(calcFunc-ln): In symbolic manipulations,

[emacs.git] / lisp / calc / calc-alg.el

;;; calc-alg.el --- algebraic functions for Calc 

;; Copyright (C) 1990, 1991, 1992, 1993, 2001 Free Software Foundation, Inc.

;; Author: David Gillespie <daveg@synaptics.com>
;; Maintainer: Colin Walters <walters@debian.org>

;; This file is part of GNU Emacs.

;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY.  No author or distributor
;; accepts responsibility to anyone for the consequences of using it
;; or for whether it serves any particular purpose or works at all,
;; unless he says so in writing.  Refer to the GNU Emacs General Public
;; License for full details.

;; Everyone is granted permission to copy, modify and redistribute
;; GNU Emacs, but only under the conditions described in the
;; GNU Emacs General Public License.   A copy of this license is
;; supposed to have been given to you along with GNU Emacs so you
;; can know your rights and responsibilities.  It should be in a
;; file named COPYING.  Among other things, the copyright notice
;; and this notice must be preserved on all copies.

;;; Commentary:

;;; Code:

;; This file is autoloaded from calc-ext.el.
(require 'calc-ext)

(require 'calc-macs)

(defun calc-Need-calc-alg () nil)

;;; Algebra commands.

(defun calc-alg-evaluate (arg)
  (interactive "p")
  (calc-slow-wrapper
   (calc-with-default-simplification
    (let ((math-simplify-only nil))
      (calc-modify-simplify-mode arg)
      (calc-enter-result 1 "dsmp" (calc-top 1))))))

(defun calc-modify-simplify-mode (arg)
  (if (= (math-abs arg) 2)
      (setq calc-simplify-mode 'alg)
    (if (>= (math-abs arg) 3)
        (setq calc-simplify-mode 'ext)))
  (if (< arg 0)
      (setq calc-simplify-mode (list calc-simplify-mode))))

(defun calc-simplify ()
  (interactive)
  (calc-slow-wrapper
   (calc-with-default-simplification
    (calc-enter-result 1 "simp" (math-simplify (calc-top-n 1))))))

(defun calc-simplify-extended ()
  (interactive)
  (calc-slow-wrapper
   (calc-with-default-simplification
    (calc-enter-result 1 "esmp" (math-simplify-extended (calc-top-n 1))))))

(defun calc-expand-formula (arg)
  (interactive "p")
  (calc-slow-wrapper
   (calc-with-default-simplification
    (let ((math-simplify-only nil))
      (calc-modify-simplify-mode arg)
      (calc-enter-result 1 "expf" 
                         (if (> arg 0)
                             (let ((math-expand-formulas t))
                               (calc-top-n 1))
                           (let ((top (calc-top-n 1)))
                             (or (math-expand-formula top)
                                 top))))))))

(defun calc-factor (arg)
  (interactive "P")
  (calc-slow-wrapper
   (calc-unary-op "fctr" (if (calc-is-hyperbolic)
                             'calcFunc-factors 'calcFunc-factor)
                  arg)))

(defun calc-expand (n)
  (interactive "P")
  (calc-slow-wrapper
   (calc-enter-result 1 "expa"
                      (append (list 'calcFunc-expand
                                    (calc-top-n 1))
                              (and n (list (prefix-numeric-value n)))))))

(defun calc-collect (&optional var)
  (interactive "sCollect terms involving: ")
  (calc-slow-wrapper
   (if (or (equal var "") (equal var "$") (null var))
       (calc-enter-result 2 "clct" (cons 'calcFunc-collect
                                         (calc-top-list-n 2)))
     (let ((var (math-read-expr var)))
       (if (eq (car-safe var) 'error)
           (error "Bad format in expression: %s" (nth 1 var)))
       (calc-enter-result 1 "clct" (list 'calcFunc-collect
                                         (calc-top-n 1)
                                         var))))))

(defun calc-apart (arg)
  (interactive "P")
  (calc-slow-wrapper
   (calc-unary-op "aprt" 'calcFunc-apart arg)))

(defun calc-normalize-rat (arg)
  (interactive "P")
  (calc-slow-wrapper
   (calc-unary-op "nrat" 'calcFunc-nrat arg)))

(defun calc-poly-gcd (arg)
  (interactive "P")
  (calc-slow-wrapper
   (calc-binary-op "pgcd" 'calcFunc-pgcd arg)))

(defun calc-poly-div (arg)
  (interactive "P")
  (calc-slow-wrapper
   (setq calc-poly-div-remainder nil)
   (calc-binary-op "pdiv" 'calcFunc-pdiv arg)
   (if (and calc-poly-div-remainder (null arg))
       (progn
         (calc-clear-command-flag 'clear-message)
         (calc-record calc-poly-div-remainder "prem")
         (if (not (Math-zerop calc-poly-div-remainder))
             (message "(Remainder was %s)"
                      (math-format-flat-expr calc-poly-div-remainder 0))
           (message "(No remainder)"))))))

(defun calc-poly-rem (arg)
  (interactive "P")
  (calc-slow-wrapper
   (calc-binary-op "prem" 'calcFunc-prem arg)))

(defun calc-poly-div-rem (arg)
  (interactive "P")
  (calc-slow-wrapper
   (if (calc-is-hyperbolic)
       (calc-binary-op "pdvr" 'calcFunc-pdivide arg)
     (calc-binary-op "pdvr" 'calcFunc-pdivrem arg))))

(defun calc-substitute (&optional oldname newname)
  (interactive "sSubstitute old: ")
  (calc-slow-wrapper
   (let (old new (num 1) expr)
     (if (or (equal oldname "") (equal oldname "$") (null oldname))
         (setq new (calc-top-n 1)
               old (calc-top-n 2)
               expr (calc-top-n 3)
               num 3)
       (or newname
           (progn (calc-unread-command ?\C-a)
                  (setq newname (read-string (concat "Substitute old: "
                                                     oldname
                                                     ", new: ")
                                             oldname))))
       (if (or (equal newname "") (equal newname "$") (null newname))
           (setq new (calc-top-n 1)
                 expr (calc-top-n 2)
                 num 2)
         (setq new (if (stringp newname) (math-read-expr newname) newname))
         (if (eq (car-safe new) 'error)
             (error "Bad format in expression: %s" (nth 1 new)))
         (setq expr (calc-top-n 1)))
       (setq old (if (stringp oldname) (math-read-expr oldname) oldname))
       (if (eq (car-safe old) 'error)
           (error "Bad format in expression: %s" (nth 1 old)))
       (or (math-expr-contains expr old)
           (error "No occurrences found")))
     (calc-enter-result num "sbst" (math-expr-subst expr old new)))))


(defun calc-has-rules (name)
  (setq name (calc-var-value name))
  (and (consp name)
       (memq (car name) '(vec calcFunc-assign calcFunc-condition))
       name))

(defun math-recompile-eval-rules ()
  (setq math-eval-rules-cache (and (calc-has-rules 'var-EvalRules)
                                   (math-compile-rewrites
                                    '(var EvalRules var-EvalRules)))
        math-eval-rules-cache-other (assq nil math-eval-rules-cache)
        math-eval-rules-cache-tag (calc-var-value 'var-EvalRules)))


;;; Try to expand a formula according to its definition.
(defun math-expand-formula (expr)
  (and (consp expr)
       (symbolp (car expr))
       (or (get (car expr) 'calc-user-defn)
           (get (car expr) 'math-expandable))
       (let ((res (let ((math-expand-formulas t))
                    (apply (car expr) (cdr expr)))))
         (and (not (eq (car-safe res) (car expr)))
              res))))




;;; True if A comes before B in a canonical ordering of expressions.  [P X X]
(defun math-beforep (a b)   ; [Public]
  (cond ((and (Math-realp a) (Math-realp b))
         (let ((comp (math-compare a b)))
           (or (eq comp -1)
               (and (eq comp 0)
                    (not (equal a b))
                    (> (length (memq (car-safe a)
                                     '(bigneg nil bigpos frac float)))
                       (length (memq (car-safe b)
                                     '(bigneg nil bigpos frac float))))))))
        ((equal b '(neg (var inf var-inf))) nil)
        ((equal a '(neg (var inf var-inf))) t)
        ((equal a '(var inf var-inf)) nil)
        ((equal b '(var inf var-inf)) t)
        ((Math-realp a)
         (if (and (eq (car-safe b) 'intv) (math-intv-constp b))
             (if (or (math-beforep a (nth 2 b)) (Math-equal a (nth 2 b)))
                 t
               nil)
           t))
        ((Math-realp b)
         (if (and (eq (car-safe a) 'intv) (math-intv-constp a))
             (if (math-beforep (nth 2 a) b)
                 t
               nil)
           nil))
        ((and (eq (car a) 'intv) (eq (car b) 'intv)
              (math-intv-constp a) (math-intv-constp b))
         (let ((comp (math-compare (nth 2 a) (nth 2 b))))
           (cond ((eq comp -1) t)
                 ((eq comp 1) nil)
                 ((and (memq (nth 1 a) '(2 3)) (memq (nth 1 b) '(0 1))) t)
                 ((and (memq (nth 1 a) '(0 1)) (memq (nth 1 b) '(2 3))) nil)
                 ((eq (setq comp (math-compare (nth 3 a) (nth 3 b))) -1) t)
                 ((eq comp 1) nil)
                 ((and (memq (nth 1 a) '(0 2)) (memq (nth 1 b) '(1 3))) t)
                 (t nil))))
        ((not (eq (not (Math-objectp a)) (not (Math-objectp b))))
         (Math-objectp a))
        ((eq (car a) 'var)
         (if (eq (car b) 'var)
             (string-lessp (symbol-name (nth 1 a)) (symbol-name (nth 1 b)))
           (not (Math-numberp b))))
        ((eq (car b) 'var) (Math-numberp a))
        ((eq (car a) (car b))
         (while (and (setq a (cdr a) b (cdr b)) a
                     (equal (car a) (car b))))
         (and b
              (or (null a)
                  (math-beforep (car a) (car b)))))
        (t (string-lessp (symbol-name (car a)) (symbol-name (car b))))))


(defsubst math-simplify-extended (a)
  (let ((math-living-dangerously t))
    (math-simplify a)))

(defalias 'calcFunc-esimplify 'math-simplify-extended)

(defun math-simplify (top-expr)
  (let ((math-simplifying t)
        (top-only (consp calc-simplify-mode))
        (simp-rules (append (and (calc-has-rules 'var-AlgSimpRules)
                                 '((var AlgSimpRules var-AlgSimpRules)))
                            (and math-living-dangerously
                                 (calc-has-rules 'var-ExtSimpRules)
                                 '((var ExtSimpRules var-ExtSimpRules)))
                            (and math-simplifying-units
                                 (calc-has-rules 'var-UnitSimpRules)
                                 '((var UnitSimpRules var-UnitSimpRules)))
                            (and math-integrating
                                 (calc-has-rules 'var-IntegSimpRules)
                                 '((var IntegSimpRules var-IntegSimpRules)))))
        res)
    (if top-only
        (let ((r simp-rules))
          (setq res (math-simplify-step (math-normalize top-expr))
                calc-simplify-mode '(nil)
                top-expr (math-normalize res))
          (while r
            (setq top-expr (math-rewrite top-expr (car r)
                                         '(neg (var inf var-inf)))
                  r (cdr r))))
      (calc-with-default-simplification
       (while (let ((r simp-rules))
                (setq res (math-normalize top-expr))
                (while r
                  (setq res (math-rewrite res (car r))
                        r (cdr r)))
                (not (equal top-expr (setq res (math-simplify-step res)))))
         (setq top-expr res)))))
  top-expr)

(defalias 'calcFunc-simplify 'math-simplify)

;;; The following has a "bug" in that if any recursive simplifications
;;; occur only the first handler will be tried; this doesn't really
;;; matter, since math-simplify-step is iterated to a fixed point anyway.
(defun math-simplify-step (a)
  (if (Math-primp a)
      a
    (let ((aa (if (or top-only
                      (memq (car a) '(calcFunc-quote calcFunc-condition
                                                     calcFunc-evalto)))
                  a
                (cons (car a) (mapcar 'math-simplify-step (cdr a))))))
      (and (symbolp (car aa))
           (let ((handler (get (car aa) 'math-simplify)))
             (and handler
                  (while (and handler
                              (equal (setq aa (or (funcall (car handler) aa)
                                                  aa))
                                     a))
                    (setq handler (cdr handler))))))
      aa)))


;; Placeholder, to synchronize autoloading.
(defun math-need-std-simps ()
  nil)

(math-defsimplify (+ -)
  (math-simplify-plus))

(defun math-simplify-plus ()
  (cond ((and (memq (car-safe (nth 1 expr)) '(+ -))
              (Math-numberp (nth 2 (nth 1 expr)))
              (not (Math-numberp (nth 2 expr))))
         (let ((x (nth 2 expr))
               (op (car expr)))
           (setcar (cdr (cdr expr)) (nth 2 (nth 1 expr)))
           (setcar expr (car (nth 1 expr)))
           (setcar (cdr (cdr (nth 1 expr))) x)
           (setcar (nth 1 expr) op)))
        ((and (eq (car expr) '+)
              (Math-numberp (nth 1 expr))
              (not (Math-numberp (nth 2 expr))))
         (let ((x (nth 2 expr)))
           (setcar (cdr (cdr expr)) (nth 1 expr))
           (setcar (cdr expr) x))))
  (let ((aa expr)
        aaa temp)
    (while (memq (car-safe (setq aaa (nth 1 aa))) '(+ -))
      (if (setq temp (math-combine-sum (nth 2 aaa) (nth 2 expr)
                                       (eq (car aaa) '-) (eq (car expr) '-) t))
          (progn
            (setcar (cdr (cdr expr)) temp)
            (setcar expr '+)
            (setcar (cdr (cdr aaa)) 0)))
      (setq aa (nth 1 aa)))
    (if (setq temp (math-combine-sum aaa (nth 2 expr)
                                     nil (eq (car expr) '-) t))
        (progn
          (setcar (cdr (cdr expr)) temp)
          (setcar expr '+)
          (setcar (cdr aa) 0)))
    expr))

(math-defsimplify *
  (math-simplify-times))

(defun math-simplify-times ()
  (if (eq (car-safe (nth 2 expr)) '*)
      (and (math-beforep (nth 1 (nth 2 expr)) (nth 1 expr))
           (or (math-known-scalarp (nth 1 expr) t)
               (math-known-scalarp (nth 1 (nth 2 expr)) t))
           (let ((x (nth 1 expr)))
             (setcar (cdr expr) (nth 1 (nth 2 expr)))
             (setcar (cdr (nth 2 expr)) x)))
    (and (math-beforep (nth 2 expr) (nth 1 expr))
         (or (math-known-scalarp (nth 1 expr) t)
             (math-known-scalarp (nth 2 expr) t))
         (let ((x (nth 2 expr)))
           (setcar (cdr (cdr expr)) (nth 1 expr))
           (setcar (cdr expr) x))))
  (let ((aa expr)
        aaa temp
        (safe t) (scalar (math-known-scalarp (nth 1 expr))))
    (if (and (Math-ratp (nth 1 expr))
             (setq temp (math-common-constant-factor (nth 2 expr))))
        (progn
          (setcar (cdr (cdr expr))
                  (math-cancel-common-factor (nth 2 expr) temp))
          (setcar (cdr expr) (math-mul (nth 1 expr) temp))))
    (while (and (eq (car-safe (setq aaa (nth 2 aa))) '*)
                safe)
      (if (setq temp (math-combine-prod (nth 1 expr) (nth 1 aaa) nil nil t))
          (progn
            (setcar (cdr expr) temp)
            (setcar (cdr aaa) 1)))
      (setq safe (or scalar (math-known-scalarp (nth 1 aaa) t))
            aa (nth 2 aa)))
    (if (and (setq temp (math-combine-prod aaa (nth 1 expr) nil nil t))
             safe)
        (progn
          (setcar (cdr expr) temp)
          (setcar (cdr (cdr aa)) 1)))
    (if (and (eq (car-safe (nth 1 expr)) 'frac)
             (memq (nth 1 (nth 1 expr)) '(1 -1)))
        (math-div (math-mul (nth 2 expr) (nth 1 (nth 1 expr)))
                  (nth 2 (nth 1 expr)))
      expr)))

(math-defsimplify /
  (math-simplify-divide))

(defun math-simplify-divide ()
  (let ((np (cdr expr))
        (nover nil)
        (nn (and (or (eq (car expr) '/) (not (Math-realp (nth 2 expr))))
                 (math-common-constant-factor (nth 2 expr))))
        n op)
    (if nn
        (progn
          (setq n (and (or (eq (car expr) '/) (not (Math-realp (nth 1 expr))))
                       (math-common-constant-factor (nth 1 expr))))
          (if (and (eq (car-safe nn) 'frac) (eq (nth 1 nn) 1) (not n))
              (progn
                (setcar (cdr expr) (math-mul (nth 2 nn) (nth 1 expr)))
                (setcar (cdr (cdr expr))
                        (math-cancel-common-factor (nth 2 expr) nn))
                (if (and (math-negp nn)
                         (setq op (assq (car expr) calc-tweak-eqn-table)))
                    (setcar expr (nth 1 op))))
            (if (and n (not (eq (setq n (math-frac-gcd n nn)) 1)))
                (progn
                  (setcar (cdr expr)
                          (math-cancel-common-factor (nth 1 expr) n))
                  (setcar (cdr (cdr expr))
                          (math-cancel-common-factor (nth 2 expr) n))
                  (if (and (math-negp n)
                           (setq op (assq (car expr) calc-tweak-eqn-table)))
                      (setcar expr (nth 1 op))))))))
    (if (and (eq (car-safe (car np)) '/)
             (math-known-scalarp (nth 2 expr) t))
        (progn
          (setq np (cdr (nth 1 expr)))
          (while (eq (car-safe (setq n (car np))) '*)
            (and (math-known-scalarp (nth 2 n) t)
                 (math-simplify-divisor (cdr n) (cdr (cdr expr)) nil t))
            (setq np (cdr (cdr n))))
          (math-simplify-divisor np (cdr (cdr expr)) nil t)
          (setq nover t
                np (cdr (cdr (nth 1 expr))))))
    (while (eq (car-safe (setq n (car np))) '*)
      (and (math-known-scalarp (nth 2 n) t)
           (math-simplify-divisor (cdr n) (cdr (cdr expr)) nover t))
      (setq np (cdr (cdr n))))
    (math-simplify-divisor np (cdr (cdr expr)) nover t)
    expr))

(defun math-simplify-divisor (np dp nover dover)
  (cond ((eq (car-safe (car dp)) '/)
         (math-simplify-divisor np (cdr (car dp)) nover dover)
         (and (math-known-scalarp (nth 1 (car dp)) t)
              (math-simplify-divisor np (cdr (cdr (car dp)))
                                     nover (not dover))))
        ((or (or (eq (car expr) '/)
                 (let ((signs (math-possible-signs (car np))))
                   (or (memq signs '(1 4))
                       (and (memq (car expr) '(calcFunc-eq calcFunc-neq))
                            (eq signs 5))
                       math-living-dangerously)))
             (math-numberp (car np)))
         (let ((n (car np))
               d dd temp op
               (safe t) (scalar (math-known-scalarp n)))
           (while (and (eq (car-safe (setq d (car dp))) '*)
                       safe)
             (math-simplify-one-divisor np (cdr d))
             (setq safe (or scalar (math-known-scalarp (nth 1 d) t))
                   dp (cdr (cdr d))))
           (if safe
               (math-simplify-one-divisor np dp))))))

(defun math-simplify-one-divisor (np dp)
  (if (setq temp (math-combine-prod (car np) (car dp) nover dover t))
      (progn
        (and (not (memq (car expr) '(/ calcFunc-eq calcFunc-neq)))
             (math-known-negp (car dp))
             (setq op (assq (car expr) calc-tweak-eqn-table))
             (setcar expr (nth 1 op)))
        (setcar np (if nover (math-div 1 temp) temp))
        (setcar dp 1))
    (and dover (not nover) (eq (car expr) '/)
         (eq (car-safe (car dp)) 'calcFunc-sqrt)
         (Math-integerp (nth 1 (car dp)))
         (progn
           (setcar np (math-mul (car np)
                                (list 'calcFunc-sqrt (nth 1 (car dp)))))
           (setcar dp (nth 1 (car dp)))))))

(defun math-common-constant-factor (expr)
  (if (Math-realp expr)
      (if (Math-ratp expr)
          (and (not (memq expr '(0 1 -1)))
               (math-abs expr))
        (if (math-ratp (setq expr (math-to-simple-fraction expr)))
            (math-common-constant-factor expr)))
    (if (memq (car expr) '(+ - cplx sdev))
        (let ((f1 (math-common-constant-factor (nth 1 expr)))
              (f2 (math-common-constant-factor (nth 2 expr))))
          (and f1 f2
               (not (eq (setq f1 (math-frac-gcd f1 f2)) 1))
               f1))
      (if (memq (car expr) '(* polar))
          (math-common-constant-factor (nth 1 expr))
        (if (eq (car expr) '/)
            (or (math-common-constant-factor (nth 1 expr))
                (and (Math-integerp (nth 2 expr))
                     (list 'frac 1 (math-abs (nth 2 expr))))))))))

(defun math-cancel-common-factor (expr val)
  (if (memq (car-safe expr) '(+ - cplx sdev))
      (progn
        (setcar (cdr expr) (math-cancel-common-factor (nth 1 expr) val))
        (setcar (cdr (cdr expr)) (math-cancel-common-factor (nth 2 expr) val))
        expr)
    (if (eq (car-safe expr) '*)
        (math-mul (math-cancel-common-factor (nth 1 expr) val) (nth 2 expr))
      (math-div expr val))))

(defun math-frac-gcd (a b)
  (if (Math-zerop a)
      b
    (if (Math-zerop b)
        a
      (if (and (Math-integerp a)
               (Math-integerp b))
          (math-gcd a b)
        (and (Math-integerp a) (setq a (list 'frac a 1)))
        (and (Math-integerp b) (setq b (list 'frac b 1)))
        (math-make-frac (math-gcd (nth 1 a) (nth 1 b))
                        (math-gcd (nth 2 a) (nth 2 b)))))))

(math-defsimplify %
  (math-simplify-mod))

(defun math-simplify-mod ()
  (and (Math-realp (nth 2 expr))
       (Math-posp (nth 2 expr))
       (let ((lin (math-is-linear (nth 1 expr)))
             t1 t2 t3)
         (or (and lin
                  (or (math-negp (car lin))
                      (not (Math-lessp (car lin) (nth 2 expr))))
                  (list '%
                        (list '+
                              (math-mul (nth 1 lin) (nth 2 lin))
                              (math-mod (car lin) (nth 2 expr)))
                        (nth 2 expr)))
             (and lin
                  (not (math-equal-int (nth 1 lin) 1))
                  (math-num-integerp (nth 1 lin))
                  (math-num-integerp (nth 2 expr))
                  (setq t1 (calcFunc-gcd (nth 1 lin) (nth 2 expr)))
                  (not (math-equal-int t1 1))
                  (list '*
                        t1
                        (list '%
                              (list '+
                                    (math-mul (math-div (nth 1 lin) t1)
                                              (nth 2 lin))
                                    (let ((calc-prefer-frac t))
                                      (math-div (car lin) t1)))
                              (math-div (nth 2 expr) t1))))
             (and (math-equal-int (nth 2 expr) 1)
                  (math-known-integerp (if lin
                                           (math-mul (nth 1 lin) (nth 2 lin))
                                         (nth 1 expr)))
                  (if lin (math-mod (car lin) 1) 0))))))

(math-defsimplify (calcFunc-eq calcFunc-neq calcFunc-lt
                               calcFunc-gt calcFunc-leq calcFunc-geq)
  (if (= (length expr) 3)
      (math-simplify-ineq)))

(defun math-simplify-ineq ()
  (let ((np (cdr expr))
        n)
    (while (memq (car-safe (setq n (car np))) '(+ -))
      (math-simplify-add-term (cdr (cdr n)) (cdr (cdr expr))
                              (eq (car n) '-) nil)
      (setq np (cdr n)))
    (math-simplify-add-term np (cdr (cdr expr)) nil (eq np (cdr expr)))
    (math-simplify-divide)
    (let ((signs (math-possible-signs (cons '- (cdr expr)))))
      (or (cond ((eq (car expr) 'calcFunc-eq)
                 (or (and (eq signs 2) 1)
                     (and (memq signs '(1 4 5)) 0)))
                ((eq (car expr) 'calcFunc-neq)
                 (or (and (eq signs 2) 0)
                     (and (memq signs '(1 4 5)) 1)))
                ((eq (car expr) 'calcFunc-lt)
                 (or (and (eq signs 1) 1)
                     (and (memq signs '(2 4 6)) 0)))
                ((eq (car expr) 'calcFunc-gt)
                 (or (and (eq signs 4) 1)
                     (and (memq signs '(1 2 3)) 0)))
                ((eq (car expr) 'calcFunc-leq)
                 (or (and (eq signs 4) 0)
                     (and (memq signs '(1 2 3)) 1)))
                ((eq (car expr) 'calcFunc-geq)
                 (or (and (eq signs 1) 0)
                     (and (memq signs '(2 4 6)) 1))))
          expr))))

(defun math-simplify-add-term (np dp minus lplain)
  (or (math-vectorp (car np))
      (let ((rplain t)
            n d dd temp)
        (while (memq (car-safe (setq n (car np) d (car dp))) '(+ -))
          (setq rplain nil)
          (if (setq temp (math-combine-sum n (nth 2 d)
                                           minus (eq (car d) '+) t))
              (if (or lplain (eq (math-looks-negp temp) minus))
                  (progn
                    (setcar np (setq n (if minus (math-neg temp) temp)))
                    (setcar (cdr (cdr d)) 0))
                (progn
                  (setcar np 0)
                  (setcar (cdr (cdr d)) (setq n (if (eq (car d) '+)
                                                    (math-neg temp)
                                                  temp))))))
          (setq dp (cdr d)))
        (if (setq temp (math-combine-sum n d minus t t))
            (if (or lplain
                    (and (not rplain)
                         (eq (math-looks-negp temp) minus)))
                (progn
                  (setcar np (setq n (if minus (math-neg temp) temp)))
                  (setcar dp 0))
              (progn
                (setcar np 0)
                (setcar dp (setq n (math-neg temp)))))))))

(math-defsimplify calcFunc-sin
  (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
           (nth 1 (nth 1 expr)))
      (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-sin (math-neg (nth 1 expr)))))
      (and (eq calc-angle-mode 'rad)
           (let ((n (math-linear-in (nth 1 expr) '(var pi var-pi))))
             (and n
                  (math-known-sin (car n) (nth 1 n) 120 0))))
      (and (eq calc-angle-mode 'deg)
           (let ((n (math-integer-plus (nth 1 expr))))
             (and n
                  (math-known-sin (car n) (nth 1 n) '(frac 2 3) 0))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
           (list 'calcFunc-sqrt (math-sub 1 (math-sqr (nth 1 (nth 1 expr))))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
           (math-div (nth 1 (nth 1 expr))
                     (list 'calcFunc-sqrt
                           (math-add 1 (math-sqr (nth 1 (nth 1 expr)))))))
      (let ((m (math-should-expand-trig (nth 1 expr))))
        (and m (integerp (car m))
             (let ((n (car m)) (a (nth 1 m)))
               (list '+
                     (list '* (list 'calcFunc-sin (list '* (1- n) a))
                           (list 'calcFunc-cos a))
                     (list '* (list 'calcFunc-cos (list '* (1- n) a))
                           (list 'calcFunc-sin a))))))))

(math-defsimplify calcFunc-cos
  (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
           (nth 1 (nth 1 expr)))
      (and (math-looks-negp (nth 1 expr))
           (list 'calcFunc-cos (math-neg (nth 1 expr))))
      (and (eq calc-angle-mode 'rad)
           (let ((n (math-linear-in (nth 1 expr) '(var pi var-pi))))
             (and n
                  (math-known-sin (car n) (nth 1 n) 120 300))))
      (and (eq calc-angle-mode 'deg)
           (let ((n (math-integer-plus (nth 1 expr))))
             (and n
                  (math-known-sin (car n) (nth 1 n) '(frac 2 3) 300))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
           (list 'calcFunc-sqrt (math-sub 1 (math-sqr (nth 1 (nth 1 expr))))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
           (math-div 1
                     (list 'calcFunc-sqrt
                           (math-add 1 (math-sqr (nth 1 (nth 1 expr)))))))
      (let ((m (math-should-expand-trig (nth 1 expr))))
        (and m (integerp (car m))
             (let ((n (car m)) (a (nth 1 m)))
               (list '-
                     (list '* (list 'calcFunc-cos (list '* (1- n) a))
                           (list 'calcFunc-cos a))
                     (list '* (list 'calcFunc-sin (list '* (1- n) a))
                           (list 'calcFunc-sin a))))))))

(defun math-should-expand-trig (x &optional hyperbolic)
  (let ((m (math-is-multiple x)))
    (and math-living-dangerously
         m (or (and (integerp (car m)) (> (car m) 1))
               (equal (car m) '(frac 1 2)))
         (or math-integrating
             (memq (car-safe (nth 1 m))
                   (if hyperbolic
                       '(calcFunc-arcsinh calcFunc-arccosh calcFunc-arctanh)
                     '(calcFunc-arcsin calcFunc-arccos calcFunc-arctan)))
             (and (eq (car-safe (nth 1 m)) 'calcFunc-ln)
                  (eq hyperbolic 'exp)))
         m)))

(defun math-known-sin (plus n mul off)
  (setq n (math-mul n mul))
  (and (math-num-integerp n)
       (setq n (math-mod (math-add (math-trunc n) off) 240))
       (if (>= n 120)
           (and (setq n (math-known-sin plus (- n 120) 1 0))
                (math-neg n))
         (if (> n 60)
             (setq n (- 120 n)))
         (if (math-zerop plus)
             (and (or calc-symbolic-mode
                      (memq n '(0 20 60)))
                  (cdr (assq n
                             '( (0 . 0)
                                (10 . (/ (calcFunc-sqrt
                                          (- 2 (calcFunc-sqrt 3))) 2))
                                (12 . (/ (- (calcFunc-sqrt 5) 1) 4))
                                (15 . (/ (calcFunc-sqrt
                                          (- 2 (calcFunc-sqrt 2))) 2))
                                (20 . (/ 1 2))
                                (24 . (* (^ (/ 1 2) (/ 3 2))
                                         (calcFunc-sqrt
                                          (- 5 (calcFunc-sqrt 5)))))
                                (30 . (/ (calcFunc-sqrt 2) 2))
                                (36 . (/ (+ (calcFunc-sqrt 5) 1) 4))
                                (40 . (/ (calcFunc-sqrt 3) 2))
                                (45 . (/ (calcFunc-sqrt
                                          (+ 2 (calcFunc-sqrt 2))) 2))
                                (48 . (* (^ (/ 1 2) (/ 3 2))
                                         (calcFunc-sqrt
                                          (+ 5 (calcFunc-sqrt 5)))))
                                (50 . (/ (calcFunc-sqrt
                                          (+ 2 (calcFunc-sqrt 3))) 2))
                                (60 . 1)))))
           (cond ((eq n 0) (math-normalize (list 'calcFunc-sin plus)))
                 ((eq n 60) (math-normalize (list 'calcFunc-cos plus)))
                 (t nil))))))

(math-defsimplify calcFunc-tan
  (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
           (nth 1 (nth 1 expr)))
      (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-tan (math-neg (nth 1 expr)))))
      (and (eq calc-angle-mode 'rad)
           (let ((n (math-linear-in (nth 1 expr) '(var pi var-pi))))
             (and n
                  (math-known-tan (car n) (nth 1 n) 120))))
      (and (eq calc-angle-mode 'deg)
           (let ((n (math-integer-plus (nth 1 expr))))
             (and n
                  (math-known-tan (car n) (nth 1 n) '(frac 2 3)))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
           (math-div (nth 1 (nth 1 expr))
                     (list 'calcFunc-sqrt
                           (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
           (math-div (list 'calcFunc-sqrt
                           (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))
                     (nth 1 (nth 1 expr))))
      (let ((m (math-should-expand-trig (nth 1 expr))))
        (and m
             (if (equal (car m) '(frac 1 2))
                 (math-div (math-sub 1 (list 'calcFunc-cos (nth 1 m)))
                           (list 'calcFunc-sin (nth 1 m)))
               (math-div (list 'calcFunc-sin (nth 1 expr))
                         (list 'calcFunc-cos (nth 1 expr))))))))

(defun math-known-tan (plus n mul)
  (setq n (math-mul n mul))
  (and (math-num-integerp n)
       (setq n (math-mod (math-trunc n) 120))
       (if (> n 60)
           (and (setq n (math-known-tan plus (- 120 n) 1))
                (math-neg n))
         (if (math-zerop plus)
             (and (or calc-symbolic-mode
                      (memq n '(0 30 60)))
                  (cdr (assq n '( (0 . 0)
                                  (10 . (- 2 (calcFunc-sqrt 3)))
                                  (12 . (calcFunc-sqrt
                                         (- 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
                                  (15 . (- (calcFunc-sqrt 2) 1))
                                  (20 . (/ (calcFunc-sqrt 3) 3))
                                  (24 . (calcFunc-sqrt
                                         (- 5 (* 2 (calcFunc-sqrt 5)))))
                                  (30 . 1)
                                  (36 . (calcFunc-sqrt
                                         (+ 1 (* (/ 2 5) (calcFunc-sqrt 5)))))
                                  (40 . (calcFunc-sqrt 3))
                                  (45 . (+ (calcFunc-sqrt 2) 1))
                                  (48 . (calcFunc-sqrt
                                         (+ 5 (* 2 (calcFunc-sqrt 5)))))
                                  (50 . (+ 2 (calcFunc-sqrt 3)))
                                  (60 . (var uinf var-uinf))))))
           (cond ((eq n 0) (math-normalize (list 'calcFunc-tan plus)))
                 ((eq n 60) (math-normalize (list '/ -1
                                                  (list 'calcFunc-tan plus))))
                 (t nil))))))

(math-defsimplify calcFunc-sinh
  (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsinh)
           (nth 1 (nth 1 expr)))
      (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-sinh (math-neg (nth 1 expr)))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccosh)
           math-living-dangerously
           (list 'calcFunc-sqrt (math-sub (math-sqr (nth 1 (nth 1 expr))) 1)))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctanh)
           math-living-dangerously
           (math-div (nth 1 (nth 1 expr))
                     (list 'calcFunc-sqrt
                           (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))))
      (let ((m (math-should-expand-trig (nth 1 expr) t)))
        (and m (integerp (car m))
             (let ((n (car m)) (a (nth 1 m)))
               (if (> n 1)
                   (list '+
                         (list '* (list 'calcFunc-sinh (list '* (1- n) a))
                               (list 'calcFunc-cosh a))
                         (list '* (list 'calcFunc-cosh (list '* (1- n) a))
                               (list 'calcFunc-sinh a)))))))))

(math-defsimplify calcFunc-cosh
  (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccosh)
           (nth 1 (nth 1 expr)))
      (and (math-looks-negp (nth 1 expr))
           (list 'calcFunc-cosh (math-neg (nth 1 expr))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsinh)
           math-living-dangerously
           (list 'calcFunc-sqrt (math-add (math-sqr (nth 1 (nth 1 expr))) 1)))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctanh)
           math-living-dangerously
           (math-div 1
                     (list 'calcFunc-sqrt
                           (math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))))
      (let ((m (math-should-expand-trig (nth 1 expr) t)))
        (and m (integerp (car m))
             (let ((n (car m)) (a (nth 1 m)))
               (if (> n 1)
                   (list '+
                         (list '* (list 'calcFunc-cosh (list '* (1- n) a))
                               (list 'calcFunc-cosh a))
                         (list '* (list 'calcFunc-sinh (list '* (1- n) a))
                               (list 'calcFunc-sinh a)))))))))

(math-defsimplify calcFunc-tanh
  (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-arctanh)
           (nth 1 (nth 1 expr)))
      (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-tanh (math-neg (nth 1 expr)))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsinh)
           math-living-dangerously
           (math-div (nth 1 (nth 1 expr))
                     (list 'calcFunc-sqrt
                           (math-add (math-sqr (nth 1 (nth 1 expr))) 1))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-arccosh)
           math-living-dangerously
           (math-div (list 'calcFunc-sqrt
                           (math-sub (math-sqr (nth 1 (nth 1 expr))) 1))
                     (nth 1 (nth 1 expr))))
      (let ((m (math-should-expand-trig (nth 1 expr) t)))
        (and m
             (if (equal (car m) '(frac 1 2))
                 (math-div (math-sub (list 'calcFunc-cosh (nth 1 m)) 1)
                           (list 'calcFunc-sinh (nth 1 m)))
               (math-div (list 'calcFunc-sinh (nth 1 expr))
                         (list 'calcFunc-cosh (nth 1 expr))))))))

(math-defsimplify calcFunc-arcsin
  (or (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-arcsin (math-neg (nth 1 expr)))))
      (and (eq (nth 1 expr) 1)
           (math-quarter-circle t))
      (and (equal (nth 1 expr) '(frac 1 2))
           (math-div (math-half-circle t) 6))
      (and math-living-dangerously
           (eq (car-safe (nth 1 expr)) 'calcFunc-sin)
           (nth 1 (nth 1 expr)))
      (and math-living-dangerously
           (eq (car-safe (nth 1 expr)) 'calcFunc-cos)
           (math-sub (math-quarter-circle t)
                     (nth 1 (nth 1 expr))))))

(math-defsimplify calcFunc-arccos
  (or (and (eq (nth 1 expr) 0)
           (math-quarter-circle t))
      (and (eq (nth 1 expr) -1)
           (math-half-circle t))
      (and (equal (nth 1 expr) '(frac 1 2))
           (math-div (math-half-circle t) 3))
      (and (equal (nth 1 expr) '(frac -1 2))
           (math-div (math-mul (math-half-circle t) 2) 3))
      (and math-living-dangerously
           (eq (car-safe (nth 1 expr)) 'calcFunc-cos)
           (nth 1 (nth 1 expr)))
      (and math-living-dangerously
           (eq (car-safe (nth 1 expr)) 'calcFunc-sin)
           (math-sub (math-quarter-circle t)
                     (nth 1 (nth 1 expr))))))

(math-defsimplify calcFunc-arctan
  (or (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-arctan (math-neg (nth 1 expr)))))
      (and (eq (nth 1 expr) 1)
           (math-div (math-half-circle t) 4))
      (and math-living-dangerously
           (eq (car-safe (nth 1 expr)) 'calcFunc-tan)
           (nth 1 (nth 1 expr)))))

(math-defsimplify calcFunc-arcsinh
  (or (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-arcsinh (math-neg (nth 1 expr)))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-sinh)
           (or math-living-dangerously
               (math-known-realp (nth 1 (nth 1 expr))))
           (nth 1 (nth 1 expr)))))

(math-defsimplify calcFunc-arccosh
  (and (eq (car-safe (nth 1 expr)) 'calcFunc-cosh)
       (or math-living-dangerously
           (math-known-realp (nth 1 (nth 1 expr))))
       (nth 1 (nth 1 expr))))

(math-defsimplify calcFunc-arctanh
  (or (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-arctanh (math-neg (nth 1 expr)))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-tanh)
           (or math-living-dangerously
               (math-known-realp (nth 1 (nth 1 expr))))
           (nth 1 (nth 1 expr)))))

(math-defsimplify calcFunc-sqrt
  (math-simplify-sqrt))

(defun math-simplify-sqrt ()
  (or (and (eq (car-safe (nth 1 expr)) 'frac)
           (math-div (list 'calcFunc-sqrt (math-mul (nth 1 (nth 1 expr))
                                                    (nth 2 (nth 1 expr))))
                     (nth 2 (nth 1 expr))))
      (let ((fac (if (math-objectp (nth 1 expr))
                     (math-squared-factor (nth 1 expr))
                   (math-common-constant-factor (nth 1 expr)))))
        (and fac (not (eq fac 1))
             (math-mul (math-normalize (list 'calcFunc-sqrt fac))
                       (math-normalize
                        (list 'calcFunc-sqrt
                              (math-cancel-common-factor (nth 1 expr) fac))))))
      (and math-living-dangerously
           (or (and (eq (car-safe (nth 1 expr)) '-)
                    (math-equal-int (nth 1 (nth 1 expr)) 1)
                    (eq (car-safe (nth 2 (nth 1 expr))) '^)
                    (math-equal-int (nth 2 (nth 2 (nth 1 expr))) 2)
                    (or (and (eq (car-safe (nth 1 (nth 2 (nth 1 expr))))
                                 'calcFunc-sin)
                             (list 'calcFunc-cos
                                   (nth 1 (nth 1 (nth 2 (nth 1 expr))))))
                        (and (eq (car-safe (nth 1 (nth 2 (nth 1 expr))))
                                 'calcFunc-cos)
                             (list 'calcFunc-sin
                                   (nth 1 (nth 1 (nth 2 (nth 1 expr))))))))
               (and (eq (car-safe (nth 1 expr)) '-)
                    (math-equal-int (nth 2 (nth 1 expr)) 1)
                    (eq (car-safe (nth 1 (nth 1 expr))) '^)
                    (math-equal-int (nth 2 (nth 1 (nth 1 expr))) 2)
                    (and (eq (car-safe (nth 1 (nth 1 (nth 1 expr))))
                             'calcFunc-cosh)
                         (list 'calcFunc-sinh
                               (nth 1 (nth 1 (nth 1 (nth 1 expr)))))))
               (and (eq (car-safe (nth 1 expr)) '+)
                    (let ((a (nth 1 (nth 1 expr)))
                          (b (nth 2 (nth 1 expr))))
                      (and (or (and (math-equal-int a 1)
                                    (setq a b b (nth 1 (nth 1 expr))))
                               (math-equal-int b 1))
                           (eq (car-safe a) '^)
                           (math-equal-int (nth 2 a) 2)
                           (or (and (eq (car-safe (nth 1 a)) 'calcFunc-sinh)
                                    (list 'calcFunc-cosh (nth 1 (nth 1 a))))
                               (and (eq (car-safe (nth 1 a)) 'calcFunc-tan)
                                    (list '/ 1 (list 'calcFunc-cos
                                                     (nth 1 (nth 1 a)))))))))
               (and (eq (car-safe (nth 1 expr)) '^)
                    (list '^
                          (nth 1 (nth 1 expr))
                          (math-div (nth 2 (nth 1 expr)) 2)))
               (and (eq (car-safe (nth 1 expr)) 'calcFunc-sqrt)
                    (list '^ (nth 1 (nth 1 expr)) (math-div 1 4)))
               (and (memq (car-safe (nth 1 expr)) '(* /))
                    (list (car (nth 1 expr))
                          (list 'calcFunc-sqrt (nth 1 (nth 1 expr)))
                          (list 'calcFunc-sqrt (nth 2 (nth 1 expr)))))
               (and (memq (car-safe (nth 1 expr)) '(+ -))
                    (not (math-any-floats (nth 1 expr)))
                    (let ((f (calcFunc-factors (calcFunc-expand
                                                (nth 1 expr)))))
                      (and (math-vectorp f)
                           (or (> (length f) 2)
                               (> (nth 2 (nth 1 f)) 1))
                           (let ((out 1) (rest 1) (sums 1) fac pow)
                             (while (setq f (cdr f))
                               (setq fac (nth 1 (car f))
                                     pow (nth 2 (car f)))
                               (if (> pow 1)
                                   (setq out (math-mul out (math-pow
                                                            fac (/ pow 2)))
                                         pow (% pow 2)))
                               (if (> pow 0)
                                   (if (memq (car-safe fac) '(+ -))
                                       (setq sums (math-mul-thru sums fac))
                                     (setq rest (math-mul rest fac)))))
                             (and (not (and (eq out 1) (memq rest '(1 -1))))
                                  (math-mul
                                   out
                                   (list 'calcFunc-sqrt
                                         (math-mul sums rest))))))))))))

;;; Rather than factoring x into primes, just check for the first ten primes.
(defun math-squared-factor (x)
  (if (Math-integerp x)
      (let ((prsqr '(4 9 25 49 121 169 289 361 529 841))
            (fac 1)
            res)
        (while prsqr
          (if (eq (cdr (setq res (math-idivmod x (car prsqr)))) 0)
              (setq x (car res)
                    fac (math-mul fac (car prsqr)))
            (setq prsqr (cdr prsqr))))
        fac)))

(math-defsimplify calcFunc-exp
  (math-simplify-exp (nth 1 expr)))

(defun math-simplify-exp (x)
  (or (and (eq (car-safe x) 'calcFunc-ln)
           (nth 1 x))
      (and math-living-dangerously
           (or (and (eq (car-safe x) 'calcFunc-arcsinh)
                    (math-add (nth 1 x)
                              (list 'calcFunc-sqrt
                                    (math-add (math-sqr (nth 1 x)) 1))))
               (and (eq (car-safe x) 'calcFunc-arccosh)
                    (math-add (nth 1 x)
                              (list 'calcFunc-sqrt
                                    (math-sub (math-sqr (nth 1 x)) 1))))
               (and (eq (car-safe x) 'calcFunc-arctanh)
                    (math-div (list 'calcFunc-sqrt (math-add 1 (nth 1 x)))
                              (list 'calcFunc-sqrt (math-sub 1 (nth 1 x)))))
               (let ((m (math-should-expand-trig x 'exp)))
                 (and m (integerp (car m))
                      (list '^ (list 'calcFunc-exp (nth 1 m)) (car m))))))
      (and calc-symbolic-mode
           (math-known-imagp x)
           (let* ((ip (calcFunc-im x))
                  (n (math-linear-in ip '(var pi var-pi)))
                  s c)
             (and n
                  (setq s (math-known-sin (car n) (nth 1 n) 120 0))
                  (setq c (math-known-sin (car n) (nth 1 n) 120 300))
                  (list '+ c (list '* s '(var i var-i))))))))

(math-defsimplify calcFunc-ln
  (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-exp)
           (or math-living-dangerously
               (math-known-realp (nth 1 (nth 1 expr))))
           (nth 1 (nth 1 expr)))
      (and (eq (car-safe (nth 1 expr)) '^)
           (equal (nth 1 (nth 1 expr)) '(var e var-e))
           (or math-living-dangerously
               (math-known-realp (nth 2 (nth 1 expr))))
           (nth 2 (nth 1 expr)))
      (and calc-symbolic-mode
           (math-known-negp (nth 1 expr))
           (math-add (list 'calcFunc-ln (math-neg (nth 1 expr)))
                     '(* (var pi var-pi) (var i var-i))))
      (and calc-symbolic-mode
           (math-known-imagp (nth 1 expr))
           (let* ((ip (calcFunc-im (nth 1 expr)))
                  (ips (math-possible-signs ip)))
             (or (and (memq ips '(4 6))
                      (math-add (list 'calcFunc-ln ip)
                                '(/ (* (var pi var-pi) (var i var-i)) 2)))
                 (and (memq ips '(1 3))
                      (math-sub (list 'calcFunc-ln (math-neg ip))
                                '(/ (* (var pi var-pi) (var i var-i)) 2))))))))

(math-defsimplify ^
  (math-simplify-pow))

(defun math-simplify-pow ()
  (or (and math-living-dangerously
           (or (and (eq (car-safe (nth 1 expr)) '^)
                    (list '^
                          (nth 1 (nth 1 expr))
                          (math-mul (nth 2 expr) (nth 2 (nth 1 expr)))))
               (and (eq (car-safe (nth 1 expr)) 'calcFunc-sqrt)
                    (list '^
                          (nth 1 (nth 1 expr))
                          (math-div (nth 2 expr) 2)))
               (and (memq (car-safe (nth 1 expr)) '(* /))
                    (list (car (nth 1 expr))
                          (list '^ (nth 1 (nth 1 expr)) (nth 2 expr))
                          (list '^ (nth 2 (nth 1 expr)) (nth 2 expr))))))
      (and (math-equal-int (nth 1 expr) 10)
           (eq (car-safe (nth 2 expr)) 'calcFunc-log10)
           (nth 1 (nth 2 expr)))
      (and (equal (nth 1 expr) '(var e var-e))
           (math-simplify-exp (nth 2 expr)))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-exp)
           (not math-integrating)
           (list 'calcFunc-exp (math-mul (nth 1 (nth 1 expr)) (nth 2 expr))))
      (and (equal (nth 1 expr) '(var i var-i))
           (math-imaginary-i)
           (math-num-integerp (nth 2 expr))
           (let ((x (math-mod (math-trunc (nth 2 expr)) 4)))
             (cond ((eq x 0) 1)
                   ((eq x 1) (nth 1 expr))
                   ((eq x 2) -1)
                   ((eq x 3) (math-neg (nth 1 expr))))))
      (and math-integrating
           (integerp (nth 2 expr))
           (>= (nth 2 expr) 2)
           (or (and (eq (car-safe (nth 1 expr)) 'calcFunc-cos)
                    (math-mul (math-pow (nth 1 expr) (- (nth 2 expr) 2))
                              (math-sub 1
                                        (math-sqr
                                         (list 'calcFunc-sin
                                               (nth 1 (nth 1 expr)))))))
               (and (eq (car-safe (nth 1 expr)) 'calcFunc-cosh)
                    (math-mul (math-pow (nth 1 expr) (- (nth 2 expr) 2))
                              (math-add 1
                                        (math-sqr
                                         (list 'calcFunc-sinh
                                               (nth 1 (nth 1 expr)))))))))
      (and (eq (car-safe (nth 2 expr)) 'frac)
           (Math-ratp (nth 1 expr))
           (Math-posp (nth 1 expr))
           (if (equal (nth 2 expr) '(frac 1 2))
               (list 'calcFunc-sqrt (nth 1 expr))
             (let ((flr (math-floor (nth 2 expr))))
               (and (not (Math-zerop flr))
                    (list '* (list '^ (nth 1 expr) flr)
                          (list '^ (nth 1 expr)
                                (math-sub (nth 2 expr) flr)))))))
      (and (eq (math-quarter-integer (nth 2 expr)) 2)
           (let ((temp (math-simplify-sqrt)))
             (and temp
                  (list '^ temp (math-mul (nth 2 expr) 2)))))))

(math-defsimplify calcFunc-log10
  (and (eq (car-safe (nth 1 expr)) '^)
       (math-equal-int (nth 1 (nth 1 expr)) 10)
       (or math-living-dangerously
           (math-known-realp (nth 2 (nth 1 expr))))
       (nth 2 (nth 1 expr))))


(math-defsimplify calcFunc-erf
  (or (and (math-looks-negp (nth 1 expr))
           (math-neg (list 'calcFunc-erf (math-neg (nth 1 expr)))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-conj)
           (list 'calcFunc-conj (list 'calcFunc-erf (nth 1 (nth 1 expr)))))))

(math-defsimplify calcFunc-erfc
  (or (and (math-looks-negp (nth 1 expr))
           (math-sub 2 (list 'calcFunc-erfc (math-neg (nth 1 expr)))))
      (and (eq (car-safe (nth 1 expr)) 'calcFunc-conj)
           (list 'calcFunc-conj (list 'calcFunc-erfc (nth 1 (nth 1 expr)))))))


(defun math-linear-in (expr term &optional always)
  (if (math-expr-contains expr term)
      (let* ((calc-prefer-frac t)
             (p (math-is-polynomial expr term 1)))
        (and (cdr p)
             p))
    (and always (list expr 0))))

(defun math-multiple-of (expr term)
  (let ((p (math-linear-in expr term)))
    (and p
         (math-zerop (car p))
         (nth 1 p))))

; not perfect, but it'll do
(defun math-integer-plus (expr)
  (cond ((Math-integerp expr)
         (list 0 expr))
        ((and (memq (car expr) '(+ -))
              (Math-integerp (nth 1 expr)))
         (list (if (eq (car expr) '+) (nth 2 expr) (math-neg (nth 2 expr)))
               (nth 1 expr)))
        ((and (memq (car expr) '(+ -))
              (Math-integerp (nth 2 expr)))
         (list (nth 1 expr)
               (if (eq (car expr) '+) (nth 2 expr) (math-neg (nth 2 expr)))))
        (t nil)))

(defun math-is-linear (expr &optional always)
  (let ((offset nil)
        (coef nil))
    (if (eq (car-safe expr) '+)
        (if (Math-objectp (nth 1 expr))
            (setq offset (nth 1 expr)
                  expr (nth 2 expr))
          (if (Math-objectp (nth 2 expr))
              (setq offset (nth 2 expr)
                    expr (nth 1 expr))))
      (if (eq (car-safe expr) '-)
          (if (Math-objectp (nth 1 expr))
              (setq offset (nth 1 expr)
                    expr (math-neg (nth 2 expr)))
            (if (Math-objectp (nth 2 expr))
                (setq offset (math-neg (nth 2 expr))
                      expr (nth 1 expr))))))
    (setq coef (math-is-multiple expr always))
    (if offset
        (list offset (or (car coef) 1) (or (nth 1 coef) expr))
      (if coef
          (cons 0 coef)))))

(defun math-is-multiple (expr &optional always)
  (or (if (eq (car-safe expr) '*)
          (if (Math-objectp (nth 1 expr))
              (list (nth 1 expr) (nth 2 expr)))
        (if (eq (car-safe expr) '/)
            (if (and (Math-objectp (nth 1 expr))
                     (not (math-equal-int (nth 1 expr) 1)))
                (list (nth 1 expr) (math-div 1 (nth 2 expr)))
              (if (Math-objectp (nth 2 expr))
                  (list (math-div 1 (nth 2 expr)) (nth 1 expr))
                (let ((res (math-is-multiple (nth 1 expr))))
                  (if res
                      (list (car res)
                            (math-div (nth 2 (nth 1 expr)) (nth 2 expr)))
                    (setq res (math-is-multiple (nth 2 expr)))
                    (if res
                        (list (math-div 1 (car res))
                              (math-div (nth 1 expr)
                                        (nth 2 (nth 2 expr)))))))))
          (if (eq (car-safe expr) 'neg)
              (list -1 (nth 1 expr)))))
      (if (Math-objvecp expr)
          (and (eq always 1)
               (list expr 1))
        (and always 
             (list 1 expr)))))

(defun calcFunc-lin (expr &optional var)
  (if var
      (let ((res (math-linear-in expr var t)))
        (or res (math-reject-arg expr "Linear term expected"))
        (list 'vec (car res) (nth 1 res) var))
    (let ((res (math-is-linear expr t)))
      (or res (math-reject-arg expr "Linear term expected"))
      (cons 'vec res))))

(defun calcFunc-linnt (expr &optional var)
  (if var
      (let ((res (math-linear-in expr var)))
        (or res (math-reject-arg expr "Linear term expected"))
        (list 'vec (car res) (nth 1 res) var))
    (let ((res (math-is-linear expr)))
      (or res (math-reject-arg expr "Linear term expected"))
      (cons 'vec res))))

(defun calcFunc-islin (expr &optional var)
  (if (and (Math-objvecp expr) (not var))
      0
    (calcFunc-lin expr var)
    1))

(defun calcFunc-islinnt (expr &optional var)
  (if (Math-objvecp expr)
      0
    (calcFunc-linnt expr var)
    1))




;;; Simple operations on expressions.

;;; Return number of occurrences of thing in expr, or nil if none.
(defun math-expr-contains-count (expr thing)
  (cond ((equal expr thing) 1)
        ((Math-primp expr) nil)
        (t
         (let ((num 0))
           (while (setq expr (cdr expr))
             (setq num (+ num (or (math-expr-contains-count
                                   (car expr) thing) 0))))
           (and (> num 0)
                num)))))

(defun math-expr-contains (expr thing)
  (cond ((equal expr thing) 1)
        ((Math-primp expr) nil)
        (t
         (while (and (setq expr (cdr expr))
                     (not (math-expr-contains (car expr) thing))))
         expr)))

;;; Return non-nil if any variable of thing occurs in expr.
(defun math-expr-depends (expr thing)
  (if (Math-primp thing)
      (and (eq (car-safe thing) 'var)
           (math-expr-contains expr thing))
    (while (and (setq thing (cdr thing))
                (not (math-expr-depends expr (car thing)))))
    thing))

;;; Substitute all occurrences of old for new in expr (non-destructive).
(defun math-expr-subst (expr old new)
  (math-expr-subst-rec expr))

(defalias 'calcFunc-subst 'math-expr-subst)

(defun math-expr-subst-rec (expr)
  (cond ((equal expr old) new)
        ((Math-primp expr) expr)
        ((memq (car expr) '(calcFunc-deriv
                            calcFunc-tderiv))
         (if (= (length expr) 2)
             (if (equal (nth 1 expr) old)
                 (append expr (list new))
               expr)
           (list (car expr) (nth 1 expr)
                 (math-expr-subst-rec (nth 2 expr)))))
        (t
         (cons (car expr)
               (mapcar 'math-expr-subst-rec (cdr expr))))))

;;; Various measures of the size of an expression.
(defun math-expr-weight (expr)
  (if (Math-primp expr)
      1
    (let ((w 1))
      (while (setq expr (cdr expr))
        (setq w (+ w (math-expr-weight (car expr)))))
      w)))

(defun math-expr-height (expr)
  (if (Math-primp expr)
      0
    (let ((h 0))
      (while (setq expr (cdr expr))
        (setq h (max h (math-expr-height (car expr)))))
      (1+ h))))




;;; Polynomial operations (to support the integrator and solve-for).

(defun calcFunc-collect (expr base)
  (let ((p (math-is-polynomial expr base 50 t)))
    (if (cdr p)
        (math-normalize   ; fix selection bug
         (math-build-polynomial-expr p base))
      expr)))

;;; If expr is of the form "a + bx + cx^2 + ...", return the list (a b c ...),
;;; else return nil if not in polynomial form.  If "loose", coefficients
;;; may contain x, e.g., sin(x) + cos(x) x^2 is a loose polynomial in x.
(defun math-is-polynomial (expr var &optional degree loose)
  (let* ((math-poly-base-variable (if loose
                                      (if (eq loose 'gen) var '(var XXX XXX))
                                    math-poly-base-variable))
         (poly (math-is-poly-rec expr math-poly-neg-powers)))
    (and (or (null degree)
             (<= (length poly) (1+ degree)))
         poly)))

(defun math-is-poly-rec (expr negpow)
  (math-poly-simplify
   (or (cond ((or (equal expr var)
                  (eq (car-safe expr) '^))
              (let ((pow 1)
                    (expr expr))
                (or (equal expr var)
                    (setq pow (nth 2 expr)
                          expr (nth 1 expr)))
                (or (eq math-poly-mult-powers 1)
                    (setq pow (let ((m (math-is-multiple pow 1)))
                                (and (eq (car-safe (car m)) 'cplx)
                                     (Math-zerop (nth 1 (car m)))
                                     (setq m (list (nth 2 (car m))
                                                   (math-mul (nth 1 m)
                                                             '(var i var-i)))))
                                (and (if math-poly-mult-powers
                                         (equal math-poly-mult-powers
                                                (nth 1 m))
                                       (setq math-poly-mult-powers (nth 1 m)))
                                     (or (equal expr var)
                                         (eq math-poly-mult-powers 1))
                                     (car m)))))
                (if (consp pow)
                    (progn
                      (setq pow (math-to-simple-fraction pow))
                      (and (eq (car-safe pow) 'frac)
                           math-poly-frac-powers
                           (equal expr var)
                           (setq math-poly-frac-powers
                                 (calcFunc-lcm math-poly-frac-powers
                                               (nth 2 pow))))))
                (or (memq math-poly-frac-powers '(1 nil))
                    (setq pow (math-mul pow math-poly-frac-powers)))
                (if (integerp pow)
                    (if (and (= pow 1)
                             (equal expr var))
                        (list 0 1)
                      (if (natnump pow)
                          (let ((p1 (if (equal expr var)
                                        (list 0 1)
                                      (math-is-poly-rec expr nil)))
                                (n pow)
                                (accum (list 1)))
                            (and p1
                                 (or (null degree)
                                     (<= (* (1- (length p1)) n) degree))
                                 (progn
                                   (while (>= n 1)
                                     (setq accum (math-poly-mul accum p1)
                                           n (1- n)))
                                   accum)))
                        (and negpow
                             (math-is-poly-rec expr nil)
                             (setq math-poly-neg-powers
                                   (cons (math-pow expr (- pow))
                                         math-poly-neg-powers))
                             (list (list '^ expr pow))))))))
             ((Math-objectp expr)
              (list expr))
             ((memq (car expr) '(+ -))
              (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
                (and p1
                     (let ((p2 (math-is-poly-rec (nth 2 expr) negpow)))
                       (and p2
                            (math-poly-mix p1 1 p2
                                           (if (eq (car expr) '+) 1 -1)))))))
             ((eq (car expr) 'neg)
              (mapcar 'math-neg (math-is-poly-rec (nth 1 expr) negpow)))
             ((eq (car expr) '*)
              (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
                (and p1
                     (let ((p2 (math-is-poly-rec (nth 2 expr) negpow)))
                       (and p2
                            (or (null degree)
                                (<= (- (+ (length p1) (length p2)) 2) degree))
                            (math-poly-mul p1 p2))))))
             ((eq (car expr) '/)
              (and (or (not (math-poly-depends (nth 2 expr) var))
                       (and negpow
                            (math-is-poly-rec (nth 2 expr) nil)
                            (setq math-poly-neg-powers
                                  (cons (nth 2 expr) math-poly-neg-powers))))
                   (not (Math-zerop (nth 2 expr)))
                   (let ((p1 (math-is-poly-rec (nth 1 expr) negpow)))
                     (mapcar (function (lambda (x) (math-div x (nth 2 expr))))
                             p1))))
             ((and (eq (car expr) 'calcFunc-exp)
                   (equal var '(var e var-e)))
              (math-is-poly-rec (list '^ var (nth 1 expr)) negpow))
             ((and (eq (car expr) 'calcFunc-sqrt)
                   math-poly-frac-powers)
              (math-is-poly-rec (list '^ (nth 1 expr) '(frac 1 2)) negpow))
             (t nil))
       (and (or (not (math-poly-depends expr var))
                loose)
            (not (eq (car expr) 'vec))
            (list expr)))))

;;; Check if expr is a polynomial in var; if so, return its degree.
(defun math-polynomial-p (expr var)
  (cond ((equal expr var) 1)
        ((Math-primp expr) 0)
        ((memq (car expr) '(+ -))
         (let ((p1 (math-polynomial-p (nth 1 expr) var))
               p2)
           (and p1 (setq p2 (math-polynomial-p (nth 2 expr) var))
                (max p1 p2))))
        ((eq (car expr) '*)
         (let ((p1 (math-polynomial-p (nth 1 expr) var))
               p2)
           (and p1 (setq p2 (math-polynomial-p (nth 2 expr) var))
                (+ p1 p2))))
        ((eq (car expr) 'neg)
         (math-polynomial-p (nth 1 expr) var))
        ((and (eq (car expr) '/)
              (not (math-poly-depends (nth 2 expr) var)))
         (math-polynomial-p (nth 1 expr) var))
        ((and (eq (car expr) '^)
              (natnump (nth 2 expr)))
         (let ((p1 (math-polynomial-p (nth 1 expr) var)))
           (and p1 (* p1 (nth 2 expr)))))
        ((math-poly-depends expr var) nil)
        (t 0)))

(defun math-poly-depends (expr var)
  (if math-poly-base-variable
      (math-expr-contains expr math-poly-base-variable)
    (math-expr-depends expr var)))

;;; Find the variable (or sub-expression) which is the base of polynomial expr.
(defun math-polynomial-base (mpb-top-expr &optional mpb-pred)
  (or mpb-pred
      (setq mpb-pred (function (lambda (base) (math-polynomial-p
                                               mpb-top-expr base)))))
  (or (let ((const-ok nil))
        (math-polynomial-base-rec mpb-top-expr))
      (let ((const-ok t))
        (math-polynomial-base-rec mpb-top-expr))))

(defun math-polynomial-base-rec (mpb-expr)
  (and (not (Math-objvecp mpb-expr))
       (or (and (memq (car mpb-expr) '(+ - *))
                (or (math-polynomial-base-rec (nth 1 mpb-expr))
                    (math-polynomial-base-rec (nth 2 mpb-expr))))
           (and (memq (car mpb-expr) '(/ neg))
                (math-polynomial-base-rec (nth 1 mpb-expr)))
           (and (eq (car mpb-expr) '^)
                (math-polynomial-base-rec (nth 1 mpb-expr)))
           (and (eq (car mpb-expr) 'calcFunc-exp)
                (math-polynomial-base-rec '(var e var-e)))
           (and (or const-ok (math-expr-contains-vars mpb-expr))
                (funcall mpb-pred mpb-expr)
                mpb-expr))))

;;; Return non-nil if expr refers to any variables.
(defun math-expr-contains-vars (expr)
  (or (eq (car-safe expr) 'var)
      (and (not (Math-primp expr))
           (progn
             (while (and (setq expr (cdr expr))
                         (not (math-expr-contains-vars (car expr)))))
             expr))))

;;; Simplify a polynomial in list form by stripping off high-end zeros.
;;; This always leaves the constant part, i.e., nil->nil and nonnil->nonnil.
(defun math-poly-simplify (p)
  (and p
       (if (Math-zerop (nth (1- (length p)) p))
           (let ((pp (copy-sequence p)))
             (while (and (cdr pp)
                         (Math-zerop (nth (1- (length pp)) pp)))
               (setcdr (nthcdr (- (length pp) 2) pp) nil))
             pp)
         p)))

;;; Compute ac*a + bc*b for polynomials in list form a, b and
;;; coefficients ac, bc.  Result may be unsimplified.
(defun math-poly-mix (a ac b bc)
  (and (or a b)
       (cons (math-add (math-mul (or (car a) 0) ac)
                       (math-mul (or (car b) 0) bc))
             (math-poly-mix (cdr a) ac (cdr b) bc))))

(defun math-poly-zerop (a)
  (or (null a)
      (and (null (cdr a)) (Math-zerop (car a)))))

;;; Multiply two polynomials in list form.
(defun math-poly-mul (a b)
  (and a b
       (math-poly-mix b (car a)
                      (math-poly-mul (cdr a) (cons 0 b)) 1)))

;;; Build an expression from a polynomial list.
(defun math-build-polynomial-expr (p var)
  (if p
      (if (Math-numberp var)
          (math-with-extra-prec 1
            (let* ((rp (reverse p))
                   (accum (car rp)))
              (while (setq rp (cdr rp))
                (setq accum (math-add (car rp) (math-mul accum var))))
              accum))
        (let* ((rp (reverse p))
               (n (1- (length rp)))
               (accum (math-mul (car rp) (math-pow var n)))
               term)
          (while (setq rp (cdr rp))
            (setq n (1- n))
            (or (math-zerop (car rp))
                (setq accum (list (if (math-looks-negp (car rp)) '- '+)
                                  accum
                                  (math-mul (if (math-looks-negp (car rp))
                                                (math-neg (car rp))
                                              (car rp))
                                            (math-pow var n))))))
          accum))
    0))


(defun math-to-simple-fraction (f)
  (or (and (eq (car-safe f) 'float)
           (or (and (>= (nth 2 f) 0)
                    (math-scale-int (nth 1 f) (nth 2 f)))
               (and (integerp (nth 1 f))
                    (> (nth 1 f) -1000)
                    (< (nth 1 f) 1000)
                    (math-make-frac (nth 1 f)
                                    (math-scale-int 1 (- (nth 2 f)))))))
      f))

;;; calc-alg.el ends here