1;;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;; 2;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; 3;;; The data in this file contains enhancments. ;;;;; 4;;; ;;;;; 5;;; Copyright (c) 1984,1987 by William Schelter,University of Texas ;;;;; 6;;; All rights reserved ;;;;; 7;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; 8;;; (c) Copyright 1982 Massachusetts Institute of Technology ;;; 9;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; 10 11(in-package :maxima) 12 13(macsyma-module trigi) 14 15(load-macsyma-macros mrgmac) 16 17(declare-top (special errorsw $demoivre 1//2 -1//2)) 18 19(defmvar $%piargs t) 20(defmvar $%iargs t) 21(defmvar $triginverses t) 22(defmvar $trigexpand nil) 23(defmvar $trigexpandplus t) 24(defmvar $trigexpandtimes t) 25(defmvar $trigsign t) 26(defmvar $exponentialize nil) 27(defmvar $logarc nil) 28(defmvar $halfangles nil) 29 30;; Simplified shortcuts for constant expressions. 31(defvar %pi//4 '((mtimes simp) ((rat simp) 1 4.) $%pi)) 32(defvar %pi//2 '((mtimes simp) ((rat simp) 1 2) $%pi)) 33(defvar sqrt3//2 '((mtimes simp) 34 ((rat simp) 1 2) 35 ((mexpt simp) 3 ((rat simp) 1 2)))) 36(defvar -sqrt3//2 '((mtimes simp) 37 ((rat simp) -1 2) 38 ((mexpt simp) 3 ((rat simp) 1 2)))) 39 40;;; Arithmetic utilities. 41 42(defun sqrt1-x^2 (x) 43 (power (sub 1 (power x 2)) 1//2)) 44 45(defun sqrt1+x^2 (x) 46 (power (add 1 (power x 2)) 1//2)) 47 48(defun sqrtx^2-1 (x) 49 (power (add (power x 2) -1) 1//2)) 50 51(defun sq-sumsq (x y) 52 (power (add (power x 2) (power y 2)) 1//2)) 53 54(defun trigp (func) 55 (member func '(%sin %cos %tan %csc %sec %cot %sinh %cosh %tanh %csch %sech %coth) 56 :test #'eq)) 57 58(defun arcp (func) 59 (member func '(%asin %acos %atan %acsc %asec %acot %asinh %acosh %atanh %acsch %asech %acoth) 60 :test #'eq)) 61 62(defprop %sin simp-%sin operators) 63(defprop %cos simp-%cos operators) 64(defprop %tan simp-%tan operators) 65(defprop %cot simp-%cot operators) 66(defprop %csc simp-%csc operators) 67(defprop %sec simp-%sec operators) 68(defprop %sinh simp-%sinh operators) 69(defprop %cosh simp-%cosh operators) 70(defprop %tanh simp-%tanh operators) 71(defprop %coth simp-%coth operators) 72(defprop %csch simp-%csch operators) 73(defprop %sech simp-%sech operators) 74(defprop %asin simp-%asin operators) 75(defprop %acos simp-%acos operators) 76(defprop %atan simp-%atan operators) 77(defprop %acot simp-%acot operators) 78(defprop %acsc simp-%acsc operators) 79(defprop %asec simp-%asec operators) 80(defprop %asinh simp-%asinh operators) 81(defprop %acosh simp-%acosh operators) 82(defprop %atanh simp-%atanh operators) 83(defprop %acoth simp-%acoth operators) 84(defprop %acsch simp-%acsch operators) 85(defprop %asech simp-%asech operators) 86 87;;; The trigonometric functions distribute of lists, matrices and equations. 88 89(dolist (x '(%sin %cos %tan %cot %csc %sec 90 %sinh %cosh %tanh %coth %csch %sech 91 %asin %acos %atan %acot %acsc %asec 92 %asinh %acosh %atanh %acoth %acsch %asech)) 93 (setf (get x 'distribute_over) '(mlist $matrix mequal))) 94 95(defun domain-error (x f) 96 (merror (intl:gettext "~A: argument ~:M isn't in the domain of ~A.") f (complexify x) f)) 97 98;; Build hash tables '*flonum-op*' and '*big-float-op*' that map Maxima 99;; function names to their corresponding Lisp functions. 100 101(defvar *flonum-op* (make-hash-table :size 64) 102 "Hash table mapping a maxima function to a corresponding Lisp 103 function to evaluate the maxima function numerically with 104 flonum precision.") 105 106(defvar *big-float-op* (make-hash-table) 107 "Hash table mapping a maxima function to a corresponding Lisp 108 function to evaluate the maxima function numerically with 109 big-float precision.") 110 111;; Some Lisp implementations goof up branch cuts for ASIN, ACOS, and/or ATANH. 112;; Here are definitions which have the right branch cuts 113;; (assuming LOG, PHASE, and SQRT have the right branch cuts). 114;; Don't bother trying to sort out which implementations get it right or wrong; 115;; we'll make all implementations use these functions. 116 117;; Apply formula from CLHS if X falls on a branch cut. 118;; Otherwise punt to CL:ASIN. 119(defun maxima-branch-asin (x) 120 ;; Test for (IMAGPART X) is EQUAL because signed zero is EQUAL to zero. 121 (if (and (> (abs (realpart x)) 1.0) (equal (imagpart x) 0.0)) 122 ;; The formula from CLHS is asin(x) = -%i*log(%i*x+sqrt(1-x^2)). 123 ;; This has problems with overflow for large x. 124 ;; 125 ;; Let's rewrite it, where abs(x)>1 126 ;; 127 ;; asin(x) = -%i*log(%i*x+abs(x)*sqrt(1-1/x^2)) 128 ;; = -%i*log(%i*x*(1+abs(x)/x*sqrt(1-1/x^2))) 129 ;; = -%i*[log(abs(x)*abs(1+abs(x)/x*sqrt(1-1/x^2))) 130 ;; + %i*arg(%i*x*(1+abs(x)/x*sqrt(1-1/x^2)))] 131 ;; = -%i*[log(abs(x)*(1+abs(x)/x*sqrt(1-1/x^2))) 132 ;; + %i*%pi/2*sign(x)] 133 ;; = %pi/2*sign(x) - %i*[log(abs(x)*(1+abs(x)/x*sqrt(1-1/x^2))] 134 ;; 135 ;; Now, look at log part. If x > 0, we have 136 ;; 137 ;; log(x*(1+sqrt(1-1/x^2))) 138 ;; 139 ;; which is just fine. For x < 0, we have 140 ;; 141 ;; log(abs(x)*(1-sqrt(1-1/x^2))). 142 ;; 143 ;; But 144 ;; 1-sqrt(1-1/x^2) = (1-sqrt(1-1/x^2))*(1+sqrt(1-1/x^2))/(1+sqrt(1-1/x^2)) 145 ;; = (1-(1-1/x^2))/(1+sqrt(1-1/x^2)) 146 ;; = 1/x^2/(1+sqrt(1-1/x^2)) 147 ;; 148 ;; So 149 ;; 150 ;; log(abs(x)*(1-sqrt(1-1/x^2))) 151 ;; = log(abs(x)/x^2/(1+sqrt(1-1/x^2))) 152 ;; = -log(x^2/abs(x)*(1+sqrt(1-1/x^2)) 153 ;; = -log(abs(x)*(1+sqrt(1-1/x^2))) 154 ;; 155 ;; Thus, for x < 0, 156 ;; 157 ;; asin(x) = -%pi/2+%i*log(abs(x)*(1+sqrt(1-1/x^2))) 158 ;; = -asin(-x) 159 ;; 160 ;; If we had an accurate f(x) = log(1+x) function, we should 161 ;; probably evaluate log(1+sqrt(1-1/x^2)) via f(x) instead of 162 ;; log. One other accuracy change is to evaluate sqrt(1-1/x^2) 163 ;; as sqrt(1-1/x)*sqrt(1+1/x), because 1/x^2 won't underflow as 164 ;; soon as 1/x. 165 (let* ((absx (abs x)) 166 (recip (/ absx)) 167 (result (complex (/ #.(float pi) 2) 168 (- (log (* absx 169 (1+ (* (sqrt (+ 1 recip)) 170 (sqrt (- 1 recip)))))))))) 171 (if (minusp x) 172 (- result) 173 result)) 174 (cl:asin x))) 175 176;; Apply formula from CLHS if X falls on a branch cut. 177;; Otherwise punt to CL:ACOS. 178(defun maxima-branch-acos (x) 179 ; Test for (IMAGPART X) is EQUAL because signed zero is EQUAL to zero. 180 (if (and (> (abs (realpart x)) 1.0) (equal (imagpart x) 0.0)) 181 (- #.(/ (float pi) 2) (maxima-branch-asin x)) 182 (cl:acos x))) 183 184(defun maxima-branch-acot (x) 185 ;; Allow 0.0 in domain of acot, otherwise use atan(1/x) 186 (if (and (equal (realpart x) 0.0) (equal (imagpart x) 0.0)) 187 #.(/ (float pi) 2) 188 (cl:atan (/ 1 x)))) 189 190;; Apply formula from CLHS if X falls on a branch cut. 191;; Otherwise punt to CL:ATANH. 192(defun maxima-branch-atanh (x) 193 ; Test for (IMAGPART X) is EQUAL because signed zero is EQUAL to zero. 194 (if (and (> (abs (realpart x)) 1.0) (equal (imagpart x) 0.0)) 195 (/ (- (cl:log (+ 1 x)) (cl:log (- 1 x))) 2) 196 (cl:atanh x))) 197 198;; Fill the hash table. 199(macrolet ((frob (mfun dfun) `(setf (gethash ',mfun *flonum-op*) ,dfun))) 200 (frob mplus #'+) 201 (frob mtimes #'*) 202 (frob mquotient #'/) 203 (frob mminus #'-) 204 205 (frob %cos #'cl:cos) 206 (frob %sin #'cl:sin) 207 (frob %tan #'cl:tan) 208 209 (frob %sec #'(lambda (x) 210 (let ((y (ignore-errors (/ 1 (cl:cos x))))) 211 (if y y (domain-error x 'sec))))) 212 213 (frob %csc #'(lambda (x) 214 (let ((y (ignore-errors (/ 1 (cl:sin x))))) 215 (if y y (domain-error x 'csc))))) 216 217 (frob %cot #'(lambda (x) 218 (let ((y (ignore-errors (/ 1 (cl:tan x))))) 219 (if y y (domain-error x 'cot))))) 220 221 (frob %acos #'maxima-branch-acos) 222 (frob %asin #'maxima-branch-asin) 223 224 (frob %atan #'cl:atan) 225 226 (frob %asec #'(lambda (x) 227 (let ((y (ignore-errors (maxima-branch-acos (/ 1 x))))) 228 (if y y (domain-error x 'asec))))) 229 230 (frob %acsc #'(lambda (x) 231 (let ((y (ignore-errors (maxima-branch-asin (/ 1 x))))) 232 (if y y (domain-error x 'acsc))))) 233 234 (frob %acot #'(lambda (x) 235 (let ((y (ignore-errors (maxima-branch-acot x)))) 236 (if y y (domain-error x 'acot))))) 237 238 (frob %cosh #'cl:cosh) 239 (frob %sinh #'cl:sinh) 240 (frob %tanh #'cl:tanh) 241 242 (frob %sech #'(lambda (x) 243 (let ((y (ignore-errors (/ 1 (cl:cosh x))))) 244 (if y y (domain-error x 'sech))))) 245 246 (frob %csch #'(lambda (x) 247 (let ((y (ignore-errors (/ 1 (cl:sinh x))))) 248 (if y y (domain-error x 'csch))))) 249 250 (frob %coth #'(lambda (x) 251 (let ((y (ignore-errors (/ 1 (cl:tanh x))))) 252 (if y y (domain-error x 'coth))))) 253 254 (frob %acosh #'cl:acosh) 255 (frob %asinh #'cl:asinh) 256 257 (frob %atanh #'maxima-branch-atanh) 258 259 (frob %asech #'(lambda (x) 260 (let ((y (ignore-errors (cl:acosh (/ 1 x))))) 261 (if y y (domain-error x 'asech))))) 262 263 (frob %acsch #'(lambda (x) 264 (let ((y (ignore-errors (cl:asinh (/ 1 x))))) 265 (if y y (domain-error x 'acsch))))) 266 267 (frob %acoth #'(lambda (x) 268 (let ((y (ignore-errors (maxima-branch-atanh (/ 1 x))))) 269 (if y y (domain-error x 'acoth))))) 270 271 (frob mabs #'cl:abs) 272 (frob %exp #'cl:exp) 273 (frob mexpt #'cl:expt) 274 (frob %sqrt #'cl:sqrt) 275 (frob %log #'(lambda (x) 276 (let ((y (ignore-errors (cl:log x)))) 277 (if y y (domain-error x 'log))))) 278 279 (frob %plog #'(lambda (x) 280 (let ((y (ignore-errors (cl:log x)))) 281 (if y y (domain-error x 'log))))) 282 283 (frob $conjugate #'cl:conjugate) 284 (frob $floor #'cl:ffloor) 285 (frob $ceiling #'cl:fceiling) 286 (frob $realpart #'cl:realpart) 287 (frob $imagpart #'cl:imagpart) 288 (frob $max #'cl:max) 289 (frob $min #'cl:min) 290 (frob %signum #'cl:signum) 291 (frob $atan2 #'cl:atan) 292 (frob %log #'(lambda (x) 293 (let ((y (ignore-errors (cl:log x)))) 294 (if y y (domain-error x 'log))))) 295 (frob %sqrt #'cl:sqrt)) 296 297(macrolet ((frob (mfun dfun) `(setf (gethash ',mfun *big-float-op*) ,dfun))) 298 ;; All big-float implementation functions MUST support a required x 299 ;; arg and an optional y arg for the real and imaginary parts. The 300 ;; imaginary part does not have to be given. 301 (frob %asin #'big-float-asin) 302 (frob %sinh #'big-float-sinh) 303 (frob %asinh #'big-float-asinh) 304 (frob %tanh #'big-float-tanh) 305 (frob %atanh #'big-float-atanh) 306 (frob %acos 'big-float-acos) 307 (frob %log 'big-float-log) 308 (frob %sqrt 'big-float-sqrt)) 309 310;; Here is a general scheme for defining and applying reflection rules. A 311;; reflection rule is something like f(-x) --> f(x), or f(-x) --> %pi - f(x). 312 313;; We define functions for the two most common reflection rules; these 314;; are the odd function rule (f(-x) --> -f(x)) and the even function rule 315;; (f(-x) --> f(x)). A reflection rule takes two arguments (the operator and 316;; the operand). 317 318(defun odd-function-reflect (op x) 319 (neg (take (list op) (neg x)))) 320 321(defun even-function-reflect (op x) 322 (take (list op) (neg x))) 323 324;; Put the reflection rule on the property list of the exponential-like 325;; functions. 326 327(setf (get '%cos 'reflection-rule) #'even-function-reflect) 328(setf (get '%sin 'reflection-rule) #'odd-function-reflect) 329(setf (get '%tan 'reflection-rule) #'odd-function-reflect) 330(setf (get '%sec 'reflection-rule) #'even-function-reflect) 331(setf (get '%csc 'reflection-rule) #'odd-function-reflect) 332(setf (get '%cot 'reflection-rule) #'odd-function-reflect) 333 334;; See A&S 4.4.14--4.4.19 335 336(setf (get '%acos 'reflection-rule) #'(lambda (op x) (sub '$%pi (take (list op) (neg x))))) 337(setf (get '%asin 'reflection-rule) #'odd-function-reflect) 338(setf (get '%atan 'reflection-rule) #'odd-function-reflect) 339(setf (get '%asec 'reflection-rule) #'(lambda (op x) (sub '$%pi (take (list op) (neg x))))) 340(setf (get '%acsc 'reflection-rule) #'odd-function-reflect) 341(setf (get '%acot 'reflection-rule) #'odd-function-reflect) 342 343(setf (get '%cosh 'reflection-rule) #'even-function-reflect) 344(setf (get '%sinh 'reflection-rule) #'odd-function-reflect) 345(setf (get '%tanh 'reflection-rule) #'odd-function-reflect) 346(setf (get '%sech 'reflection-rule) #'even-function-reflect) 347(setf (get '%csch 'reflection-rule) #'odd-function-reflect) 348(setf (get '%coth 'reflection-rule) #'odd-function-reflect) 349 350(setf (get '%asinh 'reflection-rule) #'odd-function-reflect) 351(setf (get '%atanh 'reflection-rule) #'odd-function-reflect) 352(setf (get '%asech 'reflection-rule) #'even-function-reflect) 353(setf (get '%acsch 'reflection-rule) #'odd-function-reflect) 354(setf (get '%acoth 'reflection-rule) #'odd-function-reflect) 355 356;; When b is nil, do not apply the reflection rule. For trigonometric like 357;; functions, b is $trigsign. This function uses 'great' to decide when to 358;; apply the rule. Another possibility is to apply the rule when (mminusp* x) 359;; evaluates to true. Maxima <= 5.9.3 uses this scheme; with this method, we have 360;; assume(z < 0), cos(z) --> cos(-z). I (Barton Willis) think this goofy. 361 362;; The function 'great' is non-transitive. I don't think this bug will cause 363;; trouble for this function. If there is an expression such that both 364;; (great (neg x) x) and (great x (neg x)) evaluate to true, this function 365;; could cause an infinite loop. I could protect against this possibility with 366;; (and b f (great (neg x) x) (not (great x (neg x))). 367 368(defun apply-reflection-simp (op x &optional (b t)) 369 (let ((f (get op 'reflection-rule))) 370 (if (and b f (great (neg x) x)) (funcall f op x) nil))) 371 372(defun taylorize (op x) 373 (if ($taylorp x) 374 (mfuncall '$apply '$taylor `((mlist) ((,op) ,($ratdisrep x)) ,@(cdr ($taylorinfo x)))) nil)) 375 376(defun float-or-rational-p (x) 377 (or (floatp x) ($ratnump x))) 378 379(defun bigfloat-or-number-p (x) 380 (or ($bfloatp x) (numberp x) ($ratnump x))) 381 382;; When z is a Maxima complex float or when 'numer' is true and z is a 383;; Maxima complex number, evaluate (op z) by applying the mapping from 384;; the Maxima operator 'op' to the operator in the hash table 385;; '*flonum-op*'. When z isn't a Maxima complex number, return 386;; nil. 387 388(defun flonum-eval (op z) 389 (let ((op (gethash op *flonum-op*))) 390 (when op 391 (multiple-value-bind (bool R I) 392 (complex-number-p z #'float-or-rational-p) 393 (when (and bool (or $numer (floatp R) (floatp I))) 394 (setq R ($float R)) 395 (setq I ($float I)) 396 (complexify (funcall op (if (zerop I) R (complex R I))))))))) 397 398;; For now, big float evaluation of trig-like functions for complex 399;; big floats uses rectform. I suspect that for some functions (not 400;; all of them) rectform generates expressions that are poorly suited 401;; for numerical evaluation. For better accuracy, these functions 402;; (maybe acosh, for one) may need to be special cased. If they are 403;; special-cased, the *big-float-op* hash table contains the special 404;; cases. 405 406(defun big-float-eval (op z) 407 (when (complex-number-p z 'bigfloat-or-number-p) 408 (let ((x ($realpart z)) 409 (y ($imagpart z)) 410 (bop (gethash op *big-float-op*))) 411 ;; If bop is non-NIL, we want to try that first. If bop 412 ;; declines (by returning NIL), we silently give up and use the 413 ;; rectform version. 414 (cond ((and ($bfloatp x) (like 0 y)) 415 (or (and bop (funcall bop x)) 416 ($bfloat `((,op simp) ,x)))) 417 ((or ($bfloatp x) ($bfloatp y)) 418 (or (and bop (funcall bop ($bfloat x) ($bfloat y))) 419 (let ((z (add ($bfloat x) (mul '$%i ($bfloat y))))) 420 (setq z ($rectform `((,op simp) ,z))) 421 ($bfloat z)))))))) 422 423;; For complex big float evaluation, it's important to check the 424;; simp flag -- otherwise Maxima can get stuck in an infinite loop: 425;; asin(1.23b0 + %i * 4.56b0) ---> (simp-%asin ((%asin) ...) --> 426;; (big-float-eval ((%asin) ...) --> (risplit ((%asin simp) ...) --> 427;; (simp-%asin ((%asin simp) ...). If the simp flag is ignored, we've 428;; got trouble. 429 430(defun simp-%sin (form y z) 431 (oneargcheck form) 432 (setq y (simpcheck (cadr form) z)) 433 (cond ((flonum-eval (mop form) y)) 434 ((and (not (member 'simp (car form))) (big-float-eval (mop form) y))) 435 ((taylorize (mop form) (second form))) 436 ((and $%piargs (cond ((zerop1 y) 0) 437 ((has-const-or-int-term y '$%pi) (%piargs-sin/cos y))))) 438 ((and $%iargs (multiplep y '$%i)) (mul '$%i (cons-exp '%sinh (coeff y '$%i 1)))) 439 ((and $triginverses (not (atom y)) 440 (cond ((eq '%asin (setq z (caar y))) (cadr y)) 441 ((eq '%acos z) (sqrt1-x^2 (cadr y))) 442 ((eq '%atan z) (div (cadr y) (sqrt1+x^2 (cadr y)))) 443 ((eq '%acot z) (div 1 (sqrt1+x^2 (cadr y)))) 444 ((eq '%asec z) (div (sqrtx^2-1 (cadr y)) (cadr y))) 445 ((eq '%acsc z) (div 1 (cadr y))) 446 ((eq '$atan2 z) (div (cadr y) (sq-sumsq (cadr y) (caddr y))))))) 447 ((and $trigexpand (trigexpand '%sin y))) 448 ($exponentialize (exponentialize '%sin y)) 449 ((and $halfangles (halfangle '%sin y))) 450 ((apply-reflection-simp (mop form) y $trigsign)) 451 ;((and $trigsign (mminusp* y)) (neg (cons-exp '%sin (neg y)))) 452 (t (eqtest (list '(%sin) y) form)))) 453 454(defun simp-%cos (form y z) 455 (oneargcheck form) 456 (setq y (simpcheck (cadr form) z)) 457 (cond ((flonum-eval (mop form) y)) 458 ((and (not (member 'simp (car form))) (big-float-eval (mop form) y))) 459 ((taylorize (mop form) (second form))) 460 ((and $%piargs (cond ((zerop1 y) 1) 461 ((has-const-or-int-term y '$%pi) 462 (%piargs-sin/cos (add %pi//2 y)))))) 463 ((and $%iargs (multiplep y '$%i)) (cons-exp '%cosh (coeff y '$%i 1))) 464 ((and $triginverses (not (atom y)) 465 (cond ((eq '%acos (setq z (caar y))) (cadr y)) 466 ((eq '%asin z) (sqrt1-x^2 (cadr y))) 467 ((eq '%atan z) (div 1 (sqrt1+x^2 (cadr y)))) 468 ((eq '%acot z) (div (cadr y) (sqrt1+x^2 (cadr y)))) 469 ((eq '%asec z) (div 1 (cadr y))) 470 ((eq '%acsc z) (div (sqrtx^2-1 (cadr y)) (cadr y))) 471 ((eq '$atan2 z) (div (caddr y) (sq-sumsq (cadr y) (caddr y))))))) 472 ((and $trigexpand (trigexpand '%cos y))) 473 ($exponentialize (exponentialize '%cos y)) 474 ((and $halfangles (halfangle '%cos y))) 475 ((apply-reflection-simp (mop form) y $trigsign)) 476 ;((and $trigsign (mminusp* y)) (cons-exp '%cos (neg y))) 477 (t (eqtest (list '(%cos) y) form)))) 478 479(defun %piargs-sin/cos (x) 480 (let ($float coeff ratcoeff zl-rem) 481 (setq ratcoeff (get-const-or-int-terms x '$%pi) 482 coeff (linearize ratcoeff) 483 zl-rem (get-not-const-or-int-terms x '$%pi)) 484 (cond ((zerop1 zl-rem) (%piargs coeff ratcoeff)) 485 ((not (mevenp (car coeff))) nil) 486 ((equal 0 (setq x (mmod (cdr coeff) 2))) (cons-exp '%sin zl-rem)) 487 ((equal 1 x) (neg (cons-exp '%sin zl-rem))) 488 ((alike1 1//2 x) (cons-exp '%cos zl-rem)) 489 ((alike1 '((rat) 3 2) x) (neg (cons-exp '%cos zl-rem)))))) 490 491 492(defun filter-sum (pred form simp-flag) 493 "Takes form to be a sum and a sum of the summands for which pred is 494 true. Passes simp-flag through to addn if there is more than one 495 term in the sum." 496 (if (mplusp form) 497 (addn (mapcan 498 #'(lambda (term) 499 (when (funcall pred term) (list term))) (cdr form)) 500 simp-flag) 501 (if (funcall pred form) form 0))) 502 503;; collect terms of form A*var where A is a constant or integer. 504;; returns sum of all such A. 505;; does not expand form, so does not find constant term in (x+1)*var. 506;; thus we cannot simplify sin(2*%pi*(1+x)) => sin(2*%pi*x) unless 507;; the user calls expand. this could be extended to look a little 508;; more deeply into the expression, but we don't want to call expand 509;; in the core simplifier for reasons of speed and predictability. 510(defun get-const-or-int-terms (form var) 511 (coeff 512 (filter-sum (lambda (term) 513 (let ((coeff (coeff term var 1))) 514 (and (not (zerop1 coeff)) 515 (or ($constantp coeff) 516 (maxima-integerp coeff))))) 517 form 518 0) 519 var 1)) 520 521;; collect terms skipped by get-const-or-int-terms 522(defun get-not-const-or-int-terms (form var) 523 (filter-sum (lambda (term) 524 (let ((coeff (coeff term var 1))) 525 (not (and (not (zerop1 coeff)) 526 (or ($constantp coeff) 527 (maxima-integerp coeff)))))) 528 form 529 0)) 530 531(defun has-const-or-int-term (form var) 532 "Tests whether form has at least some term of the form a*var where a 533 is constant or integer" 534 (not (zerop1 (get-const-or-int-terms form var)))) 535 536(defun simp-%tan (form y z) 537 (oneargcheck form) 538 (setq y (simpcheck (cadr form) z)) 539 (cond ((flonum-eval (mop form) y)) 540 ((and (not (member 'simp (car form))) (big-float-eval (mop form) y))) 541 ((taylorize (mop form) (second form))) 542 ((and $%piargs (cond ((zerop1 y) 0) 543 ((has-const-or-int-term y '$%pi) (%piargs-tan/cot y))))) 544 ((and $%iargs (multiplep y '$%i)) (mul '$%i (cons-exp '%tanh (coeff y '$%i 1)))) 545 ((and $triginverses (not (atom y)) 546 (cond ((eq '%atan (setq z (caar y))) (cadr y)) 547 ((eq '%asin z) (div (cadr y) (sqrt1-x^2 (cadr y)))) 548 ((eq '%acos z) (div (sqrt1-x^2 (cadr y)) (cadr y))) 549 ((eq '%acot z) (div 1 (cadr y))) 550 ((eq '%asec z) (sqrtx^2-1 (cadr y))) 551 ((eq '%acsc z) (div 1 (sqrtx^2-1 (cadr y)))) 552 ((eq '$atan2 z) (div (cadr y) (caddr y)))))) 553 ((and $trigexpand (trigexpand '%tan y))) 554 ($exponentialize (exponentialize '%tan y)) 555 ((and $halfangles (halfangle '%tan y))) 556 ((apply-reflection-simp (mop form) y $trigsign)) 557 ;((and $trigsign (mminusp* y)) (neg (cons-exp '%tan (neg y)))) 558 (t (eqtest (list '(%tan) y) form)))) 559 560(defun simp-%cot (form y z) 561 (oneargcheck form) 562 (setq y (simpcheck (cadr form) z)) 563 564 (cond ((flonum-eval (mop form) y)) 565 ((and (not (member 'simp (car form))) (big-float-eval (mop form) y))) 566 ((taylorize (mop form) (second form))) 567 ((and $%piargs (cond ((zerop1 y) (domain-error y 'cot)) 568 ((and (has-const-or-int-term y '$%pi) 569 (setq z (%piargs-tan/cot (add %pi//2 y)))) 570 (neg z))))) 571 ((and $%iargs (multiplep y '$%i)) (mul -1 '$%i (cons-exp '%coth (coeff y '$%i 1)))) 572 ((and $triginverses (not (atom y)) 573 (cond ((eq '%acot (setq z (caar y))) (cadr y)) 574 ((eq '%asin z) (div (sqrt1-x^2 (cadr y)) (cadr y))) 575 ((eq '%acos z) (div (cadr y) (sqrt1-x^2 (cadr y)))) 576 ((eq '%atan z) (div 1 (cadr y))) 577 ((eq '%asec z) (div 1 (sqrtx^2-1 (cadr y)))) 578 ((eq '%acsc z) (sqrtx^2-1 (cadr y))) 579 ((eq '$atan2 z) (div (caddr y) (cadr y)))))) 580 ((and $trigexpand (trigexpand '%cot y))) 581 ($exponentialize (exponentialize '%cot y)) 582 ((and $halfangles (halfangle '%cot y))) 583 ((apply-reflection-simp (mop form) y $trigsign)) 584 ;((and $trigsign (mminusp* y)) (neg (cons-exp '%cot (neg y)))) 585 (t (eqtest (list '(%cot) y) form)))) 586 587(defun %piargs-tan/cot (x) 588 "If x is of the form tan(u) where u has a nonzero constant linear 589 term in %pi, then %piargs-tan/cot returns a simplified version of x 590 without this constant term." 591 ;; Set coeff to be the coefficient of $%pi collecting terms with no 592 ;; other atoms, so given %pi(x+1/2), coeff = 1/2. Let zl-rem be the 593 ;; remainder (TODO: computing zl-rem could probably be prettier.) 594 (let* ((nice-terms (get-const-or-int-terms x '$%pi)) 595 (coeff (linearize nice-terms)) 596 (zl-rem (get-not-const-or-int-terms x '$%pi)) 597 (sin-of-coeff-pi) 598 (cos-of-coeff-pi)) 599 (cond 600 ;; sin-of-coeff-pi and cos-of-coeff-pi are only non-nil if they 601 ;; are constants that %piargs-offset could compute, and we just 602 ;; checked that cos-of-coeff-pi was nonzero. Thus we can just 603 ;; return their quotient. 604 ((and (zerop1 zl-rem) 605 (setq sin-of-coeff-pi 606 (%piargs coeff nil))) 607 (setq cos-of-coeff-pi 608 (%piargs (cons (car coeff) 609 (rplus 1//2 (cdr coeff))) nil)) 610 (cond ((zerop1 sin-of-coeff-pi) 611 0) ;; tan(integer*%pi) 612 ((zerop1 cos-of-coeff-pi) 613 (merror (intl:gettext "tan: ~M isn't in the domain of tan.") x)) 614 (cos-of-coeff-pi 615 (div sin-of-coeff-pi cos-of-coeff-pi)))) 616 617 ;; This expression sets x to the coeff of %pi (mod 1) as a side 618 ;; effect and then, if this is zero, returns tan of the 619 ;; rest, because tan has periodicity %pi. 620 ((zerop1 (setq x (mmod (cdr coeff) 1))) 621 (cons-exp '%tan zl-rem)) 622 623 ;; Similarly, if x = 1/2 then return -cot(x). 624 ((alike1 1//2 x) 625 (neg (cons-exp '%cot zl-rem)))))) 626 627(defun simp-%csc (form y z) 628 (oneargcheck form) 629 (setq y (simpcheck (cadr form) z)) 630 (cond ((flonum-eval (mop form) y)) 631 ((and (not (member 'simp (car form))) (big-float-eval (mop form) y))) 632 ((taylorize (mop form) (second form))) 633 ((and $%piargs (cond ((zerop1 y) (domain-error y 'csc)) 634 ((has-const-or-int-term y '$%pi) (%piargs-csc/sec y))))) 635 ((and $%iargs (multiplep y '$%i)) (mul -1 '$%i (cons-exp '%csch (coeff y '$%i 1)))) 636 ((and $triginverses (not (atom y)) 637 (cond ((eq '%acsc (setq z (caar y))) (cadr y)) 638 ((eq '%asin z) (div 1 (cadr y))) 639 ((eq '%acos z) (div 1 (sqrt1-x^2 (cadr y)))) 640 ((eq '%atan z) (div (sqrt1+x^2 (cadr y)) (cadr y))) 641 ((eq '%acot z) (sqrt1+x^2 (cadr y))) 642 ((eq '%asec z) (div (cadr y) (sqrtx^2-1 (cadr y)))) 643 ((eq '$atan2 z) (div (sq-sumsq (cadr y) (caddr y)) (cadr y)))))) 644 ((and $trigexpand (trigexpand '%csc y))) 645 ($exponentialize (exponentialize '%csc y)) 646 ((and $halfangles (halfangle '%csc y))) 647 ((apply-reflection-simp (mop form) y $trigsign)) 648 ;((and $trigsign (mminusp* y)) (neg (cons-exp '%csc (neg y)))) 649 650 (t (eqtest (list '(%csc) y) form)))) 651 652(defun simp-%sec (form y z) 653 (oneargcheck form) 654 (setq y (simpcheck (cadr form) z)) 655 (cond ((flonum-eval (mop form) y)) 656 ((and (not (member 'simp (car form))) (big-float-eval (mop form) y))) 657 ((taylorize (mop form) (second form))) 658 ((and $%piargs (cond ((zerop1 y) 1) 659 ((has-const-or-int-term y '$%pi) (%piargs-csc/sec (add %pi//2 y)))))) 660 ((and $%iargs (multiplep y '$%i)) (cons-exp '%sech (coeff y '$%i 1))) 661 ((and $triginverses (not (atom y)) 662 (cond ((eq '%asec (setq z (caar y))) (cadr y)) 663 ((eq '%asin z) (div 1 (sqrt1-x^2 (cadr y)))) 664 ((eq '%acos z) (div 1 (cadr y))) 665 ((eq '%atan z) (sqrt1+x^2 (cadr y))) 666 ((eq '%acot z) (div (sqrt1+x^2 (cadr y)) (cadr y))) 667 ((eq '%acsc z) (div (cadr y) (sqrtx^2-1 (cadr y)))) 668 ((eq '$atan2 z) (div (sq-sumsq (cadr y) (caddr y)) (caddr y)))))) 669 ((and $trigexpand (trigexpand '%sec y))) 670 ($exponentialize (exponentialize '%sec y)) 671 ((and $halfangles (halfangle '%sec y))) 672 ((apply-reflection-simp (mop form) y $trigsign)) 673 ;((and $trigsign (mminusp* y)) (cons-exp '%sec (neg y))) 674 675 (t (eqtest (list '(%sec) y) form)))) 676 677(defun %piargs-csc/sec (x) 678 (prog ($float coeff ratcoeff zl-rem) 679 (setq ratcoeff (get-const-or-int-terms x '$%pi) 680 coeff (linearize ratcoeff) 681 zl-rem (get-not-const-or-int-terms x '$%pi)) 682 (return (cond ((and (zerop1 zl-rem) (setq zl-rem (%piargs coeff nil))) (div 1 zl-rem)) 683 ((not (mevenp (car coeff))) nil) 684 ((equal 0 (setq x (mmod (cdr coeff) 2))) (cons-exp '%csc zl-rem)) 685 ((equal 1 x) (neg (cons-exp '%csc zl-rem))) 686 ((alike1 1//2 x) (cons-exp '%sec zl-rem)) 687 ((alike1 '((rat) 3 2) x) (neg (cons-exp '%sec zl-rem))))))) 688 689(defun simp-%atan (form y z) 690 (oneargcheck form) 691 (setq y (simpcheck (cadr form) z)) 692 (cond ((flonum-eval (mop form) y)) 693 ((and (not (member 'simp (car form))) (big-float-eval (mop form) y))) 694 ((taylorize (mop form) (second form))) 695 ;; Simplification for special values 696 ((zerop1 y) y) 697 ((or (eq y '$inf) (alike1 y '((mtimes) -1 $minf))) 698 (div '$%pi 2)) 699 ((or (eq y '$minf) (alike1 y '((mtimes) -1 $inf))) 700 (div '$%pi -2)) 701 ((and $%piargs 702 ;; Recognize more special values 703 (cond ((equal 1 y) (div '$%pi 4)) 704 ((equal -1 y) (div '$%pi -4)) 705 ;; sqrt(3) 706 ((alike1 y '((mexpt) 3 ((rat) 1 2))) 707 (div '$%pi 3)) 708 ;; -sqrt(3) 709 ((alike1 y '((mtimes) -1 ((mexpt) 3 ((rat) 1 2)))) 710 (div '$%pi -3)) 711 ;; 1/sqrt(3) 712 ((alike1 y '((mexpt) 3 ((rat) -1 2))) 713 (div '$%pi 6)) 714 ;; -1/sqrt(3) 715 ((alike1 y '((mtimes) -1 ((mexpt) 3 ((rat) -1 2)))) 716 (div '$%pi -6)) 717 ((alike1 y '((mplus) -1 ((mexpt) 2 ((rat) 1 2)))) 718 (div '$%pi 8)) 719 ((alike1 y '((mplus) 1 ((mexpt) 2 ((rat) 1 2)))) 720 (mul 3 (div '$%pi 8)))))) 721 ((and $%iargs (multiplep y '$%i)) 722 ;; atan(%i*y) -> %i*atanh(y) 723 (mul '$%i (take '(%atanh) (coeff y '$%i 1)))) 724 ((and (not (atom y)) (member (caar y) '(%cot %tan)) 725 (if ($constantp (cadr y)) 726 (let ((y-val (mfuncall '$mod 727 (if (eq (caar y) '%tan) 728 (cadr y) 729 (sub %pi//2 (cadr y))) 730 '$%pi))) 731 (cond ((eq (mlsp y-val %pi//2) t) y-val) 732 ((eq (mlsp y-val '$%pi) t) (sub y-val '$%pi))))))) 733 ((and (eq $triginverses '$all) (not (atom y)) 734 (if (eq (caar y) '%tan) (cadr y)))) 735 ((and (eq $triginverses t) (not (atom y)) (eq (caar y) '%tan) 736 ;; Check if y in [-%pi/2, %pi/2] 737 (if (and (member (csign (sub (cadr y) %pi//2)) '($nz $neg) :test #'eq) 738 (member (csign (add (cadr y) %pi//2)) '($pz $pos) :test #'eq)) 739 (cadr y)))) 740 ($logarc (logarc '%atan y)) 741 ((apply-reflection-simp (mop form) y $trigsign)) 742 (t (eqtest (list '(%atan) y) form)))) 743 744(defun %piargs (x ratcoeff) 745 (let (offset-result) 746 (cond ((and (integerp (car x)) (integerp (cdr x))) 0) 747 ((not (mevenp (car x))) 748 (cond ((null ratcoeff) nil) 749 ((and (integerp (car x)) 750 (setq offset-result (%piargs-offset (cdr x)))) 751 (mul (power -1 (sub ratcoeff (cdr x))) 752 offset-result)))) 753 ((%piargs-offset (mmod (cdr x) 2)))))) 754 755; simplifies sin(%pi * x) where x is between 0 and 1 756; returns nil if can't simplify 757(defun %piargs-offset (x) 758 (cond ((or (alike1 '((rat) 1 6) x) (alike1 '((rat) 5 6) x)) 1//2) 759 ((or (alike1 '((rat) 1 4) x) (alike1 '((rat) 3 4) x)) (div (power 2 1//2) 2)) 760 ((or (alike1 '((rat) 1 3) x) (alike1 '((rat) 2 3) x)) (div (power 3 1//2) 2)) 761 ((alike1 1//2 x) 1) 762 ((or (alike1 '((rat) 7 6) x) (alike1 '((rat) 11 6) x)) -1//2) 763 ((or (alike1 '((rat) 4 3) x) (alike1 '((rat) 5 3) x)) (div (power 3 1//2) -2)) 764 ((or (alike1 '((rat) 5 4) x) (alike1 '((rat) 7 4) x)) (mul -1//2 (power 2 1//2))) 765 ((alike1 '((rat) 3 2) x) -1))) 766 767;; identifies integer part of form 768;; returns (X . Y) if form can be written as X*some_integer + Y 769;; returns nil otherwise 770(defun linearize (form) 771 (cond ((integerp form) (cons 0 form)) 772 ((numberp form) nil) 773 ((atom form) 774 (let (dum) 775 (cond ((setq dum (evod form)) 776 (if (eq '$even dum) '(2 . 0) '(2 . 1))) 777 ((maxima-integerp form) '(1 . 0))))) 778 ((eq 'rat (caar form)) (cons 0 form)) 779 ((eq 'mplus (caar form)) (lin-mplus form)) 780 ((eq 'mtimes (caar form)) (lin-mtimes form)) 781 ((eq 'mexpt (caar form)) (lin-mexpt form)))) 782 783(defun lin-mplus (form) 784 (do ((tl (cdr form) (cdr tl)) (dummy) (coeff 0) (zl-rem 0)) 785 ((null tl) (cons coeff (mmod zl-rem coeff))) 786 (setq dummy (linearize (car tl))) 787 (if (null dummy) (return nil) 788 (setq coeff (rgcd (car dummy) coeff) zl-rem (rplus (cdr dummy) zl-rem))))) 789 790(defun lin-mtimes (form) 791 (do ((fl (cdr form) (cdr fl)) (dummy) (coeff 0) (zl-rem 1)) 792 ((null fl) (cons coeff (mmod zl-rem coeff))) 793 (setq dummy (linearize (car fl))) 794 (cond ((null dummy) (return nil)) 795 (t (setq coeff (rgcd (rtimes coeff (car dummy)) 796 (rgcd (rtimes coeff (cdr dummy)) (rtimes zl-rem (car dummy)))) 797 zl-rem (rtimes (cdr dummy) zl-rem)))))) 798 799(defun lin-mexpt (form) 800 (prog (dummy) 801 (cond ((and (integerp (caddr form)) (not (minusp (caddr form))) 802 (not (null (setq dummy (linearize (cadr form)))))) 803 (return (cons (car dummy) (mmod (cdr dummy) (caddr form)))))))) 804 805(defun rgcd (x y) 806 (cond ((integerp x) 807 (cond ((integerp y) (gcd x y)) 808 (t (list '(rat) (gcd x (cadr y)) (caddr y))))) 809 ((integerp y) (list '(rat) (gcd (cadr x) y) (caddr x))) 810 (t (list '(rat) (gcd (cadr x) (cadr y)) (lcm (caddr x) (caddr y)))))) 811 812(defun maxima-reduce (x y) 813 (prog (gcd) 814 (setq gcd (gcd x y) x (truncate x gcd) y (truncate y gcd)) 815 (if (minusp y) (setq x (- x) y (- y))) 816 (return (if (eql y 1) x (list '(rat simp) x y))))) 817 818;; The following four functions are generated in code by TRANSL. - JPG 2/1/81 819 820(defun rplus (x y) (addk x y)) 821 822(defun rdifference (x y) (addk x (timesk -1 y))) 823 824(defun rtimes (x y) (timesk x y)) 825 826(defun rremainder (x y) 827 (cond ((equal 0 y) (dbz-err)) 828 ((integerp x) 829 (cond ((integerp y) (maxima-reduce x y)) 830 (t (maxima-reduce (* x (caddr y)) (cadr y))))) 831 ((integerp y) (maxima-reduce (cadr x) (* (caddr x) y))) 832 (t (maxima-reduce (* (cadr x) (caddr y)) (* (caddr x) (cadr y)))))) 833 834(defmfun $exponentialize (exp) 835 (let ($demoivre) 836 (cond ((atom exp) exp) 837 ((trigp (caar exp)) 838 (exponentialize (caar exp) ($exponentialize (cadr exp)))) 839 (t (recur-apply #'$exponentialize exp))))) 840 841(defun exponentialize (op arg) 842 (cond ((eq '%sin op) 843 (div (sub (power '$%e (mul '$%i arg)) (power '$%e (mul -1 '$%i arg))) 844 (mul 2 '$%i))) 845 ((eq '%cos op) 846 (div (add (power '$%e (mul '$%i arg)) (power '$%e (mul -1 '$%i arg))) 2)) 847 ((eq '%tan op) 848 (div (sub (power '$%e (mul '$%i arg)) (power '$%e (mul -1 '$%i arg))) 849 (mul '$%i (add (power '$%e (mul '$%i arg)) 850 (power '$%e (mul -1 '$%i arg)))))) 851 ((eq '%cot op) 852 (div (mul '$%i (add (power '$%e (mul '$%i arg)) 853 (power '$%e (mul -1 '$%i arg)))) 854 (sub (power '$%e (mul '$%i arg)) (power '$%e (mul -1 '$%i arg))))) 855 ((eq '%csc op) 856 (div (mul 2 '$%i) 857 (sub (power '$%e (mul '$%i arg)) (power '$%e (mul -1 '$%i arg))))) 858 ((eq '%sec op) 859 (div 2 (add (power '$%e (mul '$%i arg)) (power '$%e (mul -1 '$%i arg))))) 860 ((eq '%sinh op) 861 (div (sub (power '$%e arg) (power '$%e (neg arg))) 2)) 862 ((eq '%cosh op) 863 (div (add (power '$%e arg) (power '$%e (mul -1 arg))) 2)) 864 ((eq '%tanh op) 865 (div (sub (power '$%e arg) (power '$%e (neg arg))) 866 (add (power '$%e arg) (power '$%e (mul -1 arg))))) 867 ((eq '%coth op) 868 (div (add (power '$%e arg) (power '$%e (mul -1 arg))) 869 (sub (power '$%e arg) (power '$%e (neg arg))))) 870 ((eq '%csch op) 871 (div 2 (sub (power '$%e arg) (power '$%e (neg arg))))) 872 ((eq '%sech op) 873 (div 2 (add (power '$%e arg) (power '$%e (mul -1 arg))))))) 874 875(defun coefficient (exp var pow) 876 (coeff exp var pow)) 877 878(defun mmod (x mod) 879 (cond ((and (integerp x) (integerp mod)) 880 (if (minusp (if (zerop mod) x (setq x (- x (* mod (truncate x mod)))))) 881 (+ x mod) 882 x)) 883 ((and ($ratnump x) ($ratnump mod)) 884 (let 885 ((d (lcm ($denom x) ($denom mod)))) 886 (setq x (mul* d x)) 887 (setq mod (mul* d mod)) 888 (div (mod x mod) d))) 889 (t nil))) 890 891(defun multiplep (exp var) 892 (and (not (zerop1 exp)) (zerop1 (sub exp (mul var (coeff exp var 1)))))) 893 894(defun linearp (exp var) 895 (and (setq exp (islinear exp var)) (not (equal (car exp) 0)))) 896 897(defun mminusp (x) 898 (= -1 (signum1 x))) 899 900(defun mminusp* (x) 901 (let (sign) 902 (setq sign (csign x)) 903 (or (member sign '($neg $nz) :test #'eq) 904 (and (mminusp x) (not (member sign '($pos $pz) :test #'eq)))))) 905 906;; This should give more information somehow. 907 908(defun dbz-err () 909 (cond ((not errorsw) (merror (intl:gettext "Division by zero attempted."))) 910 (t (throw 'errorsw t)))) 911 912(defun dbz-err1 (func) 913 (cond ((not errorsw) (merror (intl:gettext "~A: division by zero attempted.") func)) 914 (t (throw 'errorsw t)))) 915