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calc-alg.el
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;;; calc-alg.el --- algebraic functions for Calc -*- lexical-binding:t -*-
;; Copyright (C) 1990-1993, 2001-2020 Free Software Foundation, Inc.
;; Author: David Gillespie <[email protected]>
;; 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 of the License, 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. If not, see <https://www.gnu.org/licenses/>.
;;; Commentary:
;;; Code:
;; This file is autoloaded from calc-ext.el.
(require 'calc-ext)
(require 'calc-macs)
;;; Algebra commands.
(defvar math-simplify-only)
(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))))))
(defvar calc-simplify-mode)
(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
(let ((top (calc-top-n 1)))
(if (calc-is-inverse)
(setq top
(let ((calc-simplify-mode nil))
(math-normalize (math-trig-rewrite top)))))
(if (calc-is-hyperbolic)
(setq top
(let ((calc-simplify-mode nil))
(math-normalize (math-hyperbolic-trig-rewrite top)))))
(calc-with-default-simplification
(calc-enter-result 1 "simp" (math-simplify top))))))
(defun calc-simplify-extended ()
(interactive)
(calc-slow-wrapper
(calc-with-default-simplification
(calc-enter-result 1 "esmp" (math-simplify-extended (calc-top-n 1))))))
(defvar math-expand-formulas)
(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 "$") (null var))
(calc-enter-result 2 "clct" (cons 'calcFunc-collect
(calc-top-list-n 2)))
(let ((var (math-read-expr var)))
(if (eq (car-safe var) 'error)
(error "Bad format in expression: %s" (nth 1 var)))
(calc-enter-result 1 "clct" (list 'calcFunc-collect
(calc-top-n 1)
var))))))
(defun calc-apart (arg)
(interactive "P")
(calc-slow-wrapper
(calc-unary-op "aprt" 'calcFunc-apart arg)))
(defun calc-normalize-rat (arg)
(interactive "P")
(calc-slow-wrapper
(calc-unary-op "nrat" 'calcFunc-nrat arg)))
(defun calc-poly-gcd (arg)
(interactive "P")
(calc-slow-wrapper
(calc-binary-op "pgcd" 'calcFunc-pgcd arg)))
(defvar calc-poly-div-remainder)
(defun calc-poly-div (arg)
(interactive "P")
(calc-slow-wrapper
(let ((calc-poly-div-remainder nil))
(calc-binary-op "pdiv" 'calcFunc-pdiv arg)
(if (and calc-poly-div-remainder (null arg))
(progn
(calc-clear-command-flag 'clear-message)
(calc-record calc-poly-div-remainder "prem")
(if (not (Math-zerop calc-poly-div-remainder))
(message "(Remainder was %s)"
(math-format-flat-expr calc-poly-div-remainder 0))
(message "(No remainder)")))))))
(defun calc-poly-rem (arg)
(interactive "P")
(calc-slow-wrapper
(calc-binary-op "prem" 'calcFunc-prem arg)))
(defun calc-poly-div-rem (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(calc-binary-op "pdvr" 'calcFunc-pdivide arg)
(calc-binary-op "pdvr" 'calcFunc-pdivrem arg))))
(defun calc-substitute (&optional oldname newname)
(interactive "sSubstitute old: ")
(calc-slow-wrapper
(let (old new (num 1) expr)
(if (or (equal oldname "") (equal oldname "$") (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)
'(nil frac float)))
(length (memq (car-safe b)
'(nil 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 (nth 1 a) (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 (car a) (car b)))))
(defvar math-living-dangerously)
(defsubst math-simplify-extended (a)
(let ((math-living-dangerously t))
(math-simplify a)))
(defalias 'calcFunc-esimplify #'math-simplify-extended)
;;; Rewrite the trig functions in a form easier to simplify.
(defun math-trig-rewrite (fn)
"Rewrite trigonometric functions in terms of sines and cosines."
(cond
((not (consp fn))
fn)
((eq (car-safe fn) 'calcFunc-sec)
(list '/ 1 (cons 'calcFunc-cos (math-trig-rewrite (cdr fn)))))
((eq (car-safe fn) 'calcFunc-csc)
(list '/ 1 (cons 'calcFunc-sin (math-trig-rewrite (cdr fn)))))
((eq (car-safe fn) 'calcFunc-tan)
(let ((newfn (math-trig-rewrite (cdr fn))))
(list '/ (cons 'calcFunc-sin newfn)
(cons 'calcFunc-cos newfn))))
((eq (car-safe fn) 'calcFunc-cot)
(let ((newfn (math-trig-rewrite (cdr fn))))
(list '/ (cons 'calcFunc-cos newfn)
(cons 'calcFunc-sin newfn))))
(t
(mapcar #'math-trig-rewrite fn))))
(defun math-hyperbolic-trig-rewrite (fn)
"Rewrite hyperbolic functions in terms of sinhs and coshs."
(cond
((not (consp fn))
fn)
((eq (car-safe fn) 'calcFunc-sech)
(list '/ 1 (cons 'calcFunc-cosh (math-hyperbolic-trig-rewrite (cdr fn)))))
((eq (car-safe fn) 'calcFunc-csch)
(list '/ 1 (cons 'calcFunc-sinh (math-hyperbolic-trig-rewrite (cdr fn)))))
((eq (car-safe fn) 'calcFunc-tanh)
(let ((newfn (math-hyperbolic-trig-rewrite (cdr fn))))
(list '/ (cons 'calcFunc-sinh newfn)
(cons 'calcFunc-cosh newfn))))
((eq (car-safe fn) 'calcFunc-coth)
(let ((newfn (math-hyperbolic-trig-rewrite (cdr fn))))
(list '/ (cons 'calcFunc-cosh newfn)
(cons 'calcFunc-sinh newfn))))
(t
(mapcar #'math-hyperbolic-trig-rewrite fn))))
;; 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 calc-input-angle-units (input)
(cond ((math-expr-contains input '(var deg var-deg)) 'deg)
((math-expr-contains input '(var rad var-rad)) 'rad)
((math-expr-contains input '(var hms var-hms)) 'hms)
(t nil)))
;; math-normalize-error is declared in calc.el.
(defvar math-normalize-error)
(defvar math-simplifying)
(defvar calc-angle-mode)
(defun math-simplify (top-expr)
(let ((math-simplifying t)
(calc-angle-mode (if (calc-input-angle-units top-expr)
'rad
calc-angle-mode))
(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))
(if (not math-normalize-error)
(progn
(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)
"Define the simplification code for functions FUNCS.
Code can refer to the expression to simplify via lexical variable `expr'
and should return the simplified expression to use (or nil)."
(declare (indent 1) (debug (sexp body)))
(cons 'progn
(mapcar #'(lambda (func)
`(put ',func 'math-simplify
(nconc
(get ',func 'math-simplify)
(list
#'(lambda (expr) ,@code)))))
(if (symbolp funcs) (list funcs) funcs))))
(math-defsimplify (+ -)
(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 *
(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 expr))
(defvar math--simplify-divide-expr)
(defun math-simplify-divide (expr)
(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))
(unless (and (eq (car-safe expr) 'calcFunc-eq)
(eq (car-safe (nth 1 expr)) 'var)
(not (math-expr-contains (nth 2 expr)
(nth 1 expr))))
(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))))))))
(let ((math--simplify-divide-expr expr)) ;For use in math-simplify-divisor
(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)))
;; 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 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 math--simplify-divide-expr) '/)
(let ((signs (math-possible-signs (car np))))
(or (memq signs '(1 4))
(and (memq (car math--simplify-divide-expr)
'(calcFunc-eq calcFunc-neq))
(eq signs 5))
math-living-dangerously)))
(math-numberp (car np)))
(let (d
(safe t)
(math-simplify-divisor-nover nover)
(math-simplify-divisor-dover dover)
(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-divide-expr)
'(/ calcFunc-eq calcFunc-neq)))
(math-known-negp (car dp))
(setq op (assq (car math--simplify-divide-expr)
calc-tweak-eqn-table))
(setcar math--simplify-divide-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-divide-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)))))))
(defvar calc-prefer-frac)
(math-defsimplify %
(and (Math-realp (nth 2 expr))
(Math-posp (nth 2 expr))
(let ((lin (math-is-linear (nth 1 expr)))
t1)
(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 expr)))
(defun math-simplify-ineq (expr)
(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 expr)
(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 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))))))))
(math-defsimplify calcFunc-sec
(or (and (math-looks-negp (nth 1 expr))
(list 'calcFunc-sec (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
(let ((s (math-known-sin (car n) (nth 1 n) 120 300)))
(and s (math-div 1 s))))))
(and (eq calc-angle-mode 'deg)
(let ((n (math-integer-plus (nth 1 expr))))
(and n
(let ((s (math-known-sin (car n) (nth 1 n) '(frac 2 3) 300)))
(and s (math-div 1 s))))))
(and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
(math-div
1
(list 'calcFunc-sqrt
(math-sub 1 (math-sqr (nth 1 (nth 1 expr)))))))
(and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
(math-div
1
(nth 1 (nth 1 expr))))
(and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
(list 'calcFunc-sqrt
(math-add 1
(math-sqr (nth 1 (nth 1 expr))))))))
(math-defsimplify calcFunc-csc
(or (and (math-looks-negp (nth 1 expr))
(math-neg (list 'calcFunc-csc (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
(let ((s (math-known-sin (car n) (nth 1 n) 120 0)))
(and s (math-div 1 s))))))
(and (eq calc-angle-mode 'deg)
(let ((n (math-integer-plus (nth 1 expr))))
(and n
(let ((s (math-known-sin (car n) (nth 1 n) '(frac 2 3) 0)))
(and s (math-div 1 s))))))
(and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)
(math-div 1 (nth 1 (nth 1 expr))))
(and (eq (car-safe (nth 1 expr)) 'calcFunc-arccos)
(math-div
1
(list 'calcFunc-sqrt (math-sub 1 (math-sqr
(nth 1 (nth 1 expr)))))))
(and (eq (car-safe (nth 1 expr)) 'calcFunc-arctan)
(math-div (list 'calcFunc-sqrt
(math-add 1 (math-sqr
(nth 1 (nth 1 expr)))))
(nth 1 (nth 1 expr))))))
(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))))))))
(math-defsimplify calcFunc-cot
(or (and (math-looks-negp (nth 1 expr))
(math-neg (list 'calcFunc-cot (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
(let ((tn (math-known-tan (car n) (nth 1 n) 120)))
(and tn (math-div 1 tn))))))
(and (eq calc-angle-mode 'deg)
(let ((n (math-integer-plus (nth 1 expr))))
(and n
(let ((tn (math-known-tan (car n) (nth 1 n) '(frac 2 3))))
(and tn (math-div 1 tn))))))
(and (eq (car-safe (nth 1 expr)) 'calcFunc-arcsin)