root/trunk/lisp/calc/calc-alg.el

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

;; Copyright (C) 1990, 1991, 1992, 1993, 2001, 2002, 2003, 2004,
;;   2005, 2006, 2007, 2008 Free Software Foundation, Inc.

;; Author: David Gillespie <daveg@synaptics.com>
;; Maintainer: Jay Belanger  <jay.p.belanger@gmail.com>

;; This file is part of GNU Emacs.

;; GNU Emacs is free software; you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation; either version 3, or (at your option)
;; any later version.

;; GNU Emacs is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
;; GNU General Public License for more details.

;; You should have received a copy of the GNU General Public License
;; along with GNU Emacs; see the file COPYING.  If not, write to the
;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
;; Boston, MA 02110-1301, USA.

;;; Commentary:

;;; Code:

;; This file is autoloaded from calc-ext.el.

(require 'calc-ext)
(require 'calc-macs)

;;; 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)))))))

;;; Write out powers (a*b*...)^n as a*b*...*a*b*...
(defun calcFunc-powerexpand (expr)
  (math-normalize (math-map-tree 'math-powerexpand expr)))

(defun math-powerexpand (expr)
  (if (eq (car-safe expr) '^)
      (let ((n (nth 2 expr)))
        (cond ((and (integerp n)
                    (> n 0))
               (let ((i 1)
                     (a (nth 1 expr))
                     (prod (nth 1 expr)))
                 (while (< i n)
                   (setq prod (math-mul prod a))
                   (setq i (1+ i)))
                 prod))
              ((and (integerp n)
                    (< n 0))
               (let ((i -1)
                     (a (math-pow (nth 1 expr) -1))
                     (prod (math-pow (nth 1 expr) -1)))
                 (while (> i n)
                   (setq prod (math-mul a prod))
                   (setq i (1- i)))
                 prod))
              (t
               expr)))
    expr))

(defun calc-powerexpand ()
  (interactive)
  (calc-slow-wrapper
   (calc-enter-result 1 "pexp"
                      (calcFunc-powerexpand (calc-top-n 1)))))

(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
   (let ((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))

;; math-eval-rules-cache and math-eval-rules-cache-other are
;; declared in calc.el, but are used here by math-recompile-eval-rules.
(defvar math-eval-rules-cache)
(defvar math-eval-rules-cache-other)

(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)

;; math-top-only is local to math-simplify, but is used by 
;; math-simplify-step, which is called by math-simplify.
(defvar math-top-only)

(defun math-simplify (top-expr)
  (let ((math-simplifying t)
        (math-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 math-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 math-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)))


(defmacro math-defsimplify (funcs &rest code)
  (append '(progn)
          (mapcar (function
                   (lambda (func)
                     (list 'put (list 'quote func) ''math-simplify
                           (list 'nconc
                                 (list 'get (list 'quote func) ''math-simplify)
                                 (list 'list
                                       (list 'function
                                             (append '(lambda (math-simplify-expr))
                                                     code)))))))
                  (if (symbolp funcs) (list funcs) funcs))))
(put 'math-defsimplify 'lisp-indent-hook 1)

;; The function created by math-defsimplify uses the variable
;; math-simplify-expr, and so is used by functions in math-defsimplify
(defvar math-simplify-expr)

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

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

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

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

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

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

;; The variables math-simplify-divisor-nover and math-simplify-divisor-dover
;; are local variables for math-simplify-divisor, but are used by
;; math-simplify-one-divisor.
(defvar math-simplify-divisor-nover)
(defvar math-simplify-divisor-dover)

(defun math-simplify-divisor (np dp math-simplify-divisor-nover 
                                 math-simplify-divisor-dover)
  (cond ((eq (car-safe (car dp)) '/)
         (math-simplify-divisor np (cdr (car dp)) 
                                math-simplify-divisor-nover 
                                math-simplify-divisor-dover)
         (and (math-known-scalarp (nth 1 (car dp)) t)
              (math-simplify-divisor np (cdr (cdr (car dp)))
                                     math-simplify-divisor-nover 
                                     (not math-simplify-divisor-dover))))
        ((or (or (eq (car math-simplify-expr) '/)
                 (let ((signs (math-possible-signs (car np))))
                   (or (memq signs '(1 4))
                       (and (memq (car math-simplify-expr) '(calcFunc-eq calcFunc-neq))
                            (eq signs 5))
                       math-living-dangerously)))
             (math-numberp (car np)))
         (let (d
               (safe t) 
               (scalar (math-known-scalarp (car np))))
           (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)
  (let ((temp (math-combine-prod (car np) (car dp) math-simplify-divisor-nover 
                                 math-simplify-divisor-dover t))
        op)
    (if temp 
        (progn
          (and (not (memq (car math-simplify-expr) '(/ calcFunc-eq calcFunc-neq)))
               (math-known-negp (car dp))
               (setq op (assq (car math-simplify-expr) calc-tweak-eqn-table))
               (setcar math-simplify-expr (nth 1 op)))
          (setcar np (if math-simplify-divisor-nover (math-div 1 temp) temp))
          (setcar dp 1))
      (and math-simplify-divisor-dover (not math-simplify-divisor-nover) 
           (eq (car math-simplify-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 math-simplify-expr))
       (Math-posp (nth 2 math-simplify-expr))
       (let ((lin (math-is-linear (nth 1 math-simplify-expr)))
             t1 t2 t3)
         (or (and lin
                  (or (math-negp (car lin))
                      (not (Math-lessp (car lin) (nth 2 math-simplify-expr))))
                  (list '%
                        (list '+
                              (math-mul (nth 1 lin) (nth 2 lin))
                              (math-mod (car lin) (nth 2 math-simplify-expr)))
                        (nth 2 math-simplify-expr)))
             (and lin
                  (not (math-equal-int (nth 1 lin) 1))
                  (math-num-integerp (nth 1 lin))
                  (math-num-integerp (nth 2 math-simplify-expr))
                  (setq t1 (calcFunc-gcd (nth 1 lin) (nth 2 math-simplify-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 math-simplify-expr) t1))))
             (and (math-equal-int (nth 2 math-simplify-expr) 1)
                  (math-known-integerp (if lin
                                           (math-mul (nth 1 lin) (nth 2 lin))
                                         (nth 1 math-simplify-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 math-simplify-expr) 3)
      (math-simplify-ineq)))

(defun math-simplify-ineq ()
  (let ((np (cdr math-simplify-expr))
        n)
    (while (memq (car-safe (setq n (car np))) '(+ -))
      (math-simplify-add-term (cdr (cdr n)) (cdr (cdr math-simplify-expr))
                              (eq (car n) '-) nil)
      (setq np (cdr n)))
    (math-simplify-add-term np (cdr (cdr math-simplify-expr)) nil 
                            (eq np (cdr math-simplify-expr)))
    (math-simplify-divide)
    (let ((signs (math-possible-signs (cons '- (cdr math-simplify-expr)))))
      (or (cond ((eq (car math-simplify-expr) 'calcFunc-eq)
                 (or (and (eq signs 2) 1)
                     (and (memq signs '(1 4 5)) 0)))
                ((eq (car math-simplify-expr) 'calcFunc-neq)
                 (or (and (eq signs 2) 0)
                     (and (memq signs '(1 4 5)) 1)))
                ((eq (car math-simplify-expr) 'calcFunc-lt)
                 (or (and (eq signs 1) 1)
                     (and (memq signs '(2 4 6)) 0)))
                ((eq (car math-simplify-expr) 'calcFunc-gt)
                 (or (and (eq signs 4) 1)
                     (and (memq signs '(1 2 3)) 0)))
                ((eq (car math-simplify-expr) 'calcFunc-leq)
                 (or (and (eq signs 4) 0)
                     (and (memq signs '(1 2 3)) 1)))
                ((eq (car math-simplify-expr) 'calcFunc-geq)
                 (or (and (eq signs 1) 0)
                     (and (memq signs '(2 4 6)) 1))))
          math-simplify-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 math-simplify-expr)) 'calcFunc-arcsin)
           (nth 1 (nth 1 math-simplify-expr)))
      (and (math-looks-negp (nth 1 math-simplify-expr))
           (math-neg (list 'calcFunc-sin (math-neg (nth 1 math-simplify-expr)))))
      (and (eq calc-angle-mode 'rad)
           (let ((n (math-linear-in (nth 1 math-simplify-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 math-simplify-expr))))
             (and n
                  (math-known-sin (car n) (nth 1 n) '(frac 2 3) 0))))
      (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arccos)
           (list 'calcFunc-sqrt (math-sub 1 (math-sqr 
                                             (nth 1 (nth 1 math-simplify-expr))))))
      (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
           (math-div (nth 1 (nth 1 math-simplify-expr))
                     (list 'calcFunc-sqrt
                           (math-add 1 (math-sqr 
                                        (nth 1 (nth 1 math-simplify-expr)))))))
      (let ((m (math-should-expand-trig (nth 1 math-simplify-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 math-simplify-expr)) 'calcFunc-arccos)
           (nth 1 (nth 1 math-simplify-expr)))
      (and (math-looks-negp (nth 1 math-simplify-expr))
           (list 'calcFunc-cos (math-neg (nth 1 math-simplify-expr))))
      (and (eq calc-angle-mode 'rad)
           (let ((n (math-linear-in (nth 1 math-simplify-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 math-simplify-expr))))
             (and n
                  (math-known-sin (car n) (nth 1 n) '(frac 2 3) 300))))
      (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
           (list 'calcFunc-sqrt 
                 (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr))))))
      (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arctan)
           (math-div 1
                     (list 'calcFunc-sqrt
                           (math-add 1 
                                     (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
      (let ((m (math-should-expand-trig (nth 1 math-simplify-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))))))))

(math-defsimplify calcFunc-sec
  (or (and (math-looks-negp (nth 1 math-simplify-expr))
           (list 'calcFunc-sec (math-neg (nth 1 math-simplify-expr))))
      (and (eq calc-angle-mode 'rad)
           (let ((n (math-linear-in (nth 1 math-simplify-expr) '(var pi var-pi))))
             (and n
                  (math-div 1 (math-known-sin (car n) (nth 1 n) 120 300)))))
      (and (eq calc-angle-mode 'deg)
           (let ((n (math-integer-plus (nth 1 math-simplify-expr))))
             (and n
                  (math-div 1 (math-known-sin (car n) (nth 1 n) '(frac 2 3) 300)))))
      (and (eq (car-safe (nth 1 math-simplify-expr)) 'calcFunc-arcsin)
           (math-div 
            1
            (list 'calcFunc-sqrt 
                  (math-sub 1 (math-sqr (nth 1 (nth 1 math-simplify-expr)))))))
      (and