) (1- (math-numdigs (nth 1 x)))))) | |
102 | (and (<= math-smallest-emacs-expt xpon) | |
103 | (<= xpon math-largest-emacs-expt) | |
104 | (condition-case nil | |
105 | (math-read-number | |
106 | (number-to-string | |
107 | (funcall fn | |
7b3752ab JB |
108 | (string-to-number |
109 | (let ((calc-number-radix 10)) | |
110 | (math-format-number (math-float x))))))) | |
178b8baf JB |
111 | (error nil)))))) |
112 | ||
136211a9 EZ |
113 | (defun calc-sqrt (arg) |
114 | (interactive "P") | |
115 | (calc-slow-wrapper | |
116 | (if (calc-is-inverse) | |
117 | (calc-unary-op "^2" 'calcFunc-sqr arg) | |
491c3062 | 118 | (calc-unary-op "sqrt" 'calcFunc-sqrt arg)))) |
136211a9 EZ |
119 | |
120 | (defun calc-isqrt (arg) | |
121 | (interactive "P") | |
122 | (calc-slow-wrapper | |
123 | (if (calc-is-inverse) | |
124 | (calc-unary-op "^2" 'calcFunc-sqr arg) | |
491c3062 | 125 | (calc-unary-op "isqt" 'calcFunc-isqrt arg)))) |
136211a9 EZ |
126 | |
127 | ||
128 | (defun calc-hypot (arg) | |
129 | (interactive "P") | |
130 | (calc-slow-wrapper | |
491c3062 | 131 | (calc-binary-op "hypt" 'calcFunc-hypot arg))) |
136211a9 EZ |
132 | |
133 | (defun calc-ln (arg) | |
134 | (interactive "P") | |
135 | (calc-invert-func) | |
491c3062 | 136 | (calc-exp arg)) |
136211a9 EZ |
137 | |
138 | (defun calc-log10 (arg) | |
139 | (interactive "P") | |
140 | (calc-hyperbolic-func) | |
491c3062 | 141 | (calc-ln arg)) |
136211a9 EZ |
142 | |
143 | (defun calc-log (arg) | |
144 | (interactive "P") | |
145 | (calc-slow-wrapper | |
146 | (if (calc-is-inverse) | |
147 | (calc-binary-op "alog" 'calcFunc-alog arg) | |
491c3062 | 148 | (calc-binary-op "log" 'calcFunc-log arg)))) |
136211a9 EZ |
149 | |
150 | (defun calc-ilog (arg) | |
151 | (interactive "P") | |
152 | (calc-slow-wrapper | |
153 | (if (calc-is-inverse) | |
154 | (calc-binary-op "alog" 'calcFunc-alog arg) | |
491c3062 | 155 | (calc-binary-op "ilog" 'calcFunc-ilog arg)))) |
136211a9 EZ |
156 | |
157 | (defun calc-lnp1 (arg) | |
158 | (interactive "P") | |
159 | (calc-invert-func) | |
491c3062 | 160 | (calc-expm1 arg)) |
136211a9 EZ |
161 | |
162 | (defun calc-exp (arg) | |
163 | (interactive "P") | |
164 | (calc-slow-wrapper | |
165 | (if (calc-is-hyperbolic) | |
166 | (if (calc-is-inverse) | |
167 | (calc-unary-op "lg10" 'calcFunc-log10 arg) | |
168 | (calc-unary-op "10^" 'calcFunc-exp10 arg)) | |
169 | (if (calc-is-inverse) | |
170 | (calc-unary-op "ln" 'calcFunc-ln arg) | |
491c3062 | 171 | (calc-unary-op "exp" 'calcFunc-exp arg))))) |
136211a9 EZ |
172 | |
173 | (defun calc-expm1 (arg) | |
174 | (interactive "P") | |
175 | (calc-slow-wrapper | |
176 | (if (calc-is-inverse) | |
177 | (calc-unary-op "ln+1" 'calcFunc-lnp1 arg) | |
491c3062 | 178 | (calc-unary-op "ex-1" 'calcFunc-expm1 arg)))) |
136211a9 EZ |
179 | |
180 | (defun calc-pi () | |
181 | (interactive) | |
182 | (calc-slow-wrapper | |
183 | (if (calc-is-inverse) | |
184 | (if (calc-is-hyperbolic) | |
185 | (if calc-symbolic-mode | |
186 | (calc-pop-push-record 0 "phi" '(var phi var-phi)) | |
187 | (calc-pop-push-record 0 "phi" (math-phi))) | |
188 | (if calc-symbolic-mode | |
189 | (calc-pop-push-record 0 "gmma" '(var gamma var-gamma)) | |
190 | (calc-pop-push-record 0 "gmma" (math-gamma-const)))) | |
191 | (if (calc-is-hyperbolic) | |
192 | (if calc-symbolic-mode | |
193 | (calc-pop-push-record 0 "e" '(var e var-e)) | |
194 | (calc-pop-push-record 0 "e" (math-e))) | |
195 | (if calc-symbolic-mode | |
196 | (calc-pop-push-record 0 "pi" '(var pi var-pi)) | |
491c3062 | 197 | (calc-pop-push-record 0 "pi" (math-pi))))))) |
136211a9 EZ |
198 | |
199 | (defun calc-sin (arg) | |
200 | (interactive "P") | |
201 | (calc-slow-wrapper | |
202 | (if (calc-is-hyperbolic) | |
203 | (if (calc-is-inverse) | |
204 | (calc-unary-op "asnh" 'calcFunc-arcsinh arg) | |
205 | (calc-unary-op "sinh" 'calcFunc-sinh arg)) | |
206 | (if (calc-is-inverse) | |
207 | (calc-unary-op "asin" 'calcFunc-arcsin arg) | |
491c3062 | 208 | (calc-unary-op "sin" 'calcFunc-sin arg))))) |
136211a9 EZ |
209 | |
210 | (defun calc-arcsin (arg) | |
211 | (interactive "P") | |
212 | (calc-invert-func) | |
491c3062 | 213 | (calc-sin arg)) |
136211a9 EZ |
214 | |
215 | (defun calc-sinh (arg) | |
216 | (interactive "P") | |
217 | (calc-hyperbolic-func) | |
491c3062 | 218 | (calc-sin arg)) |
136211a9 EZ |
219 | |
220 | (defun calc-arcsinh (arg) | |
221 | (interactive "P") | |
222 | (calc-invert-func) | |
223 | (calc-hyperbolic-func) | |
491c3062 | 224 | (calc-sin arg)) |
136211a9 | 225 | |
f53e6c20 JB |
226 | (defun calc-sec (arg) |
227 | (interactive "P") | |
228 | (calc-slow-wrapper | |
229 | (if (calc-is-hyperbolic) | |
230 | (calc-unary-op "sech" 'calcFunc-sech arg) | |
231 | (calc-unary-op "sec" 'calcFunc-sec arg)))) | |
232 | ||
233 | (defun calc-sech (arg) | |
234 | (interactive "P") | |
235 | (calc-hyperbolic-func) | |
236 | (calc-sec arg)) | |
237 | ||
136211a9 EZ |
238 | (defun calc-cos (arg) |
239 | (interactive "P") | |
240 | (calc-slow-wrapper | |
241 | (if (calc-is-hyperbolic) | |
242 | (if (calc-is-inverse) | |
243 | (calc-unary-op "acsh" 'calcFunc-arccosh arg) | |
244 | (calc-unary-op "cosh" 'calcFunc-cosh arg)) | |
245 | (if (calc-is-inverse) | |
246 | (calc-unary-op "acos" 'calcFunc-arccos arg) | |
491c3062 | 247 | (calc-unary-op "cos" 'calcFunc-cos arg))))) |
136211a9 EZ |
248 | |
249 | (defun calc-arccos (arg) | |
250 | (interactive "P") | |
251 | (calc-invert-func) | |
491c3062 | 252 | (calc-cos arg)) |
136211a9 EZ |
253 | |
254 | (defun calc-cosh (arg) | |
255 | (interactive "P") | |
256 | (calc-hyperbolic-func) | |
491c3062 | 257 | (calc-cos arg)) |
136211a9 EZ |
258 | |
259 | (defun calc-arccosh (arg) | |
260 | (interactive "P") | |
261 | (calc-invert-func) | |
262 | (calc-hyperbolic-func) | |
491c3062 | 263 | (calc-cos arg)) |
136211a9 | 264 | |
f53e6c20 JB |
265 | (defun calc-csc (arg) |
266 | (interactive "P") | |
267 | (calc-slow-wrapper | |
268 | (if (calc-is-hyperbolic) | |
269 | (calc-unary-op "csch" 'calcFunc-csch arg) | |
270 | (calc-unary-op "csc" 'calcFunc-csc arg)))) | |
271 | ||
272 | (defun calc-csch (arg) | |
273 | (interactive "P") | |
274 | (calc-hyperbolic-func) | |
275 | (calc-csc arg)) | |
276 | ||
136211a9 EZ |
277 | (defun calc-sincos () |
278 | (interactive) | |
279 | (calc-slow-wrapper | |
280 | (if (calc-is-inverse) | |
281 | (calc-enter-result 1 "asnc" (list 'calcFunc-arcsincos (calc-top-n 1))) | |
491c3062 | 282 | (calc-enter-result 1 "sncs" (list 'calcFunc-sincos (calc-top-n 1)))))) |
136211a9 EZ |
283 | |
284 | (defun calc-tan (arg) | |
285 | (interactive "P") | |
286 | (calc-slow-wrapper | |
287 | (if (calc-is-hyperbolic) | |
288 | (if (calc-is-inverse) | |
289 | (calc-unary-op "atnh" 'calcFunc-arctanh arg) | |
290 | (calc-unary-op "tanh" 'calcFunc-tanh arg)) | |
291 | (if (calc-is-inverse) | |
292 | (calc-unary-op "atan" 'calcFunc-arctan arg) | |
491c3062 | 293 | (calc-unary-op "tan" 'calcFunc-tan arg))))) |
136211a9 EZ |
294 | |
295 | (defun calc-arctan (arg) | |
296 | (interactive "P") | |
297 | (calc-invert-func) | |
491c3062 | 298 | (calc-tan arg)) |
136211a9 EZ |
299 | |
300 | (defun calc-tanh (arg) | |
301 | (interactive "P") | |
302 | (calc-hyperbolic-func) | |
491c3062 | 303 | (calc-tan arg)) |
136211a9 EZ |
304 | |
305 | (defun calc-arctanh (arg) | |
306 | (interactive "P") | |
307 | (calc-invert-func) | |
308 | (calc-hyperbolic-func) | |
491c3062 | 309 | (calc-tan arg)) |
136211a9 | 310 | |
f53e6c20 JB |
311 | (defun calc-cot (arg) |
312 | (interactive "P") | |
313 | (calc-slow-wrapper | |
314 | (if (calc-is-hyperbolic) | |
315 | (calc-unary-op "coth" 'calcFunc-coth arg) | |
316 | (calc-unary-op "cot" 'calcFunc-cot arg)))) | |
317 | ||
eb1ef455 | 318 | (defun calc-coth (arg) |
f53e6c20 JB |
319 | (interactive "P") |
320 | (calc-hyperbolic-func) | |
eb1ef455 | 321 | (calc-cot arg)) |
f53e6c20 | 322 | |
136211a9 EZ |
323 | (defun calc-arctan2 () |
324 | (interactive) | |
325 | (calc-slow-wrapper | |
491c3062 | 326 | (calc-enter-result 2 "atn2" (cons 'calcFunc-arctan2 (calc-top-list-n 2))))) |
136211a9 EZ |
327 | |
328 | (defun calc-conj (arg) | |
329 | (interactive "P") | |
330 | (calc-wrapper | |
491c3062 | 331 | (calc-unary-op "conj" 'calcFunc-conj arg))) |
136211a9 EZ |
332 | |
333 | (defun calc-imaginary () | |
334 | (interactive) | |
335 | (calc-slow-wrapper | |
491c3062 | 336 | (calc-pop-push-record 1 "i*" (math-imaginary (calc-top-n 1))))) |
136211a9 | 337 | |
136211a9 EZ |
338 | (defun calc-to-degrees (arg) |
339 | (interactive "P") | |
340 | (calc-wrapper | |
491c3062 | 341 | (calc-unary-op ">deg" 'calcFunc-deg arg))) |
136211a9 EZ |
342 | |
343 | (defun calc-to-radians (arg) | |
344 | (interactive "P") | |
345 | (calc-wrapper | |
491c3062 | 346 | (calc-unary-op ">rad" 'calcFunc-rad arg))) |
136211a9 EZ |
347 | |
348 | ||
349 | (defun calc-degrees-mode (arg) | |
350 | (interactive "p") | |
351 | (cond ((= arg 1) | |
352 | (calc-wrapper | |
353 | (calc-change-mode 'calc-angle-mode 'deg) | |
3132f345 | 354 | (message "Angles measured in degrees"))) |
136211a9 EZ |
355 | ((= arg 2) (calc-radians-mode)) |
356 | ((= arg 3) (calc-hms-mode)) | |
491c3062 | 357 | (t (error "Prefix argument out of range")))) |
136211a9 EZ |
358 | |
359 | (defun calc-radians-mode () | |
360 | (interactive) | |
361 | (calc-wrapper | |
362 | (calc-change-mode 'calc-angle-mode 'rad) | |
3132f345 | 363 | (message "Angles measured in radians"))) |
136211a9 EZ |
364 | |
365 | ||
366 | ;;; Compute the integer square-root floor(sqrt(A)). A > 0. [I I] [Public] | |
367 | ;;; This method takes advantage of the fact that Newton's method starting | |
368 | ;;; with an overestimate always works, even using truncating integer division! | |
369 | (defun math-isqrt (a) | |
370 | (cond ((Math-zerop a) a) | |
371 | ((not (math-natnump a)) | |
372 | (math-reject-arg a 'natnump)) | |
373 | ((integerp a) | |
374 | (math-isqrt-small a)) | |
375 | (t | |
491c3062 | 376 | (math-normalize (cons 'bigpos (cdr (math-isqrt-bignum (cdr a)))))))) |
136211a9 EZ |
377 | |
378 | (defun calcFunc-isqrt (a) | |
379 | (if (math-realp a) | |
380 | (math-isqrt (math-floor a)) | |
491c3062 | 381 | (math-floor (math-sqrt a)))) |
136211a9 EZ |
382 | |
383 | ||
f0529b5b | 384 | ;;; This returns (flag . result) where the flag is t if A is a perfect square. |
136211a9 EZ |
385 | (defun math-isqrt-bignum (a) ; [P.l L] |
386 | (let ((len (length a))) | |
387 | (if (= (% len 2) 0) | |
388 | (let* ((top (nthcdr (- len 2) a))) | |
389 | (math-isqrt-bignum-iter | |
390 | a | |
98888d77 | 391 | (math-scale-bignum-digit-size |
136211a9 EZ |
392 | (math-bignum-big |
393 | (1+ (math-isqrt-small | |
98888d77 | 394 | (+ (* (nth 1 top) math-bignum-digit-size) (car top))))) |
136211a9 EZ |
395 | (1- (/ len 2))))) |
396 | (let* ((top (nth (1- len) a))) | |
397 | (math-isqrt-bignum-iter | |
398 | a | |
98888d77 | 399 | (math-scale-bignum-digit-size |
136211a9 | 400 | (list (1+ (math-isqrt-small top))) |
491c3062 | 401 | (/ len 2))))))) |
136211a9 EZ |
402 | |
403 | (defun math-isqrt-bignum-iter (a guess) ; [l L l] | |
404 | (math-working "isqrt" (cons 'bigpos guess)) | |
405 | (let* ((q (math-div-bignum a guess)) | |
406 | (s (math-add-bignum (car q) guess)) | |
407 | (g2 (math-div2-bignum s)) | |
408 | (comp (math-compare-bignum g2 guess))) | |
409 | (if (< comp 0) | |
410 | (math-isqrt-bignum-iter a g2) | |
411 | (cons (and (= comp 0) | |
412 | (math-zerop-bignum (cdr q)) | |
413 | (= (% (car s) 2) 0)) | |
491c3062 | 414 | guess)))) |
136211a9 EZ |
415 | |
416 | (defun math-zerop-bignum (a) | |
417 | (and (eq (car a) 0) | |
418 | (progn | |
419 | (while (eq (car (setq a (cdr a))) 0)) | |
491c3062 | 420 | (null a)))) |
136211a9 | 421 | |
98888d77 | 422 | (defun math-scale-bignum-digit-size (a n) ; [L L S] |
136211a9 EZ |
423 | (while (> n 0) |
424 | (setq a (cons 0 a) | |
425 | n (1- n))) | |
491c3062 | 426 | a) |
136211a9 EZ |
427 | |
428 | (defun math-isqrt-small (a) ; A > 0. [S S] | |
98888d77 JB |
429 | (let ((g (cond ((>= a 1000000) 10000) |
430 | ((>= a 10000) 1000) | |
136211a9 EZ |
431 | ((>= a 100) 100) |
432 | (t 10))) | |
433 | g2) | |
434 | (while (< (setq g2 (/ (+ g (/ a g)) 2)) g) | |
435 | (setq g g2)) | |
491c3062 | 436 | g)) |
136211a9 EZ |
437 | |
438 | ||
439 | ||
440 | ||
441 | ;;; Compute the square root of a number. | |
442 | ;;; [T N] if possible, else [F N] if possible, else [C N]. [Public] | |
443 | (defun math-sqrt (a) | |
444 | (or | |
445 | (and (Math-zerop a) a) | |
446 | (and (math-known-nonposp a) | |
447 | (math-imaginary (math-sqrt (math-neg a)))) | |
448 | (and (integerp a) | |
449 | (let ((sqrt (math-isqrt-small a))) | |
450 | (if (= (* sqrt sqrt) a) | |
451 | sqrt | |
452 | (if calc-symbolic-mode | |
453 | (list 'calcFunc-sqrt a) | |
454 | (math-sqrt-float (math-float a) (math-float sqrt)))))) | |
455 | (and (eq (car-safe a) 'bigpos) | |
456 | (let* ((res (math-isqrt-bignum (cdr a))) | |
457 | (sqrt (math-normalize (cons 'bigpos (cdr res))))) | |
458 | (if (car res) | |
459 | sqrt | |
460 | (if calc-symbolic-mode | |
461 | (list 'calcFunc-sqrt a) | |
462 | (math-sqrt-float (math-float a) (math-float sqrt)))))) | |
463 | (and (eq (car-safe a) 'frac) | |
464 | (let* ((num-res (math-isqrt-bignum (cdr (Math-bignum-test (nth 1 a))))) | |
465 | (num-sqrt (math-normalize (cons 'bigpos (cdr num-res)))) | |
466 | (den-res (math-isqrt-bignum (cdr (Math-bignum-test (nth 2 a))))) | |
467 | (den-sqrt (math-normalize (cons 'bigpos (cdr den-res))))) | |
468 | (if (and (car num-res) (car den-res)) | |
469 | (list 'frac num-sqrt den-sqrt) | |
470 | (if calc-symbolic-mode | |
471 | (if (or (car num-res) (car den-res)) | |
472 | (math-div (if (car num-res) | |
473 | num-sqrt (list 'calcFunc-sqrt (nth 1 a))) | |
474 | (if (car den-res) | |
475 | den-sqrt (list 'calcFunc-sqrt (nth 2 a)))) | |
476 | (list 'calcFunc-sqrt a)) | |
477 | (math-sqrt-float (math-float a) | |
478 | (math-div (math-float num-sqrt) den-sqrt)))))) | |
479 | (and (eq (car-safe a) 'float) | |
480 | (if calc-symbolic-mode | |
481 | (if (= (% (nth 2 a) 2) 0) | |
482 | (let ((res (math-isqrt-bignum | |
483 | (cdr (Math-bignum-test (nth 1 a)))))) | |
484 | (if (car res) | |
485 | (math-make-float (math-normalize | |
486 | (cons 'bigpos (cdr res))) | |
487 | (/ (nth 2 a) 2)) | |
488 | (signal 'inexact-result nil))) | |
489 | (signal 'inexact-result nil)) | |
490 | (math-sqrt-float a))) | |
491 | (and (eq (car-safe a) 'cplx) | |
492 | (math-with-extra-prec 2 | |
493 | (let* ((d (math-abs a)) | |
494 | (imag (math-sqrt (math-mul (math-sub d (nth 1 a)) | |
495 | '(float 5 -1))))) | |
496 | (list 'cplx | |
497 | (math-sqrt (math-mul (math-add d (nth 1 a)) '(float 5 -1))) | |
498 | (if (math-negp (nth 2 a)) (math-neg imag) imag))))) | |
499 | (and (eq (car-safe a) 'polar) | |
500 | (list 'polar | |
501 | (math-sqrt (nth 1 a)) | |
502 | (math-mul (nth 2 a) '(float 5 -1)))) | |
503 | (and (eq (car-safe a) 'sdev) | |
504 | (let ((sqrt (math-sqrt (nth 1 a)))) | |
505 | (math-make-sdev sqrt | |
506 | (math-div (nth 2 a) (math-mul sqrt 2))))) | |
507 | (and (eq (car-safe a) 'intv) | |
508 | (not (math-negp (nth 2 a))) | |
509 | (math-make-intv (nth 1 a) (math-sqrt (nth 2 a)) (math-sqrt (nth 3 a)))) | |
510 | (and (eq (car-safe a) '*) | |
511 | (or (math-known-nonnegp (nth 1 a)) | |
512 | (math-known-nonnegp (nth 2 a))) | |
513 | (math-mul (math-sqrt (nth 1 a)) (math-sqrt (nth 2 a)))) | |
514 | (and (eq (car-safe a) '/) | |
515 | (or (and (math-known-nonnegp (nth 2 a)) | |
516 | (math-div (math-sqrt (nth 1 a)) (math-sqrt (nth 2 a)))) | |
517 | (and (math-known-nonnegp (nth 1 a)) | |
518 | (not (math-equal-int (nth 1 a) 1)) | |
519 | (math-mul (math-sqrt (nth 1 a)) | |
520 | (math-sqrt (math-div 1 (nth 2 a))))))) | |
521 | (and (eq (car-safe a) '^) | |
522 | (math-known-evenp (nth 2 a)) | |
523 | (math-known-realp (nth 1 a)) | |
524 | (math-abs (math-pow (nth 1 a) (math-div (nth 2 a) 2)))) | |
525 | (let ((inf (math-infinitep a))) | |
526 | (and inf | |
527 | (math-mul (math-sqrt (math-infinite-dir a inf)) inf))) | |
528 | (progn | |
529 | (calc-record-why 'numberp a) | |
491c3062 | 530 | (list 'calcFunc-sqrt a)))) |
3132f345 | 531 | (defalias 'calcFunc-sqrt 'math-sqrt) |
136211a9 EZ |
532 | |
533 | (defun math-infinite-dir (a &optional inf) | |
534 | (or inf (setq inf (math-infinitep a))) | |
491c3062 | 535 | (math-normalize (math-expr-subst a inf 1))) |
136211a9 EZ |
536 | |
537 | (defun math-sqrt-float (a &optional guess) ; [F F F] | |
538 | (if calc-symbolic-mode | |
539 | (signal 'inexact-result nil) | |
491c3062 | 540 | (math-with-extra-prec 1 (math-sqrt-raw a guess)))) |
136211a9 EZ |
541 | |
542 | (defun math-sqrt-raw (a &optional guess) ; [F F F] | |
543 | (if (not (Math-posp a)) | |
544 | (math-sqrt a) | |
17cc361e JB |
545 | (cond |
546 | ((math-use-emacs-fn 'sqrt a)) | |
547 | (t | |
548 | (if (null guess) | |
549 | (let ((ldiff (- (math-numdigs (nth 1 a)) 6))) | |
550 | (or (= (% (+ (nth 2 a) ldiff) 2) 0) (setq ldiff (1+ ldiff))) | |
551 | (setq guess (math-make-float (math-isqrt-small | |
552 | (math-scale-int (nth 1 a) (- ldiff))) | |
553 | (/ (+ (nth 2 a) ldiff) 2))))) | |
554 | (math-sqrt-float-iter a guess))))) | |
136211a9 EZ |
555 | |
556 | (defun math-sqrt-float-iter (a guess) ; [F F F] | |
557 | (math-working "sqrt" guess) | |
558 | (let ((g2 (math-mul-float (math-add-float guess (math-div-float a guess)) | |
559 | '(float 5 -1)))) | |
560 | (if (math-nearly-equal-float g2 guess) | |
561 | g2 | |
491c3062 | 562 | (math-sqrt-float-iter a g2)))) |
136211a9 EZ |
563 | |
564 | ;;; True if A and B differ only in the last digit of precision. [P F F] | |
565 | (defun math-nearly-equal-float (a b) | |
566 | (let ((ediff (- (nth 2 a) (nth 2 b)))) | |
567 | (cond ((= ediff 0) ;; Expanded out for speed | |
568 | (setq ediff (math-add (Math-integer-neg (nth 1 a)) (nth 1 b))) | |
569 | (or (eq ediff 0) | |
570 | (and (not (consp ediff)) | |
571 | (< ediff 10) | |
572 | (> ediff -10) | |
573 | (= (math-numdigs (nth 1 a)) calc-internal-prec)))) | |
574 | ((= ediff 1) | |
575 | (setq ediff (math-add (Math-integer-neg (nth 1 b)) | |
576 | (math-scale-int (nth 1 a) 1))) | |
577 | (and (not (consp ediff)) | |
578 | (< ediff 10) | |
579 | (> ediff -10) | |
580 | (= (math-numdigs (nth 1 b)) calc-internal-prec))) | |
581 | ((= ediff -1) | |
582 | (setq ediff (math-add (Math-integer-neg (nth 1 a)) | |
583 | (math-scale-int (nth 1 b) 1))) | |
584 | (and (not (consp ediff)) | |
585 | (< ediff 10) | |
586 | (> ediff -10) | |
491c3062 | 587 | (= (math-numdigs (nth 1 a)) calc-internal-prec)))))) |
136211a9 EZ |
588 | |
589 | (defun math-nearly-equal (a b) ; [P N N] [Public] | |
590 | (setq a (math-float a)) | |
591 | (setq b (math-float b)) | |
592 | (if (eq (car a) 'polar) (setq a (math-complex a))) | |
593 | (if (eq (car b) 'polar) (setq b (math-complex b))) | |
594 | (if (eq (car a) 'cplx) | |
595 | (if (eq (car b) 'cplx) | |
596 | (and (or (math-nearly-equal-float (nth 1 a) (nth 1 b)) | |
597 | (and (math-nearly-zerop-float (nth 1 a) (nth 2 a)) | |
598 | (math-nearly-zerop-float (nth 1 b) (nth 2 b)))) | |
599 | (or (math-nearly-equal-float (nth 2 a) (nth 2 b)) | |
600 | (and (math-nearly-zerop-float (nth 2 a) (nth 1 a)) | |
601 | (math-nearly-zerop-float (nth 2 b) (nth 1 b))))) | |
602 | (and (math-nearly-equal-float (nth 1 a) b) | |
603 | (math-nearly-zerop-float (nth 2 a) b))) | |
604 | (if (eq (car b) 'cplx) | |
605 | (and (math-nearly-equal-float a (nth 1 b)) | |
606 | (math-nearly-zerop-float a (nth 2 b))) | |
491c3062 | 607 | (math-nearly-equal-float a b)))) |
136211a9 EZ |
608 | |
609 | ;;; True if A is nearly zero compared to B. [P F F] | |
610 | (defun math-nearly-zerop-float (a b) | |
611 | (or (eq (nth 1 a) 0) | |
612 | (<= (+ (math-numdigs (nth 1 a)) (nth 2 a)) | |
491c3062 | 613 | (1+ (- (+ (math-numdigs (nth 1 b)) (nth 2 b)) calc-internal-prec))))) |
136211a9 EZ |
614 | |
615 | (defun math-nearly-zerop (a b) ; [P N R] [Public] | |
616 | (setq a (math-float a)) | |
617 | (setq b (math-float b)) | |
618 | (if (eq (car a) 'cplx) | |
619 | (and (math-nearly-zerop-float (nth 1 a) b) | |
620 | (math-nearly-zerop-float (nth 2 a) b)) | |
621 | (if (eq (car a) 'polar) | |
622 | (math-nearly-zerop-float (nth 1 a) b) | |
491c3062 | 623 | (math-nearly-zerop-float a b)))) |
136211a9 EZ |
624 | |
625 | ;;; This implementation could be improved, accuracy-wise. | |
626 | (defun math-hypot (a b) | |
627 | (cond ((Math-zerop a) (math-abs b)) | |
628 | ((Math-zerop b) (math-abs a)) | |
629 | ((not (Math-scalarp a)) | |
630 | (if (math-infinitep a) | |
631 | (if (math-infinitep b) | |
632 | (if (equal a b) | |
633 | a | |
634 | '(var nan var-nan)) | |
635 | a) | |
636 | (calc-record-why 'scalarp a) | |
637 | (list 'calcFunc-hypot a b))) | |
638 | ((not (Math-scalarp b)) | |
639 | (if (math-infinitep b) | |
640 | b | |
641 | (calc-record-why 'scalarp b) | |
642 | (list 'calcFunc-hypot a b))) | |
643 | ((and (Math-numberp a) (Math-numberp b)) | |
644 | (math-with-extra-prec 1 | |
645 | (math-sqrt (math-add (calcFunc-abssqr a) (calcFunc-abssqr b))))) | |
646 | ((eq (car-safe a) 'hms) | |
647 | (if (eq (car-safe b) 'hms) ; this helps sdev's of hms forms | |
648 | (math-to-hms (math-hypot (math-from-hms a 'deg) | |
649 | (math-from-hms b 'deg))) | |
650 | (math-to-hms (math-hypot (math-from-hms a 'deg) b)))) | |
651 | ((eq (car-safe b) 'hms) | |
652 | (math-to-hms (math-hypot a (math-from-hms b 'deg)))) | |
491c3062 | 653 | (t nil))) |
3132f345 | 654 | (defalias 'calcFunc-hypot 'math-hypot) |
136211a9 EZ |
655 | |
656 | (defun calcFunc-sqr (x) | |
491c3062 | 657 | (math-pow x 2)) |
136211a9 EZ |
658 | |
659 | ||
660 | ||
661 | (defun math-nth-root (a n) | |
662 | (cond ((= n 2) (math-sqrt a)) | |
663 | ((Math-zerop a) a) | |
664 | ((Math-negp a) nil) | |
665 | ((Math-integerp a) | |
666 | (let ((root (math-nth-root-integer a n))) | |
667 | (if (car root) | |
668 | (cdr root) | |
669 | (and (not calc-symbolic-mode) | |
670 | (math-nth-root-float (math-float a) n | |
671 | (math-float (cdr root))))))) | |
672 | ((eq (car-safe a) 'frac) | |
673 | (let* ((num-root (math-nth-root-integer (nth 1 a) n)) | |
674 | (den-root (math-nth-root-integer (nth 2 a) n))) | |
675 | (if (and (car num-root) (car den-root)) | |
676 | (list 'frac (cdr num-root) (cdr den-root)) | |
677 | (and (not calc-symbolic-mode) | |
678 | (math-nth-root-float | |
679 | (math-float a) n | |
680 | (math-div-float (math-float (cdr num-root)) | |
681 | (math-float (cdr den-root)))))))) | |
682 | ((eq (car-safe a) 'float) | |
683 | (and (not calc-symbolic-mode) | |
684 | (math-nth-root-float a n))) | |
685 | ((eq (car-safe a) 'polar) | |
686 | (let ((root (math-nth-root (nth 1 a) n))) | |
687 | (and root (list 'polar root (math-div (nth 2 a) n))))) | |
491c3062 | 688 | (t nil))) |
136211a9 | 689 | |
86498823 JB |
690 | ;; The variables math-nrf-n, math-nrf-nf and math-nrf-nfm1 are local |
691 | ;; to math-nth-root-float, but are used by math-nth-root-float-iter, | |
692 | ;; which is called by math-nth-root-float. | |
693 | (defvar math-nrf-n) | |
694 | (defvar math-nrf-nf) | |
695 | (defvar math-nrf-nfm1) | |
696 | ||
697 | (defun math-nth-root-float (a math-nrf-n &optional guess) | |
136211a9 EZ |
698 | (math-inexact-result) |
699 | (math-with-extra-prec 1 | |
86498823 JB |
700 | (let ((math-nrf-nf (math-float math-nrf-n)) |
701 | (math-nrf-nfm1 (math-float (1- math-nrf-n)))) | |
136211a9 EZ |
702 | (math-nth-root-float-iter a (or guess |
703 | (math-make-float | |
704 | 1 (/ (+ (math-numdigs (nth 1 a)) | |
705 | (nth 2 a) | |
86498823 JB |
706 | (/ math-nrf-n 2)) |
707 | math-nrf-n))))))) | |
136211a9 | 708 | |
86498823 | 709 | (defun math-nth-root-float-iter (a guess) |
136211a9 | 710 | (math-working "root" guess) |
86498823 | 711 | (let ((g2 (math-div-float (math-add-float (math-mul math-nrf-nfm1 guess) |
136211a9 | 712 | (math-div-float |
86498823 JB |
713 | a (math-ipow guess (1- math-nrf-n)))) |
714 | math-nrf-nf))) | |
136211a9 EZ |
715 | (if (math-nearly-equal-float g2 guess) |
716 | g2 | |
491c3062 | 717 | (math-nth-root-float-iter a g2)))) |
136211a9 | 718 | |
86498823 JB |
719 | ;; The variable math-nri-n is local to math-nth-root-integer, but |
720 | ;; is used by math-nth-root-int-iter, which is called by | |
721 | ;; math-nth-root-int. | |
722 | (defvar math-nri-n) | |
723 | ||
724 | (defun math-nth-root-integer (a math-nri-n &optional guess) ; [I I S] | |
136211a9 EZ |
725 | (math-nth-root-int-iter a (or guess |
726 | (math-scale-int 1 (/ (+ (math-numdigs a) | |
86498823 JB |
727 | (1- math-nri-n)) |
728 | math-nri-n))))) | |
136211a9 | 729 | |
86498823 | 730 | (defun math-nth-root-int-iter (a guess) |
136211a9 | 731 | (math-working "root" guess) |
86498823 JB |
732 | (let* ((q (math-idivmod a (math-ipow guess (1- math-nri-n)))) |
733 | (s (math-add (car q) (math-mul (1- math-nri-n) guess))) | |
734 | (g2 (math-idivmod s math-nri-n))) | |
136211a9 EZ |
735 | (if (Math-natnum-lessp (car g2) guess) |
736 | (math-nth-root-int-iter a (car g2)) | |
737 | (cons (and (equal (car g2) guess) | |
738 | (eq (cdr q) 0) | |
739 | (eq (cdr g2) 0)) | |
491c3062 | 740 | guess)))) |
136211a9 EZ |
741 | |
742 | (defun calcFunc-nroot (x n) | |
743 | (calcFunc-pow x (if (integerp n) | |
744 | (math-make-frac 1 n) | |
491c3062 | 745 | (math-div 1 n)))) |
136211a9 EZ |
746 | |
747 | ||
748 | ||
749 | ||
750 | ;;;; Transcendental functions. | |
751 | ||
752 | ;;; All of these functions are defined on the complex plane. | |
753 | ;;; (Branch cuts, etc. follow Steele's Common Lisp book.) | |
754 | ||
755 | ;;; Most functions increase calc-internal-prec by 2 digits, then round | |
756 | ;;; down afterward. "-raw" functions use the current precision, require | |
757 | ;;; their arguments to be in float (or complex float) format, and always | |
758 | ;;; work in radians (where applicable). | |
759 | ||
760 | (defun math-to-radians (a) ; [N N] | |
761 | (cond ((eq (car-safe a) 'hms) | |
762 | (math-from-hms a 'rad)) | |
763 | ((memq calc-angle-mode '(deg hms)) | |
764 | (math-mul a (math-pi-over-180))) | |
491c3062 | 765 | (t a))) |
136211a9 EZ |
766 | |
767 | (defun math-from-radians (a) ; [N N] | |
768 | (cond ((eq calc-angle-mode 'deg) | |
769 | (if (math-constp a) | |
770 | (math-div a (math-pi-over-180)) | |
771 | (list 'calcFunc-deg a))) | |
772 | ((eq calc-angle-mode 'hms) | |
773 | (math-to-hms a 'rad)) | |
491c3062 | 774 | (t a))) |
136211a9 EZ |
775 | |
776 | (defun math-to-radians-2 (a) ; [N N] | |
777 | (cond ((eq (car-safe a) 'hms) | |
778 | (math-from-hms a 'rad)) | |
779 | ((memq calc-angle-mode '(deg hms)) | |
780 | (if calc-symbolic-mode | |
781 | (math-div (math-mul a '(var pi var-pi)) 180) | |
782 | (math-mul a (math-pi-over-180)))) | |
491c3062 | 783 | (t a))) |
136211a9 EZ |
784 | |
785 | (defun math-from-radians-2 (a) ; [N N] | |
786 | (cond ((memq calc-angle-mode '(deg hms)) | |
787 | (if calc-symbolic-mode | |
788 | (math-div (math-mul 180 a) '(var pi var-pi)) | |
789 | (math-div a (math-pi-over-180)))) | |
491c3062 | 790 | (t a))) |
136211a9 EZ |
791 | |
792 | ||
793 | ||
794 | ;;; Sine, cosine, and tangent. | |
795 | ||
796 | (defun calcFunc-sin (x) ; [N N] [Public] | |
797 | (cond ((and (integerp x) | |
798 | (if (eq calc-angle-mode 'deg) | |
799 | (= (% x 90) 0) | |
800 | (= x 0))) | |
801 | (aref [0 1 0 -1] (math-mod (/ x 90) 4))) | |
802 | ((Math-scalarp x) | |
803 | (math-with-extra-prec 2 | |
804 | (math-sin-raw (math-to-radians (math-float x))))) | |
805 | ((eq (car x) 'sdev) | |
806 | (if (math-constp x) | |
807 | (math-with-extra-prec 2 | |
808 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
809 | (xs (math-to-radians (math-float (nth 2 x)))) | |
810 | (sc (math-sin-cos-raw xx))) | |
811 | (math-make-sdev (car sc) (math-mul xs (cdr sc))))) | |
812 | (math-make-sdev (calcFunc-sin (nth 1 x)) | |
813 | (math-mul (nth 2 x) (calcFunc-cos (nth 1 x)))))) | |
814 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
815 | (calcFunc-cos (math-sub x (math-quarter-circle nil)))) | |
816 | ((equal x '(var nan var-nan)) | |
817 | x) | |
818 | (t (calc-record-why 'scalarp x) | |
491c3062 | 819 | (list 'calcFunc-sin x)))) |
136211a9 EZ |
820 | |
821 | (defun calcFunc-cos (x) ; [N N] [Public] | |
822 | (cond ((and (integerp x) | |
823 | (if (eq calc-angle-mode 'deg) | |
824 | (= (% x 90) 0) | |
825 | (= x 0))) | |
826 | (aref [1 0 -1 0] (math-mod (/ x 90) 4))) | |
827 | ((Math-scalarp x) | |
828 | (math-with-extra-prec 2 | |
829 | (math-cos-raw (math-to-radians (math-float x))))) | |
830 | ((eq (car x) 'sdev) | |
831 | (if (math-constp x) | |
832 | (math-with-extra-prec 2 | |
833 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
834 | (xs (math-to-radians (math-float (nth 2 x)))) | |
835 | (sc (math-sin-cos-raw xx))) | |
836 | (math-make-sdev (cdr sc) (math-mul xs (car sc))))) | |
837 | (math-make-sdev (calcFunc-cos (nth 1 x)) | |
838 | (math-mul (nth 2 x) (calcFunc-sin (nth 1 x)))))) | |
839 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
840 | (math-with-extra-prec 2 | |
841 | (let* ((xx (math-to-radians (math-float x))) | |
842 | (na (math-floor (math-div (nth 2 xx) (math-pi)))) | |
843 | (nb (math-floor (math-div (nth 3 xx) (math-pi)))) | |
844 | (span (math-sub nb na))) | |
845 | (if (memq span '(0 1)) | |
846 | (let ((int (math-sort-intv (nth 1 x) | |
847 | (math-cos-raw (nth 2 xx)) | |
848 | (math-cos-raw (nth 3 xx))))) | |
849 | (if (eq span 1) | |
850 | (if (math-evenp na) | |
851 | (math-make-intv (logior (nth 1 x) 2) | |
852 | -1 | |
853 | (nth 3 int)) | |
854 | (math-make-intv (logior (nth 1 x) 1) | |
855 | (nth 2 int) | |
856 | 1)) | |
857 | int)) | |
858 | (list 'intv 3 -1 1))))) | |
859 | ((equal x '(var nan var-nan)) | |
860 | x) | |
861 | (t (calc-record-why 'scalarp x) | |
491c3062 | 862 | (list 'calcFunc-cos x)))) |
136211a9 EZ |
863 | |
864 | (defun calcFunc-sincos (x) ; [V N] [Public] | |
865 | (if (Math-scalarp x) | |
866 | (math-with-extra-prec 2 | |
867 | (let ((sc (math-sin-cos-raw (math-to-radians (math-float x))))) | |
868 | (list 'vec (cdr sc) (car sc)))) ; the vector [cos, sin] | |
491c3062 | 869 | (list 'vec (calcFunc-sin x) (calcFunc-cos x)))) |
136211a9 EZ |
870 | |
871 | (defun calcFunc-tan (x) ; [N N] [Public] | |
872 | (cond ((and (integerp x) | |
873 | (if (eq calc-angle-mode 'deg) | |
874 | (= (% x 180) 0) | |
875 | (= x 0))) | |
876 | 0) | |
877 | ((Math-scalarp x) | |
878 | (math-with-extra-prec 2 | |
879 | (math-tan-raw (math-to-radians (math-float x))))) | |
880 | ((eq (car x) 'sdev) | |
881 | (if (math-constp x) | |
882 | (math-with-extra-prec 2 | |
883 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
884 | (xs (math-to-radians (math-float (nth 2 x)))) | |
885 | (sc (math-sin-cos-raw xx))) | |
886 | (if (and (math-zerop (cdr sc)) (not calc-infinite-mode)) | |
887 | (progn | |
888 | (calc-record-why "*Division by zero") | |
889 | (list 'calcFunc-tan x)) | |
890 | (math-make-sdev (math-div-float (car sc) (cdr sc)) | |
891 | (math-div-float xs (math-sqr (cdr sc))))))) | |
892 | (math-make-sdev (calcFunc-tan (nth 1 x)) | |
893 | (math-div (nth 2 x) | |
894 | (math-sqr (calcFunc-cos (nth 1 x))))))) | |
895 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
896 | (or (math-with-extra-prec 2 | |
897 | (let* ((xx (math-to-radians (math-float x))) | |
898 | (na (math-floor (math-div (math-sub (nth 2 xx) | |
899 | (math-pi-over-2)) | |
900 | (math-pi)))) | |
901 | (nb (math-floor (math-div (math-sub (nth 3 xx) | |
902 | (math-pi-over-2)) | |
903 | (math-pi))))) | |
904 | (and (equal na nb) | |
905 | (math-sort-intv (nth 1 x) | |
906 | (math-tan-raw (nth 2 xx)) | |
907 | (math-tan-raw (nth 3 xx)))))) | |
908 | '(intv 3 (neg (var inf var-inf)) (var inf var-inf)))) | |
909 | ((equal x '(var nan var-nan)) | |
910 | x) | |
911 | (t (calc-record-why 'scalarp x) | |
491c3062 | 912 | (list 'calcFunc-tan x)))) |
136211a9 | 913 | |
f53e6c20 JB |
914 | (defun calcFunc-sec (x) |
915 | (cond ((and (integerp x) | |
916 | (eq calc-angle-mode 'deg) | |
917 | (= (% x 180) 0)) | |
918 | (if (= (% x 360) 0) | |
919 | 1 | |
920 | -1)) | |
921 | ((and (integerp x) | |
922 | (eq calc-angle-mode 'rad) | |
923 | (= x 0)) | |
924 | 1) | |
925 | ((Math-scalarp x) | |
926 | (math-with-extra-prec 2 | |
927 | (math-sec-raw (math-to-radians (math-float x))))) | |
928 | ((eq (car x) 'sdev) | |
929 | (if (math-constp x) | |
930 | (math-with-extra-prec 2 | |
931 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
932 | (xs (math-to-radians (math-float (nth 2 x)))) | |
933 | (sc (math-sin-cos-raw xx))) | |
934 | (if (and (math-zerop (cdr sc)) | |
935 | (not calc-infinite-mode)) | |
936 | (progn | |
937 | (calc-record-why "*Division by zero") | |
938 | (list 'calcFunc-sec x)) | |
939 | (math-make-sdev (math-div-float '(float 1 0) (cdr sc)) | |
940 | (math-div-float | |
941 | (math-mul xs (car sc)) | |
942 | (math-sqr (cdr sc))))))) | |
943 | (math-make-sdev (calcFunc-sec (nth 1 x)) | |
944 | (math-div | |
945 | (math-mul (nth 2 x) | |
946 | (calcFunc-sin (nth 1 x))) | |
947 | (math-sqr (calcFunc-cos (nth 1 x))))))) | |
948 | ((and (eq (car x) 'intv) | |
949 | (math-intv-constp x)) | |
950 | (math-with-extra-prec 2 | |
951 | (let* ((xx (math-to-radians (math-float x))) | |
952 | (na (math-floor (math-div (math-sub (nth 2 xx) | |
953 | (math-pi-over-2)) | |
954 | (math-pi)))) | |
955 | (nb (math-floor (math-div (math-sub (nth 3 xx) | |
956 | (math-pi-over-2)) | |
957 | (math-pi)))) | |
958 | (naa (math-floor (math-div (nth 2 xx) (math-pi-over-2)))) | |
959 | (nbb (math-floor (math-div (nth 3 xx) (math-pi-over-2)))) | |
960 | (span (math-sub nbb naa))) | |
961 | (if (not (equal na nb)) | |
962 | '(intv 3 (neg (var inf var-inf)) (var inf var-inf)) | |
963 | (let ((int (math-sort-intv (nth 1 x) | |
964 | (math-sec-raw (nth 2 xx)) | |
965 | (math-sec-raw (nth 3 xx))))) | |
966 | (if (eq span 1) | |
967 | (if (math-evenp (math-div (math-add naa 1) 2)) | |
968 | (math-make-intv (logior (nth 1 int) 2) | |
969 | 1 | |
970 | (nth 3 int)) | |
971 | (math-make-intv (logior (nth 1 int) 1) | |
972 | (nth 2 int) | |
973 | -1)) | |
974 | int)))))) | |
975 | ((equal x '(var nan var-nan)) | |
976 | x) | |
977 | (t (calc-record-why 'scalarp x) | |
978 | (list 'calcFunc-sec x)))) | |
979 | ||
980 | (defun calcFunc-csc (x) | |
981 | (cond ((and (integerp x) | |
982 | (eq calc-angle-mode 'deg) | |
983 | (= (% (- x 90) 180) 0)) | |
984 | (if (= (% (- x 90) 360) 0) | |
985 | 1 | |
986 | -1)) | |
987 | ((Math-scalarp x) | |
988 | (math-with-extra-prec 2 | |
989 | (math-csc-raw (math-to-radians (math-float x))))) | |
990 | ((eq (car x) 'sdev) | |
991 | (if (math-constp x) | |
992 | (math-with-extra-prec 2 | |
993 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
994 | (xs (math-to-radians (math-float (nth 2 x)))) | |
995 | (sc (math-sin-cos-raw xx))) | |
996 | (if (and (math-zerop (car sc)) | |
997 | (not calc-infinite-mode)) | |
998 | (progn | |
999 | (calc-record-why "*Division by zero") | |
1000 | (list 'calcFunc-csc x)) | |
1001 | (math-make-sdev (math-div-float '(float 1 0) (car sc)) | |
1002 | (math-div-float | |
1003 | (math-mul xs (cdr sc)) | |
1004 | (math-sqr (car sc))))))) | |
1005 | (math-make-sdev (calcFunc-csc (nth 1 x)) | |
1006 | (math-div | |
1007 | (math-mul (nth 2 x) | |
1008 | (calcFunc-cos (nth 1 x))) | |
1009 | (math-sqr (calcFunc-sin (nth 1 x))))))) | |
1010 | ((and (eq (car x) 'intv) | |
1011 | (math-intv-constp x)) | |
1012 | (math-with-extra-prec 2 | |
1013 | (let* ((xx (math-to-radians (math-float x))) | |
1014 | (na (math-floor (math-div (nth 2 xx) (math-pi)))) | |
1015 | (nb (math-floor (math-div (nth 3 xx) (math-pi)))) | |
1016 | (naa (math-floor (math-div (nth 2 xx) (math-pi-over-2)))) | |
1017 | (nbb (math-floor (math-div (nth 3 xx) (math-pi-over-2)))) | |
1018 | (span (math-sub nbb naa))) | |
1019 | (if (not (equal na nb)) | |
1020 | '(intv 3 (neg (var inf var-inf)) (var inf var-inf)) | |
1021 | (let ((int (math-sort-intv (nth 1 x) | |
1022 | (math-csc-raw (nth 2 xx)) | |
1023 | (math-csc-raw (nth 3 xx))))) | |
1024 | (if (eq span 1) | |
1025 | (if (math-evenp (math-div naa 2)) | |
1026 | (math-make-intv (logior (nth 1 int) 2) | |
1027 | 1 | |
1028 | (nth 3 int)) | |
1029 | (math-make-intv (logior (nth 1 int) 1) | |
1030 | (nth 2 int) | |
1031 | -1)) | |
1032 | int)))))) | |
1033 | ((equal x '(var nan var-nan)) | |
1034 | x) | |
1035 | (t (calc-record-why 'scalarp x) | |
1036 | (list 'calcFunc-csc x)))) | |
1037 | ||
1038 | (defun calcFunc-cot (x) ; [N N] [Public] | |
1039 | (cond ((and (integerp x) | |
1040 | (if (eq calc-angle-mode 'deg) | |
1041 | (= (% (- x 90) 180) 0) | |
1042 | (= x 0))) | |
1043 | 0) | |
1044 | ((Math-scalarp x) | |
1045 | (math-with-extra-prec 2 | |
1046 | (math-cot-raw (math-to-radians (math-float x))))) | |
1047 | ((eq (car x) 'sdev) | |
1048 | (if (math-constp x) | |
1049 | (math-with-extra-prec 2 | |
1050 | (let* ((xx (math-to-radians (math-float (nth 1 x)))) | |
1051 | (xs (math-to-radians (math-float (nth 2 x)))) | |
1052 | (sc (math-sin-cos-raw xx))) | |
1053 | (if (and (math-zerop (car sc)) (not calc-infinite-mode)) | |
1054 | (progn | |
1055 | (calc-record-why "*Division by zero") | |
1056 | (list 'calcFunc-cot x)) | |
1057 | (math-make-sdev (math-div-float (cdr sc) (car sc)) | |
1058 | (math-div-float xs (math-sqr (car sc))))))) | |
1059 | (math-make-sdev (calcFunc-cot (nth 1 x)) | |
1060 | (math-div (nth 2 x) | |
1061 | (math-sqr (calcFunc-sin (nth 1 x))))))) | |
1062 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
1063 | (or (math-with-extra-prec 2 | |
1064 | (let* ((xx (math-to-radians (math-float x))) | |
1065 | (na (math-floor (math-div (nth 2 xx) (math-pi)))) | |
eb1ef455 | 1066 | (nb (math-floor (math-div (nth 3 xx) (math-pi))))) |
f53e6c20 JB |
1067 | (and (equal na nb) |
1068 | (math-sort-intv (nth 1 x) | |
1069 | (math-cot-raw (nth 2 xx)) | |
eb1ef455 | 1070 | (math-cot-raw (nth 3 xx)))))) |
f53e6c20 JB |
1071 | '(intv 3 (neg (var inf var-inf)) (var inf var-inf)))) |
1072 | ((equal x '(var nan var-nan)) | |
1073 | x) | |
1074 | (t (calc-record-why 'scalarp x) | |
1075 | (list 'calcFunc-cot x)))) | |
1076 | ||
b0d9db86 | 1077 | (defun math-sin-raw (x &optional orgx) ; [N N] |
136211a9 EZ |
1078 | (cond ((eq (car x) 'cplx) |
1079 | (let* ((expx (math-exp-raw (nth 2 x))) | |
1080 | (expmx (math-div-float '(float 1 0) expx)) | |
1081 | (sc (math-sin-cos-raw (nth 1 x)))) | |
1082 | (list 'cplx | |
1083 | (math-mul-float (car sc) | |
1084 | (math-mul-float (math-add-float expx expmx) | |
1085 | '(float 5 -1))) | |
1086 | (math-mul-float (cdr sc) | |
1087 | (math-mul-float (math-sub-float expx expmx) | |
1088 | '(float 5 -1)))))) | |
1089 | ((eq (car x) 'polar) | |
1090 | (math-polar (math-sin-raw (math-complex x)))) | |
1091 | ((Math-integer-negp (nth 1 x)) | |
b0d9db86 | 1092 | (math-neg-float (math-sin-raw (math-neg-float x) (if orgx orgx x)))) |
136211a9 | 1093 | ((math-lessp-float '(float 7 0) x) ; avoid inf loops due to roundoff |
b0d9db86 JB |
1094 | (math-sin-raw (math-mod x (math-two-pi)) (if orgx orgx x))) |
1095 | (t (math-sin-raw-2 x (if orgx orgx x))))) | |
136211a9 EZ |
1096 | |
1097 | (defun math-cos-raw (x) ; [N N] | |
1098 | (if (eq (car-safe x) 'polar) | |
1099 | (math-polar (math-cos-raw (math-complex x))) | |
b0d9db86 | 1100 | (math-sin-raw (math-sub (math-pi-over-2) x) x))) |
136211a9 | 1101 | |
f53e6c20 JB |
1102 | (defun math-sec-raw (x) ; [N N] |
1103 | (cond ((eq (car x) 'cplx) | |
1104 | (let* ((x (math-mul x '(float 1 0))) | |
1105 | (expx (math-exp-raw (nth 2 x))) | |
1106 | (expmx (math-div-float '(float 1 0) expx)) | |
1107 | (sh (math-mul-float (math-sub-float expx expmx) '(float 5 -1))) | |
1108 | (ch (math-mul-float (math-add-float expx expmx) '(float 5 -1))) | |
1109 | (sc (math-sin-cos-raw (nth 1 x))) | |
1110 | (d (math-add-float | |
1111 | (math-mul-float (math-sqr (car sc)) | |
1112 | (math-sqr sh)) | |
1113 | (math-mul-float (math-sqr (cdr sc)) | |
1114 | (math-sqr ch))))) | |
1115 | (and (not (eq (nth 1 d) 0)) | |
1116 | (list 'cplx | |
1117 | (math-div-float (math-mul-float (cdr sc) ch) d) | |
1118 | (math-div-float (math-mul-float (car sc) sh) d))))) | |
1119 | ((eq (car x) 'polar) | |
1120 | (math-polar (math-sec-raw (math-complex x)))) | |
1121 | (t | |
1122 | (let ((cs (math-cos-raw x))) | |
1123 | (if (eq cs 0) | |
1124 | (math-div 1 0) | |
1125 | (math-div-float '(float 1 0) cs)))))) | |
1126 | ||
1127 | (defun math-csc-raw (x) ; [N N] | |
1128 | (cond ((eq (car x) 'cplx) | |
1129 | (let* ((x (math-mul x '(float 1 0))) | |
1130 | (expx (math-exp-raw (nth 2 x))) | |
1131 | (expmx (math-div-float '(float 1 0) expx)) | |
1132 | (sh (math-mul-float (math-sub-float expx expmx) '(float 5 -1))) | |
1133 | (ch (math-mul-float (math-add-float expx expmx) '(float 5 -1))) | |
1134 | (sc (math-sin-cos-raw (nth 1 x))) | |
1135 | (d (math-add-float | |
1136 | (math-mul-float (math-sqr (car sc)) | |
1137 | (math-sqr ch)) | |
1138 | (math-mul-float (math-sqr (cdr sc)) | |
1139 | (math-sqr sh))))) | |
1140 | (and (not (eq (nth 1 d) 0)) | |
1141 | (list 'cplx | |
1142 | (math-div-float (math-mul-float (car sc) ch) d) | |
1143 | (math-div-float (math-mul-float (cdr sc) sh) d))))) | |
1144 | ((eq (car x) 'polar) | |
6ec30302 | 1145 | (math-polar (math-csc-raw (math-complex x)))) |
f53e6c20 JB |
1146 | (t |
1147 | (let ((sn (math-sin-raw x))) | |
1148 | (if (eq sn 0) | |
1149 | (math-div 1 0) | |
1150 | (math-div-float '(float 1 0) sn)))))) | |
1151 | ||
1152 | (defun math-cot-raw (x) ; [N N] | |
1153 | (cond ((eq (car x) 'cplx) | |
1154 | (let* ((x (math-mul x '(float 1 0))) | |
1155 | (expx (math-exp-raw (nth 2 x))) | |
1156 | (expmx (math-div-float '(float 1 0) expx)) | |
1157 | (sh (math-mul-float (math-sub-float expx expmx) '(float 5 -1))) | |
1158 | (ch (math-mul-float (math-add-float expx expmx) '(float 5 -1))) | |
1159 | (sc (math-sin-cos-raw (nth 1 x))) | |
1160 | (d (math-add-float | |
1161 | (math-sqr (car sc)) | |
1162 | (math-sqr sh)))) | |
1163 | (and (not (eq (nth 1 d) 0)) | |
1164 | (list 'cplx | |
1165 | (math-div-float | |
1166 | (math-mul-float (car sc) (cdr sc)) | |
1167 | d) | |
1168 | (math-neg | |
1169 | (math-div-float | |
1170 | (math-mul-float sh ch) | |
1171 | d)))))) | |
1172 | ((eq (car x) 'polar) | |
1173 | (math-polar (math-cot-raw (math-complex x)))) | |
1174 | (t | |
1175 | (let ((sc (math-sin-cos-raw x))) | |
1176 | (if (eq (nth 1 (car sc)) 0) | |
1177 | (math-div (cdr sc) 0) | |
1178 | (math-div-float (cdr sc) (car sc))))))) | |
1179 | ||
1180 | ||
136211a9 EZ |
1181 | ;;; This could use a smarter method: Reduce x as in math-sin-raw, then |
1182 | ;;; compute either sin(x) or cos(x), whichever is smaller, and compute | |
1183 | ;;; the other using the identity sin(x)^2 + cos(x)^2 = 1. | |
1184 | (defun math-sin-cos-raw (x) ; [F.F F] (result is (sin x . cos x)) | |
491c3062 | 1185 | (cons (math-sin-raw x) (math-cos-raw x))) |
136211a9 EZ |
1186 | |
1187 | (defun math-tan-raw (x) ; [N N] | |
1188 | (cond ((eq (car x) 'cplx) | |
1189 | (let* ((x (math-mul x '(float 2 0))) | |
1190 | (expx (math-exp-raw (nth 2 x))) | |
1191 | (expmx (math-div-float '(float 1 0) expx)) | |
1192 | (sc (math-sin-cos-raw (nth 1 x))) | |
1193 | (d (math-add-float (cdr sc) | |
1194 | (math-mul-float (math-add-float expx expmx) | |
1195 | '(float 5 -1))))) | |
1196 | (and (not (eq (nth 1 d) 0)) | |
1197 | (list 'cplx | |
1198 | (math-div-float (car sc) d) | |
1199 | (math-div-float (math-mul-float (math-sub-float expx | |
1200 | expmx) | |
1201 | '(float 5 -1)) d))))) | |
1202 | ((eq (car x) 'polar) | |
1203 | (math-polar (math-tan-raw (math-complex x)))) | |
1204 | (t | |
1205 | (let ((sc (math-sin-cos-raw x))) | |
1206 | (if (eq (nth 1 (cdr sc)) 0) | |
1207 | (math-div (car sc) 0) | |
491c3062 | 1208 | (math-div-float (car sc) (cdr sc))))))) |
136211a9 EZ |
1209 | |
1210 | (defun math-sin-raw-2 (x orgx) ; This avoids poss of inf recursion. [F F] | |
1211 | (let ((xmpo2 (math-sub-float (math-pi-over-2) x))) | |
1212 | (cond ((Math-integer-negp (nth 1 xmpo2)) | |
1213 | (math-neg-float (math-sin-raw-2 (math-sub-float x (math-pi)) | |
1214 | orgx))) | |
1215 | ((math-lessp-float (math-pi-over-4) x) | |
1216 | (math-cos-raw-2 xmpo2 orgx)) | |
1217 | ((math-lessp-float x (math-neg (math-pi-over-4))) | |
1218 | (math-neg (math-cos-raw-2 (math-add (math-pi-over-2) x) orgx))) | |
b0d9db86 JB |
1219 | ((math-with-extra-prec -1 (math-nearly-zerop-float x orgx)) |
1220 | '(float 0 0)) | |
17cc361e | 1221 | ((math-use-emacs-fn 'sin x)) |
136211a9 | 1222 | (calc-symbolic-mode (signal 'inexact-result nil)) |
491c3062 | 1223 | (t (math-sin-series x 6 4 x (math-neg-float (math-sqr-float x))))))) |
136211a9 EZ |
1224 | |
1225 | (defun math-cos-raw-2 (x orgx) ; [F F] | |
b0d9db86 JB |
1226 | (cond ((math-with-extra-prec -1 (math-nearly-zerop-float x orgx)) |
1227 | '(float 1 0)) | |
17cc361e | 1228 | ((math-use-emacs-fn 'cos x)) |
136211a9 EZ |
1229 | (calc-symbolic-mode (signal 'inexact-result nil)) |
1230 | (t (let ((xnegsqr (math-neg-float (math-sqr-float x)))) | |
1231 | (math-sin-series | |
1232 | (math-add-float '(float 1 0) | |
1233 | (math-mul-float xnegsqr '(float 5 -1))) | |
491c3062 | 1234 | 24 5 xnegsqr xnegsqr))))) |
136211a9 EZ |
1235 | |
1236 | (defun math-sin-series (sum nfac n x xnegsqr) | |
1237 | (math-working "sin" sum) | |
1238 | (let* ((nextx (math-mul-float x xnegsqr)) | |
1239 | (nextsum (math-add-float sum (math-div-float nextx | |
1240 | (math-float nfac))))) | |
1241 | (if (math-nearly-equal-float sum nextsum) | |
1242 | sum | |
1243 | (math-sin-series nextsum (math-mul nfac (* n (1+ n))) | |
491c3062 | 1244 | (+ n 2) nextx xnegsqr)))) |
136211a9 EZ |
1245 | |
1246 | ||
1247 | ;;; Inverse sine, cosine, tangent. | |
1248 | ||
1249 | (defun calcFunc-arcsin (x) ; [N N] [Public] | |
1250 | (cond ((eq x 0) 0) | |
1251 | ((and (eq x 1) (eq calc-angle-mode 'deg)) 90) | |
1252 | ((and (eq x -1) (eq calc-angle-mode 'deg)) -90) | |
1253 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1254 | ((Math-numberp x) | |
1255 | (math-with-extra-prec 2 | |
1256 | (math-from-radians (math-arcsin-raw (math-float x))))) | |
1257 | ((eq (car x) 'sdev) | |
1258 | (math-make-sdev (calcFunc-arcsin (nth 1 x)) | |
1259 | (math-from-radians | |
1260 | (math-div (nth 2 x) | |
1261 | (math-sqrt | |
1262 | (math-sub 1 (math-sqr (nth 1 x)))))))) | |
1263 | ((eq (car x) 'intv) | |
1264 | (math-sort-intv (nth 1 x) | |
1265 | (calcFunc-arcsin (nth 2 x)) | |
1266 | (calcFunc-arcsin (nth 3 x)))) | |
1267 | ((equal x '(var nan var-nan)) | |
1268 | x) | |
1269 | (t (calc-record-why 'numberp x) | |
491c3062 | 1270 | (list 'calcFunc-arcsin x)))) |
136211a9 EZ |
1271 | |
1272 | (defun calcFunc-arccos (x) ; [N N] [Public] | |
1273 | (cond ((eq x 1) 0) | |
1274 | ((and (eq x 0) (eq calc-angle-mode 'deg)) 90) | |
1275 | ((and (eq x -1) (eq calc-angle-mode 'deg)) 180) | |
1276 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1277 | ((Math-numberp x) | |
1278 | (math-with-extra-prec 2 | |
1279 | (math-from-radians (math-arccos-raw (math-float x))))) | |
1280 | ((eq (car x) 'sdev) | |
1281 | (math-make-sdev (calcFunc-arccos (nth 1 x)) | |
1282 | (math-from-radians | |
1283 | (math-div (nth 2 x) | |
1284 | (math-sqrt | |
1285 | (math-sub 1 (math-sqr (nth 1 x)))))))) | |
1286 | ((eq (car x) 'intv) | |
1287 | (math-sort-intv (nth 1 x) | |
1288 | (calcFunc-arccos (nth 2 x)) | |
1289 | (calcFunc-arccos (nth 3 x)))) | |
1290 | ((equal x '(var nan var-nan)) | |
1291 | x) | |
1292 | (t (calc-record-why 'numberp x) | |
491c3062 | 1293 | (list 'calcFunc-arccos x)))) |
136211a9 EZ |
1294 | |
1295 | (defun calcFunc-arctan (x) ; [N N] [Public] | |
1296 | (cond ((eq x 0) 0) | |
1297 | ((and (eq x 1) (eq calc-angle-mode 'deg)) 45) | |
1298 | ((and (eq x -1) (eq calc-angle-mode 'deg)) -45) | |
1299 | ((Math-numberp x) | |
1300 | (math-with-extra-prec 2 | |
1301 | (math-from-radians (math-arctan-raw (math-float x))))) | |
1302 | ((eq (car x) 'sdev) | |
1303 | (math-make-sdev (calcFunc-arctan (nth 1 x)) | |
1304 | (math-from-radians | |
1305 | (math-div (nth 2 x) | |
1306 | (math-add 1 (math-sqr (nth 1 x))))))) | |
1307 | ((eq (car x) 'intv) | |
1308 | (math-sort-intv (nth 1 x) | |
1309 | (calcFunc-arctan (nth 2 x)) | |
1310 | (calcFunc-arctan (nth 3 x)))) | |
1311 | ((equal x '(var inf var-inf)) | |
1312 | (math-quarter-circle t)) | |
1313 | ((equal x '(neg (var inf var-inf))) | |
1314 | (math-neg (math-quarter-circle t))) | |
1315 | ((equal x '(var nan var-nan)) | |
1316 | x) | |
1317 | (t (calc-record-why 'numberp x) | |
491c3062 | 1318 | (list 'calcFunc-arctan x)))) |
136211a9 EZ |
1319 | |
1320 | (defun math-arcsin-raw (x) ; [N N] | |
1321 | (let ((a (math-sqrt-raw (math-sub '(float 1 0) (math-sqr x))))) | |
1322 | (if (or (memq (car x) '(cplx polar)) | |
1323 | (memq (car a) '(cplx polar))) | |
1324 | (math-with-extra-prec 2 ; use extra precision for difficult case | |
1325 | (math-mul '(cplx 0 -1) | |
1326 | (math-ln-raw (math-add (math-mul '(cplx 0 1) x) a)))) | |
491c3062 | 1327 | (math-arctan2-raw x a)))) |
136211a9 EZ |
1328 | |
1329 | (defun math-arccos-raw (x) ; [N N] | |
491c3062 | 1330 | (math-sub (math-pi-over-2) (math-arcsin-raw x))) |
136211a9 EZ |
1331 | |
1332 | (defun math-arctan-raw (x) ; [N N] | |
1333 | (cond ((memq (car x) '(cplx polar)) | |
1334 | (math-with-extra-prec 2 ; extra-extra | |
1335 | (math-div (math-sub | |
1336 | (math-ln-raw (math-add 1 (math-mul '(cplx 0 1) x))) | |
1337 | (math-ln-raw (math-add 1 (math-mul '(cplx 0 -1) x)))) | |
1338 | '(cplx 0 2)))) | |
1339 | ((Math-integer-negp (nth 1 x)) | |
1340 | (math-neg-float (math-arctan-raw (math-neg-float x)))) | |
1341 | ((math-zerop x) x) | |
17cc361e | 1342 | ((math-use-emacs-fn 'atan x)) |
136211a9 EZ |
1343 | (calc-symbolic-mode (signal 'inexact-result nil)) |
1344 | ((math-equal-int x 1) (math-pi-over-4)) | |
1345 | ((math-equal-int x -1) (math-neg (math-pi-over-4))) | |
1346 | ((math-lessp-float '(float 414214 -6) x) ; if x > sqrt(2) - 1, reduce | |
1347 | (if (math-lessp-float '(float 1 0) x) | |
1348 | (math-sub-float (math-mul-float (math-pi) '(float 5 -1)) | |
1349 | (math-arctan-raw (math-div-float '(float 1 0) x))) | |
1350 | (math-sub-float (math-mul-float (math-pi) '(float 25 -2)) | |
1351 | (math-arctan-raw (math-div-float | |
1352 | (math-sub-float '(float 1 0) x) | |
1353 | (math-add-float '(float 1 0) | |
1354 | x)))))) | |
491c3062 | 1355 | (t (math-arctan-series x 3 x (math-neg-float (math-sqr-float x)))))) |
136211a9 EZ |
1356 | |
1357 | (defun math-arctan-series (sum n x xnegsqr) | |
1358 | (math-working "arctan" sum) | |
1359 | (let* ((nextx (math-mul-float x xnegsqr)) | |
1360 | (nextsum (math-add-float sum (math-div-float nextx (math-float n))))) | |
1361 | (if (math-nearly-equal-float sum nextsum) | |
1362 | sum | |
491c3062 | 1363 | (math-arctan-series nextsum (+ n 2) nextx xnegsqr)))) |
136211a9 EZ |
1364 | |
1365 | (defun calcFunc-arctan2 (y x) ; [F R R] [Public] | |
1366 | (if (Math-anglep y) | |
1367 | (if (Math-anglep x) | |
1368 | (math-with-extra-prec 2 | |
1369 | (math-from-radians (math-arctan2-raw (math-float y) | |
1370 | (math-float x)))) | |
1371 | (calc-record-why 'anglep x) | |
1372 | (list 'calcFunc-arctan2 y x)) | |
1373 | (if (and (or (math-infinitep x) (math-anglep x)) | |
1374 | (or (math-infinitep y) (math-anglep y))) | |
1375 | (progn | |
1376 | (if (math-posp x) | |
1377 | (setq x 1) | |
1378 | (if (math-negp x) | |
1379 | (setq x -1) | |
1380 | (or (math-zerop x) | |
1381 | (setq x nil)))) | |
1382 | (if (math-posp y) | |
1383 | (setq y 1) | |
1384 | (if (math-negp y) | |
1385 | (setq y -1) | |
1386 | (or (math-zerop y) | |
1387 | (setq y nil)))) | |
1388 | (if (and y x) | |
1389 | (calcFunc-arctan2 y x) | |
1390 | '(var nan var-nan))) | |
1391 | (calc-record-why 'anglep y) | |
491c3062 | 1392 | (list 'calcFunc-arctan2 y x)))) |
136211a9 EZ |
1393 | |
1394 | (defun math-arctan2-raw (y x) ; [F R R] | |
1395 | (cond ((math-zerop y) | |
1396 | (if (math-negp x) (math-pi) | |
1397 | (if (or (math-floatp x) (math-floatp y)) '(float 0 0) 0))) | |
1398 | ((math-zerop x) | |
1399 | (if (math-posp y) | |
1400 | (math-pi-over-2) | |
1401 | (math-neg (math-pi-over-2)))) | |
1402 | ((math-posp x) | |
1403 | (math-arctan-raw (math-div-float y x))) | |
1404 | ((math-posp y) | |
1405 | (math-add-float (math-arctan-raw (math-div-float y x)) | |
1406 | (math-pi))) | |
1407 | (t | |
1408 | (math-sub-float (math-arctan-raw (math-div-float y x)) | |
491c3062 | 1409 | (math-pi))))) |
136211a9 EZ |
1410 | |
1411 | (defun calcFunc-arcsincos (x) ; [V N] [Public] | |
1412 | (if (and (Math-vectorp x) | |
1413 | (= (length x) 3)) | |
1414 | (calcFunc-arctan2 (nth 2 x) (nth 1 x)) | |
491c3062 | 1415 | (math-reject-arg x "*Two-element vector expected"))) |
136211a9 EZ |
1416 | |
1417 | ||
1418 | ||
1419 | ;;; Exponential function. | |
1420 | ||
1421 | (defun calcFunc-exp (x) ; [N N] [Public] | |
1422 | (cond ((eq x 0) 1) | |
1423 | ((and (memq x '(1 -1)) calc-symbolic-mode) | |
1424 | (if (eq x 1) '(var e var-e) (math-div 1 '(var e var-e)))) | |
1425 | ((Math-numberp x) | |
1426 | (math-with-extra-prec 2 (math-exp-raw (math-float x)))) | |
1427 | ((eq (car-safe x) 'sdev) | |
1428 | (let ((ex (calcFunc-exp (nth 1 x)))) | |
1429 | (math-make-sdev ex (math-mul (nth 2 x) ex)))) | |
1430 | ((eq (car-safe x) 'intv) | |
1431 | (math-make-intv (nth 1 x) (calcFunc-exp (nth 2 x)) | |
1432 | (calcFunc-exp (nth 3 x)))) | |
1433 | ((equal x '(var inf var-inf)) | |
1434 | x) | |
1435 | ((equal x '(neg (var inf var-inf))) | |
1436 | 0) | |
1437 | ((equal x '(var nan var-nan)) | |
1438 | x) | |
1439 | (t (calc-record-why 'numberp x) | |
491c3062 | 1440 | (list 'calcFunc-exp x)))) |
136211a9 EZ |
1441 | |
1442 | (defun calcFunc-expm1 (x) ; [N N] [Public] | |
1443 | (cond ((eq x 0) 0) | |
1444 | ((math-zerop x) '(float 0 0)) | |
1445 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1446 | ((Math-numberp x) | |
1447 | (math-with-extra-prec 2 | |
1448 | (let ((x (math-float x))) | |
1449 | (if (and (eq (car x) 'float) | |
1450 | (math-lessp-float x '(float 1 0)) | |
1451 | (math-lessp-float '(float -1 0) x)) | |
1452 | (math-exp-minus-1-raw x) | |
1453 | (math-add (math-exp-raw x) -1))))) | |
1454 | ((eq (car-safe x) 'sdev) | |
1455 | (if (math-constp x) | |
1456 | (let ((ex (calcFunc-expm1 (nth 1 x)))) | |
1457 | (math-make-sdev ex (math-mul (nth 2 x) (math-add ex 1)))) | |
1458 | (math-make-sdev (calcFunc-expm1 (nth 1 x)) | |
1459 | (math-mul (nth 2 x) (calcFunc-exp (nth 1 x)))))) | |
1460 | ((eq (car-safe x) 'intv) | |
1461 | (math-make-intv (nth 1 x) | |
1462 | (calcFunc-expm1 (nth 2 x)) | |
1463 | (calcFunc-expm1 (nth 3 x)))) | |
1464 | ((equal x '(var inf var-inf)) | |
1465 | x) | |
1466 | ((equal x '(neg (var inf var-inf))) | |
1467 | -1) | |
1468 | ((equal x '(var nan var-nan)) | |
1469 | x) | |
1470 | (t (calc-record-why 'numberp x) | |
491c3062 | 1471 | (list 'calcFunc-expm1 x)))) |
136211a9 EZ |
1472 | |
1473 | (defun calcFunc-exp10 (x) ; [N N] [Public] | |
1474 | (if (eq x 0) | |
1475 | 1 | |
491c3062 | 1476 | (math-pow '(float 1 1) x))) |
136211a9 EZ |
1477 | |
1478 | (defun math-exp-raw (x) ; [N N] | |
1479 | (cond ((math-zerop x) '(float 1 0)) | |
1480 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1481 | ((eq (car x) 'cplx) | |
1482 | (let ((expx (math-exp-raw (nth 1 x))) | |
1483 | (sc (math-sin-cos-raw (nth 2 x)))) | |
1484 | (list 'cplx | |
1485 | (math-mul-float expx (cdr sc)) | |
1486 | (math-mul-float expx (car sc))))) | |
1487 | ((eq (car x) 'polar) | |
1488 | (let ((xc (math-complex x))) | |
1489 | (list 'polar | |
1490 | (math-exp-raw (nth 1 xc)) | |
1491 | (math-from-radians (nth 2 xc))))) | |
178b8baf | 1492 | ((math-use-emacs-fn 'exp x)) |
136211a9 EZ |
1493 | ((or (math-lessp-float '(float 5 -1) x) |
1494 | (math-lessp-float x '(float -5 -1))) | |
1495 | (if (math-lessp-float '(float 921035 1) x) | |
1496 | (math-overflow) | |
1497 | (if (math-lessp-float x '(float -921035 1)) | |
1498 | (math-underflow))) | |
1499 | (let* ((two-x (math-mul-float x '(float 2 0))) | |
1500 | (hint (math-scale-int (nth 1 two-x) (nth 2 two-x))) | |
1501 | (hfrac (math-sub-float x (math-mul-float (math-float hint) | |
1502 | '(float 5 -1))))) | |
1503 | (math-mul-float (math-ipow (math-sqrt-e) hint) | |
1504 | (math-add-float '(float 1 0) | |
1505 | (math-exp-minus-1-raw hfrac))))) | |
491c3062 | 1506 | (t (math-add-float '(float 1 0) (math-exp-minus-1-raw x))))) |
136211a9 EZ |
1507 | |
1508 | (defun math-exp-minus-1-raw (x) ; [F F] | |
491c3062 | 1509 | (math-exp-series x 2 3 x x)) |
136211a9 EZ |
1510 | |
1511 | (defun math-exp-series (sum nfac n xpow x) | |
1512 | (math-working "exp" sum) | |
1513 | (let* ((nextx (math-mul-float xpow x)) | |
1514 | (nextsum (math-add-float sum (math-div-float nextx | |
1515 | (math-float nfac))))) | |
1516 | (if (math-nearly-equal-float sum nextsum) | |
1517 | sum | |
491c3062 | 1518 | (math-exp-series nextsum (math-mul nfac n) (1+ n) nextx x)))) |
136211a9 EZ |
1519 | |
1520 | ||
1521 | ||
1522 | ;;; Logarithms. | |
1523 | ||
1524 | (defun calcFunc-ln (x) ; [N N] [Public] | |
1525 | (cond ((math-zerop x) | |
1526 | (if calc-infinite-mode | |
1527 | '(neg (var inf var-inf)) | |
1528 | (math-reject-arg x "*Logarithm of zero"))) | |
1529 | ((eq x 1) 0) | |
1530 | ((Math-numberp x) | |
1531 | (math-with-extra-prec 2 (math-ln-raw (math-float x)))) | |
1532 | ((eq (car-safe x) 'sdev) | |
1533 | (math-make-sdev (calcFunc-ln (nth 1 x)) | |
1534 | (math-div (nth 2 x) (nth 1 x)))) | |
1535 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1536 | (Math-zerop (nth 2 x)) | |
1537 | (not (math-intv-constp x)))) | |
1538 | (let ((calc-infinite-mode t)) | |
1539 | (math-make-intv (nth 1 x) (calcFunc-ln (nth 2 x)) | |
1540 | (calcFunc-ln (nth 3 x))))) | |
1541 | ((equal x '(var e var-e)) | |
1542 | 1) | |
1543 | ((and (eq (car-safe x) '^) | |
1544 | (equal (nth 1 x) '(var e var-e)) | |
1545 | (math-known-realp (nth 2 x))) | |
1546 | (nth 2 x)) | |
1547 | ((math-infinitep x) | |
1548 | (if (equal x '(var nan var-nan)) | |
1549 | x | |
1550 | '(var inf var-inf))) | |
1551 | (t (calc-record-why 'numberp x) | |
491c3062 | 1552 | (list 'calcFunc-ln x)))) |
136211a9 EZ |
1553 | |
1554 | (defun calcFunc-log10 (x) ; [N N] [Public] | |
1555 | (cond ((math-equal-int x 1) | |
1556 | (if (math-floatp x) '(float 0 0) 0)) | |
1557 | ((and (Math-integerp x) | |
1558 | (math-posp x) | |
1559 | (let ((res (math-integer-log x 10))) | |
1560 | (and (car res) | |
1561 | (setq x (cdr res))))) | |
1562 | x) | |
1563 | ((and (eq (car-safe x) 'frac) | |
1564 | (eq (nth 1 x) 1) | |
1565 | (let ((res (math-integer-log (nth 2 x) 10))) | |
1566 | (and (car res) | |
1567 | (setq x (- (cdr res)))))) | |
1568 | x) | |
1569 | ((math-zerop x) | |
1570 | (if calc-infinite-mode | |
1571 | '(neg (var inf var-inf)) | |
1572 | (math-reject-arg x "*Logarithm of zero"))) | |
1573 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1574 | ((Math-numberp x) | |
1575 | (math-with-extra-prec 2 | |
1576 | (let ((xf (math-float x))) | |
1577 | (if (eq (nth 1 xf) 0) | |
1578 | (math-reject-arg x "*Logarithm of zero")) | |
1579 | (if (Math-integer-posp (nth 1 xf)) | |
1580 | (if (eq (nth 1 xf) 1) ; log10(1*10^n) = n | |
1581 | (math-float (nth 2 xf)) | |
1582 | (let ((xdigs (1- (math-numdigs (nth 1 xf))))) | |
1583 | (math-add-float | |
1584 | (math-div-float (math-ln-raw-2 | |
1585 | (list 'float (nth 1 xf) (- xdigs))) | |
1586 | (math-ln-10)) | |
1587 | (math-float (+ (nth 2 xf) xdigs))))) | |
1588 | (math-div (calcFunc-ln xf) (math-ln-10)))))) | |
1589 | ((eq (car-safe x) 'sdev) | |
1590 | (math-make-sdev (calcFunc-log10 (nth 1 x)) | |
1591 | (math-div (nth 2 x) | |
1592 | (math-mul (nth 1 x) (math-ln-10))))) | |
1593 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1594 | (not (math-intv-constp x)))) | |
1595 | (math-make-intv (nth 1 x) | |
1596 | (calcFunc-log10 (nth 2 x)) | |
1597 | (calcFunc-log10 (nth 3 x)))) | |
1598 | ((math-infinitep x) | |
1599 | (if (equal x '(var nan var-nan)) | |
1600 | x | |
1601 | '(var inf var-inf))) | |
1602 | (t (calc-record-why 'numberp x) | |
491c3062 | 1603 | (list 'calcFunc-log10 x)))) |
136211a9 EZ |
1604 | |
1605 | (defun calcFunc-log (x &optional b) ; [N N N] [Public] | |
1606 | (cond ((or (null b) (equal b '(var e var-e))) | |
1607 | (math-normalize (list 'calcFunc-ln x))) | |
1608 | ((or (eq b 10) (equal b '(float 1 1))) | |
1609 | (math-normalize (list 'calcFunc-log10 x))) | |
1610 | ((math-zerop x) | |
1611 | (if calc-infinite-mode | |
1612 | (math-div (calcFunc-ln x) (calcFunc-ln b)) | |
1613 | (math-reject-arg x "*Logarithm of zero"))) | |
1614 | ((math-zerop b) | |
1615 | (if calc-infinite-mode | |
1616 | (math-div (calcFunc-ln x) (calcFunc-ln b)) | |
1617 | (math-reject-arg b "*Logarithm of zero"))) | |
1618 | ((math-equal-int b 1) | |
1619 | (if calc-infinite-mode | |
1620 | (math-div (calcFunc-ln x) 0) | |
1621 | (math-reject-arg b "*Logarithm base one"))) | |
1622 | ((math-equal-int x 1) | |
86498823 | 1623 | (if (math-floatp b) '(float 0 0) 0)) |
136211a9 EZ |
1624 | ((and (Math-ratp x) (Math-ratp b) |
1625 | (math-posp x) (math-posp b) | |
1626 | (let* ((sign 1) (inv nil) | |
1627 | (xx (if (Math-lessp 1 x) | |
1628 | x | |
1629 | (setq sign -1) | |
1630 | (math-div 1 x))) | |
1631 | (bb (if (Math-lessp 1 b) | |
1632 | b | |
1633 | (setq sign (- sign)) | |
1634 | (math-div 1 b))) | |
1635 | (res (if (Math-lessp xx bb) | |
1636 | (setq inv (math-integer-log bb xx)) | |
1637 | (math-integer-log xx bb)))) | |
1638 | (and (car res) | |
1639 | (setq x (if inv | |
1640 | (math-div 1 (* sign (cdr res))) | |
1641 | (* sign (cdr res))))))) | |
1642 | x) | |
1643 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1644 | ((and (Math-numberp x) (Math-numberp b)) | |
1645 | (math-with-extra-prec 2 | |
1646 | (math-div (math-ln-raw (math-float x)) | |
1647 | (math-log-base-raw b)))) | |
1648 | ((and (eq (car-safe x) 'sdev) | |
1649 | (Math-numberp b)) | |
1650 | (math-make-sdev (calcFunc-log (nth 1 x) b) | |
1651 | (math-div (nth 2 x) | |
1652 | (math-mul (nth 1 x) | |
1653 | (math-log-base-raw b))))) | |
1654 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1655 | (not (math-intv-constp x))) | |
1656 | (math-realp b)) | |
1657 | (math-make-intv (nth 1 x) | |
1658 | (calcFunc-log (nth 2 x) b) | |
1659 | (calcFunc-log (nth 3 x) b))) | |
1660 | ((or (eq (car-safe x) 'intv) (eq (car-safe b) 'intv)) | |
1661 | (math-div (calcFunc-ln x) (calcFunc-ln b))) | |
1662 | ((or (math-infinitep x) | |
1663 | (math-infinitep b)) | |
1664 | (math-div (calcFunc-ln x) (calcFunc-ln b))) | |
1665 | (t (if (Math-numberp b) | |
1666 | (calc-record-why 'numberp x) | |
1667 | (calc-record-why 'numberp b)) | |
491c3062 | 1668 | (list 'calcFunc-log x b)))) |
136211a9 EZ |
1669 | |
1670 | (defun calcFunc-alog (x &optional b) | |
1671 | (cond ((or (null b) (equal b '(var e var-e))) | |
1672 | (math-normalize (list 'calcFunc-exp x))) | |
491c3062 | 1673 | (t (math-pow b x)))) |
136211a9 EZ |
1674 | |
1675 | (defun calcFunc-ilog (x b) | |
1676 | (if (and (math-natnump x) (not (eq x 0)) | |
1677 | (math-natnump b) (not (eq b 0))) | |
1678 | (if (eq b 1) | |
1679 | (math-reject-arg x "*Logarithm base one") | |
1680 | (if (Math-natnum-lessp x b) | |
1681 | 0 | |
1682 | (cdr (math-integer-log x b)))) | |
491c3062 | 1683 | (math-floor (calcFunc-log x b)))) |
136211a9 EZ |
1684 | |
1685 | (defun math-integer-log (x b) | |
1686 | (let ((pows (list b)) | |
1687 | (pow (math-sqr b)) | |
1688 | next | |
1689 | sum n) | |
1690 | (while (not (Math-lessp x pow)) | |
1691 | (setq pows (cons pow pows) | |
1692 | pow (math-sqr pow))) | |
1693 | (setq n (lsh 1 (1- (length pows))) | |
1694 | sum n | |
1695 | pow (car pows)) | |
1696 | (while (and (setq pows (cdr pows)) | |
1697 | (Math-lessp pow x)) | |
1698 | (setq n (/ n 2) | |
1699 | next (math-mul pow (car pows))) | |
1700 | (or (Math-lessp x next) | |
1701 | (setq pow next | |
1702 | sum (+ sum n)))) | |
491c3062 | 1703 | (cons (equal pow x) sum))) |
136211a9 EZ |
1704 | |
1705 | ||
3132f345 | 1706 | (defvar math-log-base-cache nil) |
136211a9 EZ |
1707 | (defun math-log-base-raw (b) ; [N N] |
1708 | (if (not (and (equal (car math-log-base-cache) b) | |
1709 | (eq (nth 1 math-log-base-cache) calc-internal-prec))) | |
1710 | (setq math-log-base-cache (list b calc-internal-prec | |
1711 | (math-ln-raw (math-float b))))) | |
491c3062 | 1712 | (nth 2 math-log-base-cache)) |
136211a9 EZ |
1713 | |
1714 | (defun calcFunc-lnp1 (x) ; [N N] [Public] | |
1715 | (cond ((Math-equal-int x -1) | |
1716 | (if calc-infinite-mode | |
1717 | '(neg (var inf var-inf)) | |
1718 | (math-reject-arg x "*Logarithm of zero"))) | |
1719 | ((eq x 0) 0) | |
1720 | ((math-zerop x) '(float 0 0)) | |
1721 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1722 | ((Math-numberp x) | |
1723 | (math-with-extra-prec 2 | |
1724 | (let ((x (math-float x))) | |
1725 | (if (and (eq (car x) 'float) | |
1726 | (math-lessp-float x '(float 5 -1)) | |
1727 | (math-lessp-float '(float -5 -1) x)) | |
1728 | (math-ln-plus-1-raw x) | |
1729 | (math-ln-raw (math-add-float x '(float 1 0))))))) | |
1730 | ((eq (car-safe x) 'sdev) | |
1731 | (math-make-sdev (calcFunc-lnp1 (nth 1 x)) | |
1732 | (math-div (nth 2 x) (math-add (nth 1 x) 1)))) | |
1733 | ((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x)) | |
1734 | (not (math-intv-constp x)))) | |
1735 | (math-make-intv (nth 1 x) | |
1736 | (calcFunc-lnp1 (nth 2 x)) | |
1737 | (calcFunc-lnp1 (nth 3 x)))) | |
1738 | ((math-infinitep x) | |
1739 | (if (equal x '(var nan var-nan)) | |
1740 | x | |
1741 | '(var inf var-inf))) | |
1742 | (t (calc-record-why 'numberp x) | |
491c3062 | 1743 | (list 'calcFunc-lnp1 x)))) |
136211a9 EZ |
1744 | |
1745 | (defun math-ln-raw (x) ; [N N] --- must be float format! | |
1746 | (cond ((eq (car-safe x) 'cplx) | |
1747 | (list 'cplx | |
1748 | (math-mul-float (math-ln-raw | |
1749 | (math-add-float (math-sqr-float (nth 1 x)) | |
1750 | (math-sqr-float (nth 2 x)))) | |
1751 | '(float 5 -1)) | |
1752 | (math-arctan2-raw (nth 2 x) (nth 1 x)))) | |
1753 | ((eq (car x) 'polar) | |
1754 | (math-polar (list 'cplx | |
1755 | (math-ln-raw (nth 1 x)) | |
1756 | (math-to-radians (nth 2 x))))) | |
1757 | ((Math-equal-int x 1) | |
1758 | '(float 0 0)) | |
1759 | (calc-symbolic-mode (signal 'inexact-result nil)) | |
1760 | ((math-posp (nth 1 x)) ; positive and real | |
17cc361e JB |
1761 | (cond |
1762 | ((math-use-emacs-fn 'log x)) | |
1763 | (t | |
1764 | (let ((xdigs (1- (math-numdigs (nth 1 x))))) | |
1765 | (math-add-float (math-ln-raw-2 (list 'float (nth 1 x) (- xdigs))) | |
1766 | (math-mul-float (math-float (+ (nth 2 x) xdigs)) | |
1767 | (math-ln-10))))))) | |
136211a9 EZ |
1768 | ((math-zerop x) |
1769 | (math-reject-arg x "*Logarithm of zero")) | |
1770 | ((eq calc-complex-mode 'polar) ; negative and real | |
1771 | (math-polar | |
1772 | (list 'cplx ; negative and real | |
1773 | (math-ln-raw (math-neg-float x)) | |
1774 | (math-pi)))) | |
1775 | (t (list 'cplx ; negative and real | |
1776 | (math-ln-raw (math-neg-float x)) | |
491c3062 | 1777 | (math-pi))))) |
136211a9 EZ |
1778 | |
1779 | (defun math-ln-raw-2 (x) ; [F F] | |
1780 | (cond ((math-lessp-float '(float 14 -1) x) | |
1781 | (math-add-float (math-ln-raw-2 (math-mul-float x '(float 5 -1))) | |
1782 | (math-ln-2))) | |
1783 | (t ; now .7 < x <= 1.4 | |
1784 | (math-ln-raw-3 (math-div-float (math-sub-float x '(float 1 0)) | |
491c3062 | 1785 | (math-add-float x '(float 1 0))))))) |
136211a9 EZ |
1786 | |
1787 | (defun math-ln-raw-3 (x) ; [F F] | |
1788 | (math-mul-float (math-ln-raw-series x 3 x (math-sqr-float x)) | |
491c3062 | 1789 | '(float 2 0))) |
136211a9 EZ |
1790 | |
1791 | ;;; Compute ln((1+x)/(1-x)) | |
1792 | (defun math-ln-raw-series (sum n x xsqr) | |
1793 | (math-working "log" sum) | |
1794 | (let* ((nextx (math-mul-float x xsqr)) | |
1795 | (nextsum (math-add-float sum (math-div-float nextx (math-float n))))) | |
1796 | (if (math-nearly-equal-float sum nextsum) | |
1797 | sum | |
491c3062 | 1798 | (math-ln-raw-series nextsum (+ n 2) nextx xsqr)))) |
136211a9 EZ |
1799 | |
1800 | (defun math-ln-plus-1-raw (x) | |
491c3062 | 1801 | (math-lnp1-series x 2 x (math-neg x))) |
136211a9 EZ |
1802 | |
1803 | (defun math-lnp1-series (sum n xpow x) | |
1804 | (math-working "lnp1" sum) | |
1805 | (let* ((nextx (math-mul-float xpow x)) | |
1806 | (nextsum (math-add-float sum (math-div-float nextx (math-float n))))) | |
1807 | (if (math-nearly-equal-float sum nextsum) | |
1808 | sum | |
491c3062 | 1809 | (math-lnp1-series nextsum (1+ n) nextx x)))) |
136211a9 | 1810 | |
4603f755 | 1811 | (defconst math-approx-ln-10 |
82472f45 | 1812 | (math-read-number-simple "2.302585092994045684018") |
94c95a35 | 1813 | "An approximation for ln(10).") |
4603f755 JB |
1814 | |
1815 | (math-defcache math-ln-10 math-approx-ln-10 | |
136211a9 EZ |
1816 | (math-ln-raw-2 '(float 1 1))) |
1817 | ||
4603f755 | 1818 | (defconst math-approx-ln-2 |
82472f45 | 1819 | (math-read-number-simple "0.693147180559945309417") |
94c95a35 | 1820 | "An approximation for ln(2).") |
4603f755 JB |
1821 | |
1822 | (math-defcache math-ln-2 math-approx-ln-2 | |
136211a9 EZ |
1823 | (math-ln-raw-3 (math-float '(frac 1 3)))) |
1824 | ||
1825 | ||
1826 | ||
1827 | ;;; Hyperbolic functions. | |
1828 | ||
1829 | (defun calcFunc-sinh (x) ; [N N] [Public] | |
1830 | (cond ((eq x 0) 0) | |
1831 | (math-expand-formulas | |
1832 | (math-normalize | |
1833 | (list '/ (list '- (list 'calcFunc-exp x) | |
1834 | (list 'calcFunc-exp (list 'neg x))) 2))) | |
1835 | ((Math-numberp x) | |
1836 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1837 | (math-with-extra-prec 2 | |
1838 | (let ((expx (math-exp-raw (math-float x)))) | |
1839 | (math-mul (math-add expx (math-div -1 expx)) '(float 5 -1))))) | |
1840 | ((eq (car-safe x) 'sdev) | |
1841 | (math-make-sdev (calcFunc-sinh (nth 1 x)) | |
1842 | (math-mul (nth 2 x) (calcFunc-cosh (nth 1 x))))) | |
1843 | ((eq (car x) 'intv) | |
1844 | (math-sort-intv (nth 1 x) | |
1845 | (calcFunc-sinh (nth 2 x)) | |
1846 | (calcFunc-sinh (nth 3 x)))) | |
1847 | ((or (equal x '(var inf var-inf)) | |
1848 | (equal x '(neg (var inf var-inf))) | |
1849 | (equal x '(var nan var-nan))) | |
1850 | x) | |
1851 | (t (calc-record-why 'numberp x) | |
491c3062 | 1852 | (list 'calcFunc-sinh x)))) |
136211a9 EZ |
1853 | (put 'calcFunc-sinh 'math-expandable t) |
1854 | ||
1855 | (defun calcFunc-cosh (x) ; [N N] [Public] | |
1856 | (cond ((eq x 0) 1) | |
1857 | (math-expand-formulas | |
1858 | (math-normalize | |
1859 | (list '/ (list '+ (list 'calcFunc-exp x) | |
1860 | (list 'calcFunc-exp (list 'neg x))) 2))) | |
1861 | ((Math-numberp x) | |
1862 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1863 | (math-with-extra-prec 2 | |
1864 | (let ((expx (math-exp-raw (math-float x)))) | |
1865 | (math-mul (math-add expx (math-div 1 expx)) '(float 5 -1))))) | |
1866 | ((eq (car-safe x) 'sdev) | |
1867 | (math-make-sdev (calcFunc-cosh (nth 1 x)) | |
1868 | (math-mul (nth 2 x) | |
1869 | (calcFunc-sinh (nth 1 x))))) | |
1870 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
1871 | (setq x (math-abs x)) | |
1872 | (math-sort-intv (nth 1 x) | |
1873 | (calcFunc-cosh (nth 2 x)) | |
1874 | (calcFunc-cosh (nth 3 x)))) | |
1875 | ((or (equal x '(var inf var-inf)) | |
1876 | (equal x '(neg (var inf var-inf))) | |
1877 | (equal x '(var nan var-nan))) | |
1878 | (math-abs x)) | |
1879 | (t (calc-record-why 'numberp x) | |
491c3062 | 1880 | (list 'calcFunc-cosh x)))) |
136211a9 EZ |
1881 | (put 'calcFunc-cosh 'math-expandable t) |
1882 | ||
1883 | (defun calcFunc-tanh (x) ; [N N] [Public] | |
1884 | (cond ((eq x 0) 0) | |
1885 | (math-expand-formulas | |
1886 | (math-normalize | |
1887 | (let ((expx (list 'calcFunc-exp x)) | |
1888 | (expmx (list 'calcFunc-exp (list 'neg x)))) | |
1889 | (math-normalize | |
1890 | (list '/ (list '- expx expmx) (list '+ expx expmx)))))) | |
1891 | ((Math-numberp x) | |
1892 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1893 | (math-with-extra-prec 2 | |
1894 | (let* ((expx (calcFunc-exp (math-float x))) | |
1895 | (expmx (math-div 1 expx))) | |
1896 | (math-div (math-sub expx expmx) | |
1897 | (math-add expx expmx))))) | |
1898 | ((eq (car-safe x) 'sdev) | |
1899 | (math-make-sdev (calcFunc-tanh (nth 1 x)) | |
1900 | (math-div (nth 2 x) | |
1901 | (math-sqr (calcFunc-cosh (nth 1 x)))))) | |
1902 | ((eq (car x) 'intv) | |
1903 | (math-sort-intv (nth 1 x) | |
1904 | (calcFunc-tanh (nth 2 x)) | |
1905 | (calcFunc-tanh (nth 3 x)))) | |
1906 | ((equal x '(var inf var-inf)) | |
1907 | 1) | |
1908 | ((equal x '(neg (var inf var-inf))) | |
1909 | -1) | |
1910 | ((equal x '(var nan var-nan)) | |
1911 | x) | |
1912 | (t (calc-record-why 'numberp x) | |
491c3062 | 1913 | (list 'calcFunc-tanh x)))) |
136211a9 EZ |
1914 | (put 'calcFunc-tanh 'math-expandable t) |
1915 | ||
f53e6c20 JB |
1916 | (defun calcFunc-sech (x) ; [N N] [Public] |
1917 | (cond ((eq x 0) 1) | |
1918 | (math-expand-formulas | |
1919 | (math-normalize | |
1920 | (list '/ 2 (list '+ (list 'calcFunc-exp x) | |
1921 | (list 'calcFunc-exp (list 'neg x)))))) | |
1922 | ((Math-numberp x) | |
1923 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1924 | (math-with-extra-prec 2 | |
1925 | (let ((expx (math-exp-raw (math-float x)))) | |
1926 | (math-div '(float 2 0) (math-add expx (math-div 1 expx)))))) | |
1927 | ((eq (car-safe x) 'sdev) | |
1928 | (math-make-sdev (calcFunc-sech (nth 1 x)) | |
1929 | (math-mul (nth 2 x) | |
1930 | (math-mul (calcFunc-sech (nth 1 x)) | |
1931 | (calcFunc-tanh (nth 1 x)))))) | |
1932 | ((and (eq (car x) 'intv) (math-intv-constp x)) | |
1933 | (setq x (math-abs x)) | |
1934 | (math-sort-intv (nth 1 x) | |
1935 | (calcFunc-sech (nth 2 x)) | |
1936 | (calcFunc-sech (nth 3 x)))) | |
1937 | ((or (equal x '(var inf var-inf)) | |
1938 | (equal x '(neg (var inf var-inf)))) | |
1939 | 0) | |
1940 | ((equal x '(var nan var-nan)) | |
1941 | x) | |
1942 | (t (calc-record-why 'numberp x) | |
1943 | (list 'calcFunc-sech x)))) | |
1944 | (put 'calcFunc-sech 'math-expandable t) | |
1945 | ||
1946 | (defun calcFunc-csch (x) ; [N N] [Public] | |
1947 | (cond ((eq x 0) (math-div 1 0)) | |
1948 | (math-expand-formulas | |
1949 | (math-normalize | |
1950 | (list '/ 2 (list '- (list 'calcFunc-exp x) | |
1951 | (list 'calcFunc-exp (list 'neg x)))))) | |
1952 | ((Math-numberp x) | |
1953 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1954 | (math-with-extra-prec 2 | |
1955 | (let ((expx (math-exp-raw (math-float x)))) | |
1956 | (math-div '(float 2 0) (math-add expx (math-div -1 expx)))))) | |
1957 | ((eq (car-safe x) 'sdev) | |
1958 | (math-make-sdev (calcFunc-csch (nth 1 x)) | |
1959 | (math-mul (nth 2 x) | |
1960 | (math-mul (calcFunc-csch (nth 1 x)) | |
1961 | (calcFunc-coth (nth 1 x)))))) | |
1962 | ((eq (car x) 'intv) | |
1963 | (if (and (Math-negp (nth 2 x)) | |
1964 | (Math-posp (nth 3 x))) | |
1965 | '(intv 3 (neg (var inf var-inf)) (var inf var-inf)) | |
1966 | (math-sort-intv (nth 1 x) | |
1967 | (calcFunc-csch (nth 2 x)) | |
1968 | (calcFunc-csch (nth 3 x))))) | |
1969 | ((or (equal x '(var inf var-inf)) | |
1970 | (equal x '(neg (var inf var-inf)))) | |
1971 | 0) | |
1972 | ((equal x '(var nan var-nan)) | |
1973 | x) | |
1974 | (t (calc-record-why 'numberp x) | |
1975 | (list 'calcFunc-csch x)))) | |
1976 | (put 'calcFunc-csch 'math-expandable t) | |
1977 | ||
1978 | (defun calcFunc-coth (x) ; [N N] [Public] | |
1979 | (cond ((eq x 0) (math-div 1 0)) | |
1980 | (math-expand-formulas | |
1981 | (math-normalize | |
1982 | (let ((expx (list 'calcFunc-exp x)) | |
1983 | (expmx (list 'calcFunc-exp (list 'neg x)))) | |
1984 | (math-normalize | |
1985 | (list '/ (list '+ expx expmx) (list '- expx expmx)))))) | |
1986 | ((Math-numberp x) | |
1987 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
1988 | (math-with-extra-prec 2 | |
1989 | (let* ((expx (calcFunc-exp (math-float x))) | |
1990 | (expmx (math-div 1 expx))) | |
1991 | (math-div (math-add expx expmx) | |
1992 | (math-sub expx expmx))))) | |
1993 | ((eq (car-safe x) 'sdev) | |
1994 | (math-make-sdev (calcFunc-coth (nth 1 x)) | |
1995 | (math-div (nth 2 x) | |
1996 | (math-sqr (calcFunc-sinh (nth 1 x)))))) | |
1997 | ((eq (car x) 'intv) | |
1998 | (if (and (Math-negp (nth 2 x)) | |
1999 | (Math-posp (nth 3 x))) | |
2000 | '(intv 3 (neg (var inf var-inf)) (var inf var-inf)) | |
2001 | (math-sort-intv (nth 1 x) | |
2002 | (calcFunc-coth (nth 2 x)) | |
2003 | (calcFunc-coth (nth 3 x))))) | |
2004 | ((equal x '(var inf var-inf)) | |
2005 | 1) | |
2006 | ((equal x '(neg (var inf var-inf))) | |
2007 | -1) | |
2008 | ((equal x '(var nan var-nan)) | |
2009 | x) | |
2010 | (t (calc-record-why 'numberp x) | |
2011 | (list 'calcFunc-coth x)))) | |
2012 | (put 'calcFunc-coth 'math-expandable t) | |
2013 | ||
136211a9 EZ |
2014 | (defun calcFunc-arcsinh (x) ; [N N] [Public] |
2015 | (cond ((eq x 0) 0) | |
2016 | (math-expand-formulas | |
2017 | (math-normalize | |
2018 | (list 'calcFunc-ln (list '+ x (list 'calcFunc-sqrt | |
2019 | (list '+ (list '^ x 2) 1)))))) | |
2020 | ((Math-numberp x) | |
2021 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
2022 | (math-with-extra-prec 2 | |
2023 | (math-ln-raw (math-add x (math-sqrt-raw (math-add (math-sqr x) | |
2024 | '(float 1 0))))))) | |
2025 | ((eq (car-safe x) 'sdev) | |
2026 | (math-make-sdev (calcFunc-arcsinh (nth 1 x)) | |
2027 | (math-div (nth 2 x) | |
2028 | (math-sqrt | |
2029 | (math-add (math-sqr (nth 1 x)) 1))))) | |
2030 | ((eq (car x) 'intv) | |
2031 | (math-sort-intv (nth 1 x) | |
2032 | (calcFunc-arcsinh (nth 2 x)) | |
2033 | (calcFunc-arcsinh (nth 3 x)))) | |
2034 | ((or (equal x '(var inf var-inf)) | |
2035 | (equal x '(neg (var inf var-inf))) | |
2036 | (equal x '(var nan var-nan))) | |
2037 | x) | |
2038 | (t (calc-record-why 'numberp x) | |
491c3062 | 2039 | (list 'calcFunc-arcsinh x)))) |
136211a9 EZ |
2040 | (put 'calcFunc-arcsinh 'math-expandable t) |
2041 | ||
2042 | (defun calcFunc-arccosh (x) ; [N N] [Public] | |
2043 | (cond ((eq x 1) 0) | |
2044 | ((and (eq x -1) calc-symbolic-mode) | |
2045 | '(var pi var-pi)) | |
2046 | ((and (eq x 0) calc-symbolic-mode) | |
2047 | (math-div (math-mul '(var pi var-pi) '(var i var-i)) 2)) | |
2048 | (math-expand-formulas | |
2049 | (math-normalize | |
2050 | (list 'calcFunc-ln (list '+ x (list 'calcFunc-sqrt | |
2051 | (list '- (list '^ x 2) 1)))))) | |
2052 | ((Math-numberp x) | |
2053 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
2054 | (if (Math-equal-int x -1) | |
2055 | (math-imaginary (math-pi)) | |
2056 | (math-with-extra-prec 2 | |
2057 | (if (or t ; need to do this even in the real case! | |
2058 | (memq (car-safe x) '(cplx polar))) | |
2059 | (let ((xp1 (math-add 1 x))) ; this gets the branch cuts right | |
2060 | (math-ln-raw | |
2061 | (math-add x (math-mul xp1 | |
2062 | (math-sqrt-raw | |
2063 | (math-div (math-sub | |
2064 | x | |
2065 | '(float 1 0)) | |
2066 | xp1)))))) | |
2067 | (math-ln-raw | |
2068 | (math-add x (math-sqrt-raw (math-add (math-sqr x) | |
2069 | '(float -1 0))))))))) | |
2070 | ((eq (car-safe x) 'sdev) | |
2071 | (math-make-sdev (calcFunc-arccosh (nth 1 x)) | |
2072 | (math-div (nth 2 x) | |
2073 | (math-sqrt | |
2074 | (math-add (math-sqr (nth 1 x)) -1))))) | |
2075 | ((eq (car x) 'intv) | |
2076 | (math-sort-intv (nth 1 x) | |
2077 | (calcFunc-arccosh (nth 2 x)) | |
2078 | (calcFunc-arccosh (nth 3 x)))) | |
2079 | ((or (equal x '(var inf var-inf)) | |
2080 | (equal x '(neg (var inf var-inf))) | |
2081 | (equal x '(var nan var-nan))) | |
2082 | x) | |
2083 | (t (calc-record-why 'numberp x) | |
491c3062 | 2084 | (list 'calcFunc-arccosh x)))) |
136211a9 EZ |
2085 | (put 'calcFunc-arccosh 'math-expandable t) |
2086 | ||
2087 | (defun calcFunc-arctanh (x) ; [N N] [Public] | |
2088 | (cond ((eq x 0) 0) | |
2089 | ((and (Math-equal-int x 1) calc-infinite-mode) | |
2090 | '(var inf var-inf)) | |
2091 | ((and (Math-equal-int x -1) calc-infinite-mode) | |
2092 | '(neg (var inf var-inf))) | |
2093 | (math-expand-formulas | |
2094 | (list '/ (list '- | |
2095 | (list 'calcFunc-ln (list '+ 1 x)) | |
2096 | (list 'calcFunc-ln (list '- 1 x))) 2)) | |
2097 | ((Math-numberp x) | |
2098 | (if calc-symbolic-mode (signal 'inexact-result nil)) | |
2099 | (math-with-extra-prec 2 | |
2100 | (if (or (memq (car-safe x) '(cplx polar)) | |
2101 | (Math-lessp 1 x)) | |
2102 | (math-mul (math-sub (math-ln-raw (math-add '(float 1 0) x)) | |
2103 | (math-ln-raw (math-sub '(float 1 0) x))) | |
2104 | '(float 5 -1)) | |
2105 | (if (and (math-equal-int x 1) calc-infinite-mode) | |
2106 | '(var inf var-inf) | |
2107 | (if (and (math-equal-int x -1) calc-infinite-mode) | |
2108 | '(neg (var inf var-inf)) | |
2109 | (math-mul (math-ln-raw (math-div (math-add '(float 1 0) x) | |
2110 | (math-sub 1 x))) | |
2111 | '(float 5 -1))))))) | |
2112 | ((eq (car-safe x) 'sdev) | |
2113 | (math-make-sdev (calcFunc-arctanh (nth 1 x)) | |
2114 | (math-div (nth 2 x) | |
2115 | (math-sub 1 (math-sqr (nth 1 x)))))) | |
2116 | ((eq (car x) 'intv) | |
2117 | (math-sort-intv (nth 1 x) | |
2118 | (calcFunc-arctanh (nth 2 x)) | |
2119 | (calcFunc-arctanh (nth 3 x)))) | |
2120 | ((equal x '(var nan var-nan)) | |
2121 | x) | |
2122 | (t (calc-record-why 'numberp x) | |
491c3062 | 2123 | (list 'calcFunc-arctanh x)))) |
136211a9 EZ |
2124 | (put 'calcFunc-arctanh 'math-expandable t) |
2125 | ||
2126 | ||
2127 | ;;; Convert A from HMS or degrees to radians. | |
2128 | (defun calcFunc-rad (a) ; [R R] [Public] | |
2129 | (cond ((or (Math-numberp a) | |
2130 | (eq (car a) 'intv)) | |
2131 | (math-with-extra-prec 2 | |
2132 | (math-mul a (math-pi-over-180)))) | |
2133 | ((eq (car a) 'hms) | |
2134 | (math-from-hms a 'rad)) | |
2135 | ((eq (car a) 'sdev) | |
2136 | (math-make-sdev (calcFunc-rad (nth 1 a)) | |
2137 | (calcFunc-rad (nth 2 a)))) | |
2138 | (math-expand-formulas | |
2139 | (math-div (math-mul a '(var pi var-pi)) 180)) | |
2140 | ((math-infinitep a) a) | |
491c3062 | 2141 | (t (list 'calcFunc-rad a)))) |
136211a9 EZ |
2142 | (put 'calcFunc-rad 'math-expandable t) |
2143 | ||
2144 | ;;; Convert A from HMS or radians to degrees. | |
2145 | (defun calcFunc-deg (a) ; [R R] [Public] | |
2146 | (cond ((or (Math-numberp a) | |
2147 | (eq (car a) 'intv)) | |
2148 | (math-with-extra-prec 2 | |
2149 | (math-div a (math-pi-over-180)))) | |
2150 | ((eq (car a) 'hms) | |
2151 | (math-from-hms a 'deg)) | |
2152 | ((eq (car a) 'sdev) | |
2153 | (math-make-sdev (calcFunc-deg (nth 1 a)) | |
2154 | (calcFunc-deg (nth 2 a)))) | |
2155 | (math-expand-formulas | |
2156 | (math-div (math-mul 180 a) '(var pi var-pi))) | |
2157 | ((math-infinitep a) a) | |
491c3062 | 2158 | (t (list 'calcFunc-deg a)))) |
136211a9 EZ |
2159 | (put 'calcFunc-deg 'math-expandable t) |
2160 | ||
95995a85 JB |
2161 | (provide 'calc-math) |
2162 | ||
cbee283d | 2163 | ;; arch-tag: c7367e8e-d0b8-4f70-8577-2fb3f31dbb4c |
491c3062 | 2164 | ;;; calc-math.el ends here |