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1 | ;; Calculator for GNU Emacs, part II [calc-funcs.el] |
2 | ;; Copyright (C) 1990, 1991, 1992, 1993 Free Software Foundation, Inc. | |
3 | ;; Written by Dave Gillespie, daveg@synaptics.com. | |
4 | ||
5 | ;; This file is part of GNU Emacs. | |
6 | ||
7 | ;; GNU Emacs is distributed in the hope that it will be useful, | |
8 | ;; but WITHOUT ANY WARRANTY. No author or distributor | |
9 | ;; accepts responsibility to anyone for the consequences of using it | |
10 | ;; or for whether it serves any particular purpose or works at all, | |
11 | ;; unless he says so in writing. Refer to the GNU Emacs General Public | |
12 | ;; License for full details. | |
13 | ||
14 | ;; Everyone is granted permission to copy, modify and redistribute | |
15 | ;; GNU Emacs, but only under the conditions described in the | |
16 | ;; GNU Emacs General Public License. A copy of this license is | |
17 | ;; supposed to have been given to you along with GNU Emacs so you | |
18 | ;; can know your rights and responsibilities. It should be in a | |
19 | ;; file named COPYING. Among other things, the copyright notice | |
20 | ;; and this notice must be preserved on all copies. | |
21 | ||
22 | ||
23 | ||
24 | ;; This file is autoloaded from calc-ext.el. | |
25 | (require 'calc-ext) | |
26 | ||
27 | (require 'calc-macs) | |
28 | ||
29 | (defun calc-Need-calc-funcs () nil) | |
30 | ||
31 | ||
32 | (defun calc-inc-gamma (arg) | |
33 | (interactive "P") | |
34 | (calc-slow-wrapper | |
35 | (if (calc-is-inverse) | |
36 | (if (calc-is-hyperbolic) | |
37 | (calc-binary-op "gamG" 'calcFunc-gammaG arg) | |
38 | (calc-binary-op "gamQ" 'calcFunc-gammaQ arg)) | |
39 | (if (calc-is-hyperbolic) | |
40 | (calc-binary-op "gamg" 'calcFunc-gammag arg) | |
41 | (calc-binary-op "gamP" 'calcFunc-gammaP arg)))) | |
42 | ) | |
43 | ||
44 | (defun calc-erf (arg) | |
45 | (interactive "P") | |
46 | (calc-slow-wrapper | |
47 | (if (calc-is-inverse) | |
48 | (calc-unary-op "erfc" 'calcFunc-erfc arg) | |
49 | (calc-unary-op "erf" 'calcFunc-erf arg))) | |
50 | ) | |
51 | ||
52 | (defun calc-erfc (arg) | |
53 | (interactive "P") | |
54 | (calc-invert-func) | |
55 | (calc-erf arg) | |
56 | ) | |
57 | ||
58 | (defun calc-beta (arg) | |
59 | (interactive "P") | |
60 | (calc-slow-wrapper | |
61 | (calc-binary-op "beta" 'calcFunc-beta arg)) | |
62 | ) | |
63 | ||
64 | (defun calc-inc-beta () | |
65 | (interactive) | |
66 | (calc-slow-wrapper | |
67 | (if (calc-is-hyperbolic) | |
68 | (calc-enter-result 3 "betB" (cons 'calcFunc-betaB (calc-top-list-n 3))) | |
69 | (calc-enter-result 3 "betI" (cons 'calcFunc-betaI (calc-top-list-n 3))))) | |
70 | ) | |
71 | ||
72 | (defun calc-bessel-J (arg) | |
73 | (interactive "P") | |
74 | (calc-slow-wrapper | |
75 | (calc-binary-op "besJ" 'calcFunc-besJ arg)) | |
76 | ) | |
77 | ||
78 | (defun calc-bessel-Y (arg) | |
79 | (interactive "P") | |
80 | (calc-slow-wrapper | |
81 | (calc-binary-op "besY" 'calcFunc-besY arg)) | |
82 | ) | |
83 | ||
84 | (defun calc-bernoulli-number (arg) | |
85 | (interactive "P") | |
86 | (calc-slow-wrapper | |
87 | (if (calc-is-hyperbolic) | |
88 | (calc-binary-op "bern" 'calcFunc-bern arg) | |
89 | (calc-unary-op "bern" 'calcFunc-bern arg))) | |
90 | ) | |
91 | ||
92 | (defun calc-euler-number (arg) | |
93 | (interactive "P") | |
94 | (calc-slow-wrapper | |
95 | (if (calc-is-hyperbolic) | |
96 | (calc-binary-op "eulr" 'calcFunc-euler arg) | |
97 | (calc-unary-op "eulr" 'calcFunc-euler arg))) | |
98 | ) | |
99 | ||
100 | (defun calc-stirling-number (arg) | |
101 | (interactive "P") | |
102 | (calc-slow-wrapper | |
103 | (if (calc-is-hyperbolic) | |
104 | (calc-binary-op "str2" 'calcFunc-stir2 arg) | |
105 | (calc-binary-op "str1" 'calcFunc-stir1 arg))) | |
106 | ) | |
107 | ||
108 | (defun calc-utpb () | |
109 | (interactive) | |
110 | (calc-prob-dist "b" 3) | |
111 | ) | |
112 | ||
113 | (defun calc-utpc () | |
114 | (interactive) | |
115 | (calc-prob-dist "c" 2) | |
116 | ) | |
117 | ||
118 | (defun calc-utpf () | |
119 | (interactive) | |
120 | (calc-prob-dist "f" 3) | |
121 | ) | |
122 | ||
123 | (defun calc-utpn () | |
124 | (interactive) | |
125 | (calc-prob-dist "n" 3) | |
126 | ) | |
127 | ||
128 | (defun calc-utpp () | |
129 | (interactive) | |
130 | (calc-prob-dist "p" 2) | |
131 | ) | |
132 | ||
133 | (defun calc-utpt () | |
134 | (interactive) | |
135 | (calc-prob-dist "t" 2) | |
136 | ) | |
137 | ||
138 | (defun calc-prob-dist (letter nargs) | |
139 | (calc-slow-wrapper | |
140 | (if (calc-is-inverse) | |
141 | (calc-enter-result nargs (concat "ltp" letter) | |
142 | (append (list (intern (concat "calcFunc-ltp" letter)) | |
143 | (calc-top-n 1)) | |
144 | (calc-top-list-n (1- nargs) 2))) | |
145 | (calc-enter-result nargs (concat "utp" letter) | |
146 | (append (list (intern (concat "calcFunc-utp" letter)) | |
147 | (calc-top-n 1)) | |
148 | (calc-top-list-n (1- nargs) 2))))) | |
149 | ) | |
150 | ||
151 | ||
152 | ||
153 | ||
154 | ;;; Sources: Numerical Recipes, Press et al; | |
155 | ;;; Handbook of Mathematical Functions, Abramowitz & Stegun. | |
156 | ||
157 | ||
158 | ;;; Gamma function. | |
159 | ||
160 | (defun calcFunc-gamma (x) | |
161 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
162 | (calcFunc-fact (math-add x -1)) | |
163 | ) | |
164 | ||
165 | (defun math-gammap1-raw (x &optional fprec nfprec) ; compute gamma(1 + x) | |
166 | (or fprec | |
167 | (setq fprec (math-float calc-internal-prec) | |
168 | nfprec (math-float (- calc-internal-prec)))) | |
169 | (cond ((math-lessp-float (calcFunc-re x) fprec) | |
170 | (if (math-lessp-float (calcFunc-re x) nfprec) | |
171 | (math-neg (math-div | |
172 | (math-pi) | |
173 | (math-mul (math-gammap1-raw | |
174 | (math-add (math-neg x) | |
175 | '(float -1 0)) | |
176 | fprec nfprec) | |
177 | (math-sin-raw | |
178 | (math-mul (math-pi) x))))) | |
179 | (let ((xplus1 (math-add x '(float 1 0)))) | |
180 | (math-div (math-gammap1-raw xplus1 fprec nfprec) xplus1)))) | |
181 | ((and (math-realp x) | |
182 | (math-lessp-float '(float 736276 0) x)) | |
183 | (math-overflow)) | |
184 | (t ; re(x) now >= 10.0 | |
185 | (let ((xinv (math-div 1 x)) | |
186 | (lnx (math-ln-raw x))) | |
187 | (math-mul (math-sqrt-two-pi) | |
188 | (math-exp-raw | |
189 | (math-gamma-series | |
190 | (math-sub (math-mul (math-add x '(float 5 -1)) | |
191 | lnx) | |
192 | x) | |
193 | xinv | |
194 | (math-sqr xinv) | |
195 | '(float 0 0) | |
196 | 2)))))) | |
197 | ) | |
198 | ||
199 | (defun math-gamma-series (sum x xinvsqr oterm n) | |
200 | (math-working "gamma" sum) | |
201 | (let* ((bn (math-bernoulli-number n)) | |
202 | (term (math-mul (math-div-float (math-float (nth 1 bn)) | |
203 | (math-float (* (nth 2 bn) | |
204 | (* n (1- n))))) | |
205 | x)) | |
206 | (next (math-add sum term))) | |
207 | (if (math-nearly-equal sum next) | |
208 | next | |
209 | (if (> n (* 2 calc-internal-prec)) | |
210 | (progn | |
211 | ;; Need this because series eventually diverges for large enough n. | |
212 | (calc-record-why | |
213 | "*Gamma computation stopped early, not all digits may be valid") | |
214 | next) | |
215 | (math-gamma-series next (math-mul x xinvsqr) xinvsqr term (+ n 2))))) | |
216 | ) | |
217 | ||
218 | ||
219 | ;;; Incomplete gamma function. | |
220 | ||
221 | (defun calcFunc-gammaP (a x) | |
222 | (if (equal x '(var inf var-inf)) | |
223 | '(float 1 0) | |
224 | (math-inexact-result) | |
225 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
226 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
227 | (if (and (math-num-integerp a) | |
228 | (integerp (setq a (math-trunc a))) | |
229 | (> a 0) (< a 20)) | |
230 | (math-sub 1 (calcFunc-gammaQ a x)) | |
231 | (let ((math-current-gamma-value (calcFunc-gamma a))) | |
232 | (math-div (calcFunc-gammag a x) math-current-gamma-value)))) | |
233 | ) | |
234 | ||
235 | (defun calcFunc-gammaQ (a x) | |
236 | (if (equal x '(var inf var-inf)) | |
237 | '(float 0 0) | |
238 | (math-inexact-result) | |
239 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
240 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
241 | (if (and (math-num-integerp a) | |
242 | (integerp (setq a (math-trunc a))) | |
243 | (> a 0) (< a 20)) | |
244 | (let ((n 0) | |
245 | (sum '(float 1 0)) | |
246 | (term '(float 1 0))) | |
247 | (math-with-extra-prec 1 | |
248 | (while (< (setq n (1+ n)) a) | |
249 | (setq term (math-div (math-mul term x) n) | |
250 | sum (math-add sum term)) | |
251 | (math-working "gamma" sum)) | |
252 | (math-mul sum (calcFunc-exp (math-neg x))))) | |
253 | (let ((math-current-gamma-value (calcFunc-gamma a))) | |
254 | (math-div (calcFunc-gammaG a x) math-current-gamma-value)))) | |
255 | ) | |
256 | ||
257 | (defun calcFunc-gammag (a x) | |
258 | (if (equal x '(var inf var-inf)) | |
259 | (calcFunc-gamma a) | |
260 | (math-inexact-result) | |
261 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
262 | (or (Math-numberp x) (math-reject-arg x 'numberp)) | |
263 | (math-with-extra-prec 2 | |
264 | (setq a (math-float a)) | |
265 | (setq x (math-float x)) | |
266 | (if (or (math-negp (calcFunc-re a)) | |
267 | (math-lessp-float (calcFunc-re x) | |
268 | (math-add-float (calcFunc-re a) | |
269 | '(float 1 0)))) | |
270 | (math-inc-gamma-series a x) | |
271 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
272 | (math-inc-gamma-cfrac a x))))) | |
273 | ) | |
274 | (setq math-current-gamma-value nil) | |
275 | ||
276 | (defun calcFunc-gammaG (a x) | |
277 | (if (equal x '(var inf var-inf)) | |
278 | '(float 0 0) | |
279 | (math-inexact-result) | |
280 | (or (Math-numberp a) (math-reject-arg a 'numberp)) | |
281 | (or (Math-numberp x) (math-reject-arg x 'numberp)) | |
282 | (math-with-extra-prec 2 | |
283 | (setq a (math-float a)) | |
284 | (setq x (math-float x)) | |
285 | (if (or (math-negp (calcFunc-re a)) | |
286 | (math-lessp-float (calcFunc-re x) | |
287 | (math-add-float (math-abs-approx a) | |
288 | '(float 1 0)))) | |
289 | (math-sub (or math-current-gamma-value (calcFunc-gamma a)) | |
290 | (math-inc-gamma-series a x)) | |
291 | (math-inc-gamma-cfrac a x)))) | |
292 | ) | |
293 | ||
294 | (defun math-inc-gamma-series (a x) | |
295 | (if (Math-zerop x) | |
296 | '(float 0 0) | |
297 | (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) | |
298 | (math-with-extra-prec 2 | |
299 | (let ((start (math-div '(float 1 0) a))) | |
300 | (math-inc-gamma-series-step start start a x))))) | |
301 | ) | |
302 | ||
303 | (defun math-inc-gamma-series-step (sum term a x) | |
304 | (math-working "gamma" sum) | |
305 | (setq a (math-add a '(float 1 0)) | |
306 | term (math-div (math-mul term x) a)) | |
307 | (let ((next (math-add sum term))) | |
308 | (if (math-nearly-equal sum next) | |
309 | next | |
310 | (math-inc-gamma-series-step next term a x))) | |
311 | ) | |
312 | ||
313 | (defun math-inc-gamma-cfrac (a x) | |
314 | (if (Math-zerop x) | |
315 | (or math-current-gamma-value (calcFunc-gamma a)) | |
316 | (math-mul (math-exp-raw (math-sub (math-mul a (math-ln-raw x)) x)) | |
317 | (math-inc-gamma-cfrac-step '(float 1 0) x | |
318 | '(float 0 0) '(float 1 0) | |
319 | '(float 1 0) '(float 1 0) '(float 0 0) | |
320 | a x))) | |
321 | ) | |
322 | ||
323 | (defun math-inc-gamma-cfrac-step (a0 a1 b0 b1 n fac g a x) | |
324 | (let ((ana (math-sub n a)) | |
325 | (anf (math-mul n fac))) | |
326 | (setq n (math-add n '(float 1 0)) | |
327 | a0 (math-mul (math-add a1 (math-mul a0 ana)) fac) | |
328 | b0 (math-mul (math-add b1 (math-mul b0 ana)) fac) | |
329 | a1 (math-add (math-mul x a0) (math-mul anf a1)) | |
330 | b1 (math-add (math-mul x b0) (math-mul anf b1))) | |
331 | (if (math-zerop a1) | |
332 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac g a x) | |
333 | (setq fac (math-div '(float 1 0) a1)) | |
334 | (let ((next (math-mul b1 fac))) | |
335 | (math-working "gamma" next) | |
336 | (if (math-nearly-equal next g) | |
337 | next | |
338 | (math-inc-gamma-cfrac-step a0 a1 b0 b1 n fac next a x))))) | |
339 | ) | |
340 | ||
341 | ||
342 | ;;; Error function. | |
343 | ||
344 | (defun calcFunc-erf (x) | |
345 | (if (equal x '(var inf var-inf)) | |
346 | '(float 1 0) | |
347 | (if (equal x '(neg (var inf var-inf))) | |
348 | '(float -1 0) | |
349 | (if (Math-zerop x) | |
350 | x | |
351 | (let ((math-current-gamma-value (math-sqrt-pi))) | |
352 | (math-to-same-complex-quad | |
353 | (math-div (calcFunc-gammag '(float 5 -1) | |
354 | (math-sqr (math-to-complex-quad-one x))) | |
355 | math-current-gamma-value) | |
356 | x))))) | |
357 | ) | |
358 | ||
359 | (defun calcFunc-erfc (x) | |
360 | (if (equal x '(var inf var-inf)) | |
361 | '(float 0 0) | |
362 | (if (math-posp x) | |
363 | (let ((math-current-gamma-value (math-sqrt-pi))) | |
364 | (math-div (calcFunc-gammaG '(float 5 -1) (math-sqr x)) | |
365 | math-current-gamma-value)) | |
366 | (math-sub 1 (calcFunc-erf x)))) | |
367 | ) | |
368 | ||
369 | (defun math-to-complex-quad-one (x) | |
370 | (if (eq (car-safe x) 'polar) (setq x (math-complex x))) | |
371 | (if (eq (car-safe x) 'cplx) | |
372 | (list 'cplx (math-abs (nth 1 x)) (math-abs (nth 2 x))) | |
373 | x) | |
374 | ) | |
375 | ||
376 | (defun math-to-same-complex-quad (x y) | |
377 | (if (eq (car-safe y) 'cplx) | |
378 | (if (eq (car-safe x) 'cplx) | |
379 | (list 'cplx | |
380 | (if (math-negp (nth 1 y)) (math-neg (nth 1 x)) (nth 1 x)) | |
381 | (if (math-negp (nth 2 y)) (math-neg (nth 2 x)) (nth 2 x))) | |
382 | (if (math-negp (nth 1 y)) (math-neg x) x)) | |
383 | (if (math-negp y) | |
384 | (if (eq (car-safe x) 'cplx) | |
385 | (list 'cplx (math-neg (nth 1 x)) (nth 2 x)) | |
386 | (math-neg x)) | |
387 | x)) | |
388 | ) | |
389 | ||
390 | ||
391 | ;;; Beta function. | |
392 | ||
393 | (defun calcFunc-beta (a b) | |
394 | (if (math-num-integerp a) | |
395 | (let ((am (math-add a -1))) | |
396 | (or (math-numberp b) (math-reject-arg b 'numberp)) | |
397 | (math-div 1 (math-mul b (calcFunc-choose (math-add b am) am)))) | |
398 | (if (math-num-integerp b) | |
399 | (calcFunc-beta b a) | |
400 | (math-div (math-mul (calcFunc-gamma a) (calcFunc-gamma b)) | |
401 | (calcFunc-gamma (math-add a b))))) | |
402 | ) | |
403 | ||
404 | ||
405 | ;;; Incomplete beta function. | |
406 | ||
407 | (defun calcFunc-betaI (x a b) | |
408 | (cond ((math-zerop x) | |
409 | '(float 0 0)) | |
410 | ((math-equal-int x 1) | |
411 | '(float 1 0)) | |
412 | ((or (math-zerop a) | |
413 | (and (math-num-integerp a) | |
414 | (math-negp a))) | |
415 | (if (or (math-zerop b) | |
416 | (and (math-num-integerp b) | |
417 | (math-negp b))) | |
418 | (math-reject-arg b 'range) | |
419 | '(float 1 0))) | |
420 | ((or (math-zerop b) | |
421 | (and (math-num-integerp b) | |
422 | (math-negp b))) | |
423 | '(float 0 0)) | |
424 | ((not (math-numberp a)) (math-reject-arg a 'numberp)) | |
425 | ((not (math-numberp b)) (math-reject-arg b 'numberp)) | |
426 | ((math-inexact-result)) | |
427 | (t (let ((math-current-beta-value (calcFunc-beta a b))) | |
428 | (math-div (calcFunc-betaB x a b) math-current-beta-value)))) | |
429 | ) | |
430 | ||
431 | (defun calcFunc-betaB (x a b) | |
432 | (cond | |
433 | ((math-zerop x) | |
434 | '(float 0 0)) | |
435 | ((math-equal-int x 1) | |
436 | (calcFunc-beta a b)) | |
437 | ((not (math-numberp x)) (math-reject-arg x 'numberp)) | |
438 | ((not (math-numberp a)) (math-reject-arg a 'numberp)) | |
439 | ((not (math-numberp b)) (math-reject-arg b 'numberp)) | |
440 | ((math-zerop a) (math-reject-arg a 'nonzerop)) | |
441 | ((math-zerop b) (math-reject-arg b 'nonzerop)) | |
442 | ((and (math-num-integerp b) | |
443 | (if (math-negp b) | |
444 | (math-reject-arg b 'range) | |
445 | (Math-natnum-lessp (setq b (math-trunc b)) 20))) | |
446 | (and calc-symbolic-mode (or (math-floatp a) (math-floatp b)) | |
447 | (math-inexact-result)) | |
448 | (math-mul | |
449 | (math-with-extra-prec 2 | |
450 | (let* ((i 0) | |
451 | (term 1) | |
452 | (sum (math-div term a))) | |
453 | (while (< (setq i (1+ i)) b) | |
454 | (setq term (math-mul (math-div (math-mul term (- i b)) i) x) | |
455 | sum (math-add sum (math-div term (math-add a i)))) | |
456 | (math-working "beta" sum)) | |
457 | sum)) | |
458 | (math-pow x a))) | |
459 | ((and (math-num-integerp a) | |
460 | (if (math-negp a) | |
461 | (math-reject-arg a 'range) | |
462 | (Math-natnum-lessp (setq a (math-trunc a)) 20))) | |
463 | (math-sub (or math-current-beta-value (calcFunc-beta a b)) | |
464 | (calcFunc-betaB (math-sub 1 x) b a))) | |
465 | (t | |
466 | (math-inexact-result) | |
467 | (math-with-extra-prec 2 | |
468 | (setq x (math-float x)) | |
469 | (setq a (math-float a)) | |
470 | (setq b (math-float b)) | |
471 | (let ((bt (math-exp-raw (math-add (math-mul a (math-ln-raw x)) | |
472 | (math-mul b (math-ln-raw | |
473 | (math-sub '(float 1 0) | |
474 | x))))))) | |
475 | (if (Math-lessp x (math-div (math-add a '(float 1 0)) | |
476 | (math-add (math-add a b) '(float 2 0)))) | |
477 | (math-div (math-mul bt (math-beta-cfrac a b x)) a) | |
478 | (math-sub (or math-current-beta-value (calcFunc-beta a b)) | |
479 | (math-div (math-mul bt | |
480 | (math-beta-cfrac b a (math-sub 1 x))) | |
481 | b))))))) | |
482 | ) | |
483 | (setq math-current-beta-value nil) | |
484 | ||
485 | (defun math-beta-cfrac (a b x) | |
486 | (let ((qab (math-add a b)) | |
487 | (qap (math-add a '(float 1 0))) | |
488 | (qam (math-add a '(float -1 0)))) | |
489 | (math-beta-cfrac-step '(float 1 0) | |
490 | (math-sub '(float 1 0) | |
491 | (math-div (math-mul qab x) qap)) | |
492 | '(float 1 0) '(float 1 0) | |
493 | '(float 1 0) | |
494 | qab qap qam a b x)) | |
495 | ) | |
496 | ||
497 | (defun math-beta-cfrac-step (az bz am bm m qab qap qam a b x) | |
498 | (let* ((two-m (math-mul m '(float 2 0))) | |
499 | (d (math-div (math-mul (math-mul (math-sub b m) m) x) | |
500 | (math-mul (math-add qam two-m) (math-add a two-m)))) | |
501 | (ap (math-add az (math-mul d am))) | |
502 | (bp (math-add bz (math-mul d bm))) | |
503 | (d2 (math-neg | |
504 | (math-div (math-mul (math-mul (math-add a m) (math-add qab m)) x) | |
505 | (math-mul (math-add qap two-m) (math-add a two-m))))) | |
506 | (app (math-add ap (math-mul d2 az))) | |
507 | (bpp (math-add bp (math-mul d2 bz))) | |
508 | (next (math-div app bpp))) | |
509 | (math-working "beta" next) | |
510 | (if (math-nearly-equal next az) | |
511 | next | |
512 | (math-beta-cfrac-step next '(float 1 0) | |
513 | (math-div ap bpp) (math-div bp bpp) | |
514 | (math-add m '(float 1 0)) | |
515 | qab qap qam a b x))) | |
516 | ) | |
517 | ||
518 | ||
519 | ;;; Bessel functions. | |
520 | ||
521 | ;;; Should generalize this to handle arbitrary precision! | |
522 | ||
523 | (defun calcFunc-besJ (v x) | |
524 | (or (math-numberp v) (math-reject-arg v 'numberp)) | |
525 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
526 | (let ((calc-internal-prec (min 8 calc-internal-prec))) | |
527 | (math-with-extra-prec 3 | |
528 | (setq x (math-float (math-normalize x))) | |
529 | (setq v (math-float (math-normalize v))) | |
530 | (cond ((math-zerop x) | |
531 | (if (math-zerop v) | |
532 | '(float 1 0) | |
533 | '(float 0 0))) | |
534 | ((math-inexact-result)) | |
535 | ((not (math-num-integerp v)) | |
536 | (let ((start (math-div 1 (calcFunc-fact v)))) | |
537 | (math-mul (math-besJ-series start start | |
538 | 0 | |
539 | (math-mul '(float -25 -2) | |
540 | (math-sqr x)) | |
541 | v) | |
542 | (math-pow (math-div x 2) v)))) | |
543 | ((math-negp (setq v (math-trunc v))) | |
544 | (if (math-oddp v) | |
545 | (math-neg (calcFunc-besJ (math-neg v) x)) | |
546 | (calcFunc-besJ (math-neg v) x))) | |
547 | ((eq v 0) | |
548 | (math-besJ0 x)) | |
549 | ((eq v 1) | |
550 | (math-besJ1 x)) | |
551 | ((Math-lessp v (math-abs-approx x)) | |
552 | (let ((j 0) | |
553 | (bjm (math-besJ0 x)) | |
554 | (bj (math-besJ1 x)) | |
555 | (two-over-x (math-div 2 x)) | |
556 | bjp) | |
557 | (while (< (setq j (1+ j)) v) | |
558 | (setq bjp (math-sub (math-mul (math-mul j two-over-x) bj) | |
559 | bjm) | |
560 | bjm bj | |
561 | bj bjp)) | |
562 | bj)) | |
563 | (t | |
564 | (if (Math-lessp 100 v) (math-reject-arg v 'range)) | |
565 | (let* ((j (logior (+ v (math-isqrt-small (* 40 v))) 1)) | |
566 | (two-over-x (math-div 2 x)) | |
567 | (jsum nil) | |
568 | (bjp '(float 0 0)) | |
569 | (sum '(float 0 0)) | |
570 | (bj '(float 1 0)) | |
571 | bjm ans) | |
572 | (while (> (setq j (1- j)) 0) | |
573 | (setq bjm (math-sub (math-mul (math-mul j two-over-x) bj) | |
574 | bjp) | |
575 | bjp bj | |
576 | bj bjm) | |
577 | (if (> (nth 2 (math-abs-approx bj)) 10) | |
578 | (setq bj (math-mul bj '(float 1 -10)) | |
579 | bjp (math-mul bjp '(float 1 -10)) | |
580 | ans (and ans (math-mul ans '(float 1 -10))) | |
581 | sum (math-mul sum '(float 1 -10)))) | |
582 | (or (setq jsum (not jsum)) | |
583 | (setq sum (math-add sum bj))) | |
584 | (if (= j v) | |
585 | (setq ans bjp))) | |
586 | (math-div ans (math-sub (math-mul 2 sum) bj))))))) | |
587 | ) | |
588 | ||
589 | (defun math-besJ-series (sum term k zz vk) | |
590 | (math-working "besJ" sum) | |
591 | (setq k (1+ k) | |
592 | vk (math-add 1 vk) | |
593 | term (math-div (math-mul term zz) (math-mul k vk))) | |
594 | (let ((next (math-add sum term))) | |
595 | (if (math-nearly-equal next sum) | |
596 | next | |
597 | (math-besJ-series next term k zz vk))) | |
598 | ) | |
599 | ||
600 | (defun math-besJ0 (x &optional yflag) | |
601 | (cond ((and (not yflag) (math-negp (calcFunc-re x))) | |
602 | (math-besJ0 (math-neg x))) | |
603 | ((Math-lessp '(float 8 0) (math-abs-approx x)) | |
604 | (let* ((z (math-div '(float 8 0) x)) | |
605 | (y (math-sqr z)) | |
606 | (xx (math-add x '(float (bigneg 164 398 785) -9))) | |
607 | (a1 (math-poly-eval y | |
608 | '((float (bigpos 211 887 093 2) -16) | |
609 | (float (bigneg 639 370 073 2) -15) | |
610 | (float (bigpos 407 510 734 2) -14) | |
611 | (float (bigneg 627 628 098 1) -12) | |
612 | (float 1 0)))) | |
613 | (a2 (math-poly-eval y | |
614 | '((float (bigneg 152 935 934) -16) | |
615 | (float (bigpos 161 095 621 7) -16) | |
616 | (float (bigneg 651 147 911 6) -15) | |
617 | (float (bigpos 765 488 430 1) -13) | |
618 | (float (bigneg 995 499 562 1) -11)))) | |
619 | (sc (math-sin-cos-raw xx))) | |
620 | (if yflag | |
621 | (setq sc (cons (math-neg (cdr sc)) (car sc)))) | |
622 | (math-mul (math-sqrt | |
623 | (math-div '(float (bigpos 722 619 636) -9) x)) | |
624 | (math-sub (math-mul (cdr sc) a1) | |
625 | (math-mul (car sc) (math-mul z a2)))))) | |
626 | (t | |
627 | (let ((y (math-sqr x))) | |
628 | (math-div (math-poly-eval y | |
629 | '((float (bigneg 456 052 849 1) -7) | |
630 | (float (bigpos 017 233 739 7) -5) | |
631 | (float (bigneg 418 442 121 1) -2) | |
632 | (float (bigpos 407 196 516 6) -1) | |
633 | (float (bigneg 354 590 362 13) 0) | |
634 | (float (bigpos 574 490 568 57) 0))) | |
635 | (math-poly-eval y | |
636 | '((float 1 0) | |
637 | (float (bigpos 712 532 678 2) -7) | |
638 | (float (bigpos 853 264 927 5) -5) | |
639 | (float (bigpos 718 680 494 9) -3) | |
640 | (float (bigpos 985 532 029 1) 0) | |
641 | (float (bigpos 411 490 568 57) 0))))))) | |
642 | ) | |
643 | ||
644 | (defun math-besJ1 (x &optional yflag) | |
645 | (cond ((and (math-negp (calcFunc-re x)) (not yflag)) | |
646 | (math-neg (math-besJ1 (math-neg x)))) | |
647 | ((Math-lessp '(float 8 0) (math-abs-approx x)) | |
648 | (let* ((z (math-div '(float 8 0) x)) | |
649 | (y (math-sqr z)) | |
650 | (xx (math-add x '(float (bigneg 491 194 356 2) -9))) | |
651 | (a1 (math-poly-eval y | |
652 | '((float (bigneg 019 337 240) -15) | |
653 | (float (bigpos 174 520 457 2) -15) | |
654 | (float (bigneg 496 396 516 3) -14) | |
655 | (float 183105 -8) | |
656 | (float 1 0)))) | |
657 | (a2 (math-poly-eval y | |
658 | '((float (bigpos 412 787 105) -15) | |
659 | (float (bigneg 987 228 88) -14) | |
660 | (float (bigpos 096 199 449 8) -15) | |
661 | (float (bigneg 873 690 002 2) -13) | |
662 | (float (bigpos 995 499 687 4) -11)))) | |
663 | (sc (math-sin-cos-raw xx))) | |
664 | (if yflag | |
665 | (setq sc (cons (math-neg (cdr sc)) (car sc))) | |
666 | (if (math-negp x) | |
667 | (setq sc (cons (math-neg (car sc)) (math-neg (cdr sc)))))) | |
668 | (math-mul (math-sqrt (math-div '(float (bigpos 722 619 636) -9) x)) | |
669 | (math-sub (math-mul (cdr sc) a1) | |
670 | (math-mul (car sc) (math-mul z a2)))))) | |
671 | (t | |
672 | (let ((y (math-sqr x))) | |
673 | (math-mul | |
674 | x | |
675 | (math-div (math-poly-eval y | |
676 | '((float (bigneg 606 036 016 3) -8) | |
677 | (float (bigpos 826 044 157) -4) | |
678 | (float (bigneg 439 611 972 2) -3) | |
679 | (float (bigpos 531 968 423 2) -1) | |
680 | (float (bigneg 235 059 895 7) 0) | |
681 | (float (bigpos 232 614 362 72) 0))) | |
682 | (math-poly-eval y | |
683 | '((float 1 0) | |
684 | (float (bigpos 397 991 769 3) -7) | |
685 | (float (bigpos 394 743 944 9) -5) | |
686 | (float (bigpos 474 330 858 1) -2) | |
687 | (float (bigpos 178 535 300 2) 0) | |
688 | (float (bigpos 442 228 725 144) | |
689 | 0)))))))) | |
690 | ) | |
691 | ||
692 | (defun calcFunc-besY (v x) | |
693 | (math-inexact-result) | |
694 | (or (math-numberp v) (math-reject-arg v 'numberp)) | |
695 | (or (math-numberp x) (math-reject-arg x 'numberp)) | |
696 | (let ((calc-internal-prec (min 8 calc-internal-prec))) | |
697 | (math-with-extra-prec 3 | |
698 | (setq x (math-float (math-normalize x))) | |
699 | (setq v (math-float (math-normalize v))) | |
700 | (cond ((not (math-num-integerp v)) | |
701 | (let ((sc (math-sin-cos-raw (math-mul v (math-pi))))) | |
702 | (math-div (math-sub (math-mul (calcFunc-besJ v x) (cdr sc)) | |
703 | (calcFunc-besJ (math-neg v) x)) | |
704 | (car sc)))) | |
705 | ((math-negp (setq v (math-trunc v))) | |
706 | (if (math-oddp v) | |
707 | (math-neg (calcFunc-besY (math-neg v) x)) | |
708 | (calcFunc-besY (math-neg v) x))) | |
709 | ((eq v 0) | |
710 | (math-besY0 x)) | |
711 | ((eq v 1) | |
712 | (math-besY1 x)) | |
713 | (t | |
714 | (let ((j 0) | |
715 | (bym (math-besY0 x)) | |
716 | (by (math-besY1 x)) | |
717 | (two-over-x (math-div 2 x)) | |
718 | byp) | |
719 | (while (< (setq j (1+ j)) v) | |
720 | (setq byp (math-sub (math-mul (math-mul j two-over-x) by) | |
721 | bym) | |
722 | bym by | |
723 | by byp)) | |
724 | by))))) | |
725 | ) | |
726 | ||
727 | (defun math-besY0 (x) | |
728 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
729 | (let ((y (math-sqr x))) | |
730 | (math-add | |
731 | (math-div (math-poly-eval y | |
732 | '((float (bigpos 733 622 284 2) -7) | |
733 | (float (bigneg 757 792 632 8) -5) | |
734 | (float (bigpos 129 988 087 1) -2) | |
735 | (float (bigneg 036 598 123 5) -1) | |
736 | (float (bigpos 065 834 062 7) 0) | |
737 | (float (bigneg 389 821 957 2) 0))) | |
738 | (math-poly-eval y | |
739 | '((float 1 0) | |
740 | (float (bigpos 244 030 261 2) -7) | |
741 | (float (bigpos 647 472 474) -4) | |
742 | (float (bigpos 438 466 189 7) -3) | |
743 | (float (bigpos 648 499 452 7) -1) | |
744 | (float (bigpos 269 544 076 40) 0)))) | |
745 | (math-mul '(float (bigpos 772 619 636) -9) | |
746 | (math-mul (math-besJ0 x) (math-ln-raw x)))))) | |
747 | ((math-negp (calcFunc-re x)) | |
748 | (math-add (math-besJ0 (math-neg x) t) | |
749 | (math-mul '(cplx 0 2) | |
750 | (math-besJ0 (math-neg x))))) | |
751 | (t | |
752 | (math-besJ0 x t))) | |
753 | ) | |
754 | ||
755 | (defun math-besY1 (x) | |
756 | (cond ((Math-lessp (math-abs-approx x) '(float 8 0)) | |
757 | (let ((y (math-sqr x))) | |
758 | (math-add | |
759 | (math-mul | |
760 | x | |
761 | (math-div (math-poly-eval y | |
762 | '((float (bigpos 935 937 511 8) -6) | |
763 | (float (bigneg 726 922 237 4) -3) | |
764 | (float (bigpos 551 264 349 7) -1) | |
765 | (float (bigneg 139 438 153 5) 1) | |
766 | (float (bigpos 439 527 127) 4) | |
767 | (float (bigneg 943 604 900 4) 3))) | |
768 | (math-poly-eval y | |
769 | '((float 1 0) | |
770 | (float (bigpos 885 632 549 3) -7) | |
771 | (float (bigpos 605 042 102) -3) | |
772 | (float (bigpos 002 904 245 2) -2) | |
773 | (float (bigpos 367 650 733 3) 0) | |
774 | (float (bigpos 664 419 244 4) 2) | |
775 | (float (bigpos 057 958 249) 5))))) | |
776 | (math-mul '(float (bigpos 772 619 636) -9) | |
777 | (math-sub (math-mul (math-besJ1 x) (math-ln-raw x)) | |
778 | (math-div 1 x)))))) | |
779 | ((math-negp (calcFunc-re x)) | |
780 | (math-neg | |
781 | (math-add (math-besJ1 (math-neg x) t) | |
782 | (math-mul '(cplx 0 2) | |
783 | (math-besJ1 (math-neg x)))))) | |
784 | (t | |
785 | (math-besJ1 x t))) | |
786 | ) | |
787 | ||
788 | (defun math-poly-eval (x coefs) | |
789 | (let ((accum (car coefs))) | |
790 | (while (setq coefs (cdr coefs)) | |
791 | (setq accum (math-add (car coefs) (math-mul accum x)))) | |
792 | accum) | |
793 | ) | |
794 | ||
795 | ||
796 | ;;;; Bernoulli and Euler polynomials and numbers. | |
797 | ||
798 | (defun calcFunc-bern (n &optional x) | |
799 | (if (and x (not (math-zerop x))) | |
800 | (if (and calc-symbolic-mode (math-floatp x)) | |
801 | (math-inexact-result) | |
802 | (math-build-polynomial-expr (math-bernoulli-coefs n) x)) | |
803 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
804 | (if (consp n) | |
805 | (progn | |
806 | (math-inexact-result) | |
807 | (math-float (math-bernoulli-number (math-trunc n)))) | |
808 | (math-bernoulli-number n))) | |
809 | ) | |
810 | ||
811 | (defun calcFunc-euler (n &optional x) | |
812 | (or (math-num-natnump n) (math-reject-arg n 'natnump)) | |
813 | (if x | |
814 | (let* ((n1 (math-add n 1)) | |
815 | (coefs (math-bernoulli-coefs n1)) | |
816 | (fac (math-div (math-pow 2 n1) n1)) | |
817 | (k -1) | |
818 | (x1 (math-div (math-add x 1) 2)) | |
819 | (x2 (math-div x 2))) | |
820 | (if (math-numberp x) | |
821 | (if (and calc-symbolic-mode (math-floatp x)) | |
822 | (math-inexact-result) | |
823 | (math-mul fac | |
824 | (math-sub (math-build-polynomial-expr coefs x1) | |
825 | (math-build-polynomial-expr coefs x2)))) | |
826 | (calcFunc-collect | |
827 | (math-reduce-vec | |
828 | 'math-add | |
829 | (cons 'vec | |
830 | (mapcar (function | |
831 | (lambda (c) | |
832 | (setq k (1+ k)) | |
833 | (math-mul (math-mul fac c) | |
834 | (math-sub (math-pow x1 k) | |
835 | (math-pow x2 k))))) | |
836 | coefs))) | |
837 | x))) | |
838 | (math-mul (math-pow 2 n) | |
839 | (if (consp n) | |
840 | (progn | |
841 | (math-inexact-result) | |
842 | (calcFunc-euler n '(float 5 -1))) | |
843 | (calcFunc-euler n '(frac 1 2))))) | |
844 | ) | |
845 | ||
846 | (defun math-bernoulli-coefs (n) | |
847 | (let* ((coefs (list (calcFunc-bern n))) | |
848 | (nn (math-trunc n)) | |
849 | (k nn) | |
850 | (term nn) | |
851 | coef | |
852 | (calc-prefer-frac (or (integerp n) calc-prefer-frac))) | |
853 | (while (>= (setq k (1- k)) 0) | |
854 | (setq term (math-div term (- nn k)) | |
855 | coef (math-mul term (math-bernoulli-number k)) | |
856 | coefs (cons (if (consp n) (math-float coef) coef) coefs) | |
857 | term (math-mul term k))) | |
858 | (nreverse coefs)) | |
859 | ) | |
860 | ||
861 | (defun math-bernoulli-number (n) | |
862 | (if (= (% n 2) 1) | |
863 | (if (= n 1) | |
864 | '(frac -1 2) | |
865 | 0) | |
866 | (setq n (/ n 2)) | |
867 | (while (>= n math-bernoulli-cache-size) | |
868 | (let* ((sum 0) | |
869 | (nk 1) ; nk = n-k+1 | |
870 | (fact 1) ; fact = (n-k+1)! | |
871 | ofact | |
872 | (p math-bernoulli-b-cache) | |
873 | (calc-prefer-frac t)) | |
874 | (math-working "bernoulli B" (* 2 math-bernoulli-cache-size)) | |
875 | (while p | |
876 | (setq nk (+ nk 2) | |
877 | ofact fact | |
878 | fact (math-mul fact (* nk (1- nk))) | |
879 | sum (math-add sum (math-div (car p) fact)) | |
880 | p (cdr p))) | |
881 | (setq ofact (math-mul ofact (1- nk)) | |
882 | sum (math-sub (math-div '(frac 1 2) ofact) sum) | |
883 | math-bernoulli-b-cache (cons sum math-bernoulli-b-cache) | |
884 | math-bernoulli-B-cache (cons (math-mul sum ofact) | |
885 | math-bernoulli-B-cache) | |
886 | math-bernoulli-cache-size (1+ math-bernoulli-cache-size)))) | |
887 | (nth (- math-bernoulli-cache-size n 1) math-bernoulli-B-cache)) | |
888 | ) | |
889 | ||
890 | ;;; Bn = n! bn | |
891 | ;;; bn = - sum_k=0^n-1 bk / (n-k+1)! | |
892 | ||
893 | ;;; A faster method would be to use "tangent numbers", c.f., Concrete | |
894 | ;;; Mathematics pg. 273. | |
895 | ||
896 | (setq math-bernoulli-b-cache '( (frac -174611 | |
897 | (bigpos 0 200 291 698 662 857 802)) | |
898 | (frac 43867 (bigpos 0 944 170 217 94 109 5)) | |
899 | (frac -3617 (bigpos 0 880 842 622 670 10)) | |
900 | (frac 1 (bigpos 600 249 724 74)) | |
901 | (frac -691 (bigpos 0 368 674 307 1)) | |
902 | (frac 1 (bigpos 160 900 47)) | |
903 | (frac -1 (bigpos 600 209 1)) | |
904 | (frac 1 30240) (frac -1 720) | |
905 | (frac 1 12) 1 )) | |
906 | ||
907 | (setq math-bernoulli-B-cache '( (frac -174611 330) (frac 43867 798) | |
908 | (frac -3617 510) (frac 7 6) (frac -691 2730) | |
909 | (frac 5 66) (frac -1 30) (frac 1 42) | |
910 | (frac -1 30) (frac 1 6) 1 )) | |
911 | ||
912 | (setq math-bernoulli-cache-size 11) | |
913 | ||
914 | ||
915 | ||
916 | ;;; Probability distributions. | |
917 | ||
918 | ;;; Binomial. | |
919 | (defun calcFunc-utpb (x n p) | |
920 | (if math-expand-formulas | |
921 | (math-normalize (list 'calcFunc-betaI p x (list '+ (list '- n x) 1))) | |
922 | (calcFunc-betaI p x (math-add (math-sub n x) 1))) | |
923 | ) | |
924 | (put 'calcFunc-utpb 'math-expandable t) | |
925 | ||
926 | (defun calcFunc-ltpb (x n p) | |
927 | (math-sub 1 (calcFunc-utpb x n p)) | |
928 | ) | |
929 | (put 'calcFunc-ltpb 'math-expandable t) | |
930 | ||
931 | ;;; Chi-square. | |
932 | (defun calcFunc-utpc (chisq v) | |
933 | (if math-expand-formulas | |
934 | (math-normalize (list 'calcFunc-gammaQ (list '/ v 2) (list '/ chisq 2))) | |
935 | (calcFunc-gammaQ (math-div v 2) (math-div chisq 2))) | |
936 | ) | |
937 | (put 'calcFunc-utpc 'math-expandable t) | |
938 | ||
939 | (defun calcFunc-ltpc (chisq v) | |
940 | (if math-expand-formulas | |
941 | (math-normalize (list 'calcFunc-gammaP (list '/ v 2) (list '/ chisq 2))) | |
942 | (calcFunc-gammaP (math-div v 2) (math-div chisq 2))) | |
943 | ) | |
944 | (put 'calcFunc-ltpc 'math-expandable t) | |
945 | ||
946 | ;;; F-distribution. | |
947 | (defun calcFunc-utpf (f v1 v2) | |
948 | (if math-expand-formulas | |
949 | (math-normalize (list 'calcFunc-betaI | |
950 | (list '/ v2 (list '+ v2 (list '* v1 f))) | |
951 | (list '/ v2 2) | |
952 | (list '/ v1 2))) | |
953 | (calcFunc-betaI (math-div v2 (math-add v2 (math-mul v1 f))) | |
954 | (math-div v2 2) | |
955 | (math-div v1 2))) | |
956 | ) | |
957 | (put 'calcFunc-utpf 'math-expandable t) | |
958 | ||
959 | (defun calcFunc-ltpf (f v1 v2) | |
960 | (math-sub 1 (calcFunc-utpf f v1 v2)) | |
961 | ) | |
962 | (put 'calcFunc-ltpf 'math-expandable t) | |
963 | ||
964 | ;;; Normal. | |
965 | (defun calcFunc-utpn (x mean sdev) | |
966 | (if math-expand-formulas | |
967 | (math-normalize | |
968 | (list '/ | |
969 | (list '+ 1 | |
970 | (list 'calcFunc-erf | |
971 | (list '/ (list '- mean x) | |
972 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
973 | 2)) | |
974 | (math-mul (math-add '(float 1 0) | |
975 | (calcFunc-erf | |
976 | (math-div (math-sub mean x) | |
977 | (math-mul sdev (math-sqrt-2))))) | |
978 | '(float 5 -1))) | |
979 | ) | |
980 | (put 'calcFunc-utpn 'math-expandable t) | |
981 | ||
982 | (defun calcFunc-ltpn (x mean sdev) | |
983 | (if math-expand-formulas | |
984 | (math-normalize | |
985 | (list '/ | |
986 | (list '+ 1 | |
987 | (list 'calcFunc-erf | |
988 | (list '/ (list '- x mean) | |
989 | (list '* sdev (list 'calcFunc-sqrt 2))))) | |
990 | 2)) | |
991 | (math-mul (math-add '(float 1 0) | |
992 | (calcFunc-erf | |
993 | (math-div (math-sub x mean) | |
994 | (math-mul sdev (math-sqrt-2))))) | |
995 | '(float 5 -1))) | |
996 | ) | |
997 | (put 'calcFunc-ltpn 'math-expandable t) | |
998 | ||
999 | ;;; Poisson. | |
1000 | (defun calcFunc-utpp (n x) | |
1001 | (if math-expand-formulas | |
1002 | (math-normalize (list 'calcFunc-gammaP x n)) | |
1003 | (calcFunc-gammaP x n)) | |
1004 | ) | |
1005 | (put 'calcFunc-utpp 'math-expandable t) | |
1006 | ||
1007 | (defun calcFunc-ltpp (n x) | |
1008 | (if math-expand-formulas | |
1009 | (math-normalize (list 'calcFunc-gammaQ x n)) | |
1010 | (calcFunc-gammaQ x n)) | |
1011 | ) | |
1012 | (put 'calcFunc-ltpp 'math-expandable t) | |
1013 | ||
1014 | ;;; Student's t. (As defined in Abramowitz & Stegun and Numerical Recipes.) | |
1015 | (defun calcFunc-utpt (tt v) | |
1016 | (if math-expand-formulas | |
1017 | (math-normalize (list 'calcFunc-betaI | |
1018 | (list '/ v (list '+ v (list '^ tt 2))) | |
1019 | (list '/ v 2) | |
1020 | '(float 5 -1))) | |
1021 | (calcFunc-betaI (math-div v (math-add v (math-sqr tt))) | |
1022 | (math-div v 2) | |
1023 | '(float 5 -1))) | |
1024 | ) | |
1025 | (put 'calcFunc-utpt 'math-expandable t) | |
1026 | ||
1027 | (defun calcFunc-ltpt (tt v) | |
1028 | (math-sub 1 (calcFunc-utpt tt v)) | |
1029 | ) | |
1030 | (put 'calcFunc-ltpt 'math-expandable t) | |
1031 | ||
1032 | ||
1033 | ||
1034 |