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emacs/lisp/calc/calc-math.el
Paul Eggert 0e963201d0 Update copyright year to 2016 
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2016-01-01 01:34:24 -08:00

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;;; calc-math.el --- mathematical functions for Calc
;; Copyright (C) 1990-1993, 2001-2016 Free Software Foundation, Inc.
;; Author: David Gillespie <daveg@synaptics.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 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 <http://www.gnu.org/licenses/>.
;;; Commentary:
;;; Code:
;; This file is autoloaded from calc-ext.el.
(require 'calc-ext)
(require 'calc-macs)
;;; Find out how many 9s in 9.9999... will give distinct Emacs floats,
;;; then back off by one.
(defvar math-emacs-precision
(let* ((n 1)
(x 9)
(xx (+ x (* 9 (expt 10 (- n))))))
(while (/= x xx)
(progn
(setq n (1+ n))
(setq x xx)
(setq xx (+ x (* 9 (expt 10 (- n)))))))
(1- n))
"The number of digits in an Emacs float.")
;;; Find the largest power of 10 which is an Emacs float,
;;; then back off by one so that any float d.dddd...eN
;;; is an Emacs float, for acceptable d.dddd....
(defvar math-largest-emacs-expt
(let ((x 1)
(pow 1e2))
;; The following loop is for efficiency; it should stop when
;; 10^(2x) is too large. This could be indicated by a range
;; error when computing 10^(2x) or an infinite value for 10^(2x).
(while (and
pow
(< pow 1.0e+INF))
(setq x (* 2 x))
(setq pow (condition-case nil
(expt 10.0 (* 2 x))
(error nil))))
;; The following loop should stop when 10^(x+1) is too large.
(setq pow (condition-case nil
(expt 10.0 (1+ x))
(error nil)))
(while (and
pow
(< pow 1.0e+INF))
(setq x (1+ x))
(setq pow (condition-case nil
(expt 10.0 (1+ x))
(error nil))))
(1- x))
"The largest exponent which Calc will convert to an Emacs float.")
(defvar math-smallest-emacs-expt
(let ((x -1))
(while (condition-case nil
(> (expt 10.0 x) 0.0)
(error nil))
(setq x (* 2 x)))
(setq x (/ x 2))
(while (condition-case nil
(> (expt 10.0 x) 0.0)
(error nil))
(setq x (1- x)))
(+ x 2))
"The smallest exponent which Calc will convert to an Emacs float.")
(defun math-use-emacs-fn (fn x)
"Use the native Emacs function FN to evaluate the Calc number X.
If this can't be done, return NIL."
(and
(<= calc-internal-prec math-emacs-precision)
(math-realp x)
(let* ((fx (math-float x))
(xpon (+ (nth 2 x) (1- (math-numdigs (nth 1 x))))))
(and (<= math-smallest-emacs-expt xpon)
(<= xpon math-largest-emacs-expt)
(condition-case nil
(math-read-number
(number-to-string
(funcall fn
(string-to-number
(let
((calc-number-radix 10)
(calc-twos-complement-mode nil)
(calc-float-format (list 'float calc-internal-prec))
(calc-group-digits nil)
(calc-point-char "."))
(math-format-number (math-float x)))))))
(error nil))))))
(defun calc-sqrt (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-inverse)
(calc-unary-op "^2" 'calcFunc-sqr arg)
(calc-unary-op "sqrt" 'calcFunc-sqrt arg))))
(defun calc-isqrt (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-inverse)
(calc-unary-op "^2" 'calcFunc-sqr arg)
(calc-unary-op "isqt" 'calcFunc-isqrt arg))))
(defun calc-hypot (arg)
(interactive "P")
(calc-slow-wrapper
(calc-binary-op "hypt" 'calcFunc-hypot arg)))
(defun calc-ln (arg)
(interactive "P")
(calc-invert-func)
(calc-exp arg))
(defun calc-log10 (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-ln arg))
(defun calc-log (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-inverse)
(calc-binary-op "alog" 'calcFunc-alog arg)
(calc-binary-op "log" 'calcFunc-log arg))))
(defun calc-ilog (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-inverse)
(calc-binary-op "alog" 'calcFunc-alog arg)
(calc-binary-op "ilog" 'calcFunc-ilog arg))))
(defun calc-lnp1 (arg)
(interactive "P")
(calc-invert-func)
(calc-expm1 arg))
(defun calc-exp (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(if (calc-is-inverse)
(calc-unary-op "lg10" 'calcFunc-log10 arg)
(calc-unary-op "10^" 'calcFunc-exp10 arg))
(if (calc-is-inverse)
(calc-unary-op "ln" 'calcFunc-ln arg)
(calc-unary-op "exp" 'calcFunc-exp arg)))))
(defun calc-expm1 (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-inverse)
(calc-unary-op "ln+1" 'calcFunc-lnp1 arg)
(calc-unary-op "ex-1" 'calcFunc-expm1 arg))))
(defun calc-pi ()
(interactive)
(calc-slow-wrapper
(if (calc-is-inverse)
(if (calc-is-hyperbolic)
(if calc-symbolic-mode
(calc-pop-push-record 0 "phi" '(var phi var-phi))
(calc-pop-push-record 0 "phi" (math-phi)))
(if calc-symbolic-mode
(calc-pop-push-record 0 "gmma" '(var gamma var-gamma))
(calc-pop-push-record 0 "gmma" (math-gamma-const))))
(if (calc-is-hyperbolic)
(if calc-symbolic-mode
(calc-pop-push-record 0 "e" '(var e var-e))
(calc-pop-push-record 0 "e" (math-e)))
(if calc-symbolic-mode
(calc-pop-push-record 0 "pi" '(var pi var-pi))
(calc-pop-push-record 0 "pi" (math-pi)))))))
(defun calc-sin (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(if (calc-is-inverse)
(calc-unary-op "asnh" 'calcFunc-arcsinh arg)
(calc-unary-op "sinh" 'calcFunc-sinh arg))
(if (calc-is-inverse)
(calc-unary-op "asin" 'calcFunc-arcsin arg)
(calc-unary-op "sin" 'calcFunc-sin arg)))))
(defun calc-arcsin (arg)
(interactive "P")
(calc-invert-func)
(calc-sin arg))
(defun calc-sinh (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-sin arg))
(defun calc-arcsinh (arg)
(interactive "P")
(calc-invert-func)
(calc-hyperbolic-func)
(calc-sin arg))
(defun calc-sec (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(calc-unary-op "sech" 'calcFunc-sech arg)
(calc-unary-op "sec" 'calcFunc-sec arg))))
(defun calc-sech (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-sec arg))
(defun calc-cos (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(if (calc-is-inverse)
(calc-unary-op "acsh" 'calcFunc-arccosh arg)
(calc-unary-op "cosh" 'calcFunc-cosh arg))
(if (calc-is-inverse)
(calc-unary-op "acos" 'calcFunc-arccos arg)
(calc-unary-op "cos" 'calcFunc-cos arg)))))
(defun calc-arccos (arg)
(interactive "P")
(calc-invert-func)
(calc-cos arg))
(defun calc-cosh (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-cos arg))
(defun calc-arccosh (arg)
(interactive "P")
(calc-invert-func)
(calc-hyperbolic-func)
(calc-cos arg))
(defun calc-csc (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(calc-unary-op "csch" 'calcFunc-csch arg)
(calc-unary-op "csc" 'calcFunc-csc arg))))
(defun calc-csch (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-csc arg))
(defun calc-sincos ()
(interactive)
(calc-slow-wrapper
(if (calc-is-inverse)
(calc-enter-result 1 "asnc" (list 'calcFunc-arcsincos (calc-top-n 1)))
(calc-enter-result 1 "sncs" (list 'calcFunc-sincos (calc-top-n 1))))))
(defun calc-tan (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(if (calc-is-inverse)
(calc-unary-op "atnh" 'calcFunc-arctanh arg)
(calc-unary-op "tanh" 'calcFunc-tanh arg))
(if (calc-is-inverse)
(calc-unary-op "atan" 'calcFunc-arctan arg)
(calc-unary-op "tan" 'calcFunc-tan arg)))))
(defun calc-arctan (arg)
(interactive "P")
(calc-invert-func)
(calc-tan arg))
(defun calc-tanh (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-tan arg))
(defun calc-arctanh (arg)
(interactive "P")
(calc-invert-func)
(calc-hyperbolic-func)
(calc-tan arg))
(defun calc-cot (arg)
(interactive "P")
(calc-slow-wrapper
(if (calc-is-hyperbolic)
(calc-unary-op "coth" 'calcFunc-coth arg)
(calc-unary-op "cot" 'calcFunc-cot arg))))
(defun calc-coth (arg)
(interactive "P")
(calc-hyperbolic-func)
(calc-cot arg))
(defun calc-arctan2 ()
(interactive)
(calc-slow-wrapper
(calc-enter-result 2 "atn2" (cons 'calcFunc-arctan2 (calc-top-list-n 2)))))
(defun calc-conj (arg)
(interactive "P")
(calc-wrapper
(calc-unary-op "conj" 'calcFunc-conj arg)))
(defun calc-imaginary ()
(interactive)
(calc-slow-wrapper
(calc-pop-push-record 1 "i*" (math-imaginary (calc-top-n 1)))))
(defun calc-to-degrees (arg)
(interactive "P")
(calc-wrapper
(calc-unary-op ">deg" 'calcFunc-deg arg)))
(defun calc-to-radians (arg)
(interactive "P")
(calc-wrapper
(calc-unary-op ">rad" 'calcFunc-rad arg)))
(defun calc-degrees-mode (arg)
(interactive "p")
(cond ((= arg 1)
(calc-wrapper
(calc-change-mode 'calc-angle-mode 'deg)
(message "Angles measured in degrees")))
((= arg 2) (calc-radians-mode))
((= arg 3) (calc-hms-mode))
(t (error "Prefix argument out of range"))))
(defun calc-radians-mode ()
(interactive)
(calc-wrapper
(calc-change-mode 'calc-angle-mode 'rad)
(message "Angles measured in radians")))
;;; Compute the integer square-root floor(sqrt(A)). A > 0. [I I] [Public]
;;; This method takes advantage of the fact that Newton's method starting
;;; with an overestimate always works, even using truncating integer division!
(defun math-isqrt (a)
(cond ((Math-zerop a) a)
((not (math-natnump a))
(math-reject-arg a 'natnump))
((integerp a)
(math-isqrt-small a))
(t
(math-normalize (cons 'bigpos (cdr (math-isqrt-bignum (cdr a))))))))
(defun calcFunc-isqrt (a)
(if (math-realp a)
(math-isqrt (math-floor a))
(math-floor (math-sqrt a))))
;;; This returns (flag . result) where the flag is t if A is a perfect square.
(defun math-isqrt-bignum (a) ; [P.l L]
(let ((len (length a)))
(if (= (% len 2) 0)
(let* ((top (nthcdr (- len 2) a)))
(math-isqrt-bignum-iter
a
(math-scale-bignum-digit-size
(math-bignum-big
(1+ (math-isqrt-small
(+ (* (nth 1 top) math-bignum-digit-size) (car top)))))
(1- (/ len 2)))))
(let* ((top (nth (1- len) a)))
(math-isqrt-bignum-iter
a
(math-scale-bignum-digit-size
(list (1+ (math-isqrt-small top)))
(/ len 2)))))))
(defun math-isqrt-bignum-iter (a guess) ; [l L l]
(math-working "isqrt" (cons 'bigpos guess))
(let* ((q (math-div-bignum a guess))
(s (math-add-bignum (car q) guess))
(g2 (math-div2-bignum s))
(comp (math-compare-bignum g2 guess)))
(if (< comp 0)
(math-isqrt-bignum-iter a g2)
(cons (and (= comp 0)
(math-zerop-bignum (cdr q))
(= (% (car s) 2) 0))
guess))))
(defun math-zerop-bignum (a)
(and (eq (car a) 0)
(progn
(while (eq (car (setq a (cdr a))) 0))
(null a))))
(defun math-scale-bignum-digit-size (a n) ; [L L S]
(while (> n 0)
(setq a (cons 0 a)
n (1- n)))
a)
(defun math-isqrt-small (a) ; A > 0. [S S]
(let ((g (cond ((>= a 1000000) 10000)
((>= a 10000) 1000)
((>= a 100) 100)
(t 10)))
g2)
(while (< (setq g2 (/ (+ g (/ a g)) 2)) g)
(setq g g2))
g))
;;; Compute the square root of a number.
;;; [T N] if possible, else [F N] if possible, else [C N]. [Public]
(defun math-sqrt (a)
(or
(and (Math-zerop a) a)
(and (math-known-nonposp a)
(math-imaginary (math-sqrt (math-neg a))))
(and (integerp a)
(let ((sqrt (math-isqrt-small a)))
(if (= (* sqrt sqrt) a)
sqrt
(if calc-symbolic-mode
(list 'calcFunc-sqrt a)
(math-sqrt-float (math-float a) (math-float sqrt))))))
(and (eq (car-safe a) 'bigpos)
(let* ((res (math-isqrt-bignum (cdr a)))
(sqrt (math-normalize (cons 'bigpos (cdr res)))))
(if (car res)
sqrt
(if calc-symbolic-mode
(list 'calcFunc-sqrt a)
(math-sqrt-float (math-float a) (math-float sqrt))))))
(and (eq (car-safe a) 'frac)
(let* ((num-res (math-isqrt-bignum (cdr (Math-bignum-test (nth 1 a)))))
(num-sqrt (math-normalize (cons 'bigpos (cdr num-res))))
(den-res (math-isqrt-bignum (cdr (Math-bignum-test (nth 2 a)))))
(den-sqrt (math-normalize (cons 'bigpos (cdr den-res)))))
(if (and (car num-res) (car den-res))
(list 'frac num-sqrt den-sqrt)
(if calc-symbolic-mode
(if (or (car num-res) (car den-res))
(math-div (if (car num-res)
num-sqrt (list 'calcFunc-sqrt (nth 1 a)))
(if (car den-res)
den-sqrt (list 'calcFunc-sqrt (nth 2 a))))
(list 'calcFunc-sqrt a))
(math-sqrt-float (math-float a)
(math-div (math-float num-sqrt) den-sqrt))))))
(and (eq (car-safe a) 'float)
(if calc-symbolic-mode
(if (= (% (nth 2 a) 2) 0)
(let ((res (math-isqrt-bignum
(cdr (Math-bignum-test (nth 1 a))))))
(if (car res)
(math-make-float (math-normalize
(cons 'bigpos (cdr res)))
(/ (nth 2 a) 2))
(signal 'inexact-result nil)))
(signal 'inexact-result nil))
(math-sqrt-float a)))
(and (eq (car-safe a) 'cplx)
(math-with-extra-prec 2
(let* ((d (math-abs a))
(imag (math-sqrt (math-mul (math-sub d (nth 1 a))
'(float 5 -1)))))
(list 'cplx
(math-sqrt (math-mul (math-add d (nth 1 a)) '(float 5 -1)))
(if (math-negp (nth 2 a)) (math-neg imag) imag)))))
(and (eq (car-safe a) 'polar)
(list 'polar
(math-sqrt (nth 1 a))
(math-mul (nth 2 a) '(float 5 -1))))
(and (eq (car-safe a) 'sdev)
(let ((sqrt (math-sqrt (nth 1 a))))
(math-make-sdev sqrt
(math-div (nth 2 a) (math-mul sqrt 2)))))
(and (eq (car-safe a) 'intv)
(not (math-negp (nth 2 a)))
(math-make-intv (nth 1 a) (math-sqrt (nth 2 a)) (math-sqrt (nth 3 a))))
(and (eq (car-safe a) '*)
(or (math-known-nonnegp (nth 1 a))
(math-known-nonnegp (nth 2 a)))
(math-mul (math-sqrt (nth 1 a)) (math-sqrt (nth 2 a))))
(and (eq (car-safe a) '/)
(or (and (math-known-nonnegp (nth 2 a))
(math-div (math-sqrt (nth 1 a)) (math-sqrt (nth 2 a))))
(and (math-known-nonnegp (nth 1 a))
(not (math-equal-int (nth 1 a) 1))
(math-mul (math-sqrt (nth 1 a))
(math-sqrt (math-div 1 (nth 2 a)))))))
(and (eq (car-safe a) '^)
(math-known-evenp (nth 2 a))
(math-known-realp (nth 1 a))
(math-abs (math-pow (nth 1 a) (math-div (nth 2 a) 2))))
(let ((inf (math-infinitep a)))
(and inf
(math-mul (math-sqrt (math-infinite-dir a inf)) inf)))
(progn
(calc-record-why 'numberp a)
(list 'calcFunc-sqrt a))))
(defalias 'calcFunc-sqrt 'math-sqrt)
(defun math-infinite-dir (a &optional inf)
(or inf (setq inf (math-infinitep a)))
(math-normalize (math-expr-subst a inf 1)))
(defun math-sqrt-float (a &optional guess) ; [F F F]
(if calc-symbolic-mode
(signal 'inexact-result nil)
(math-with-extra-prec 1 (math-sqrt-raw a guess))))
(defun math-sqrt-raw (a &optional guess) ; [F F F]
(if (not (Math-posp a))
(math-sqrt a)
(cond
((math-use-emacs-fn 'sqrt a))
(t
(if (null guess)
(let ((ldiff (- (math-numdigs (nth 1 a)) 6)))
(or (= (% (+ (nth 2 a) ldiff) 2) 0) (setq ldiff (1+ ldiff)))
(setq guess (math-make-float (math-isqrt-small
(math-scale-int (nth 1 a) (- ldiff)))
(/ (+ (nth 2 a) ldiff) 2)))))
(math-sqrt-float-iter a guess)))))
(defun math-sqrt-float-iter (a guess) ; [F F F]
(math-working "sqrt" guess)
(let ((g2 (math-mul-float (math-add-float guess (math-div-float a guess))
'(float 5 -1))))
(if (math-nearly-equal-float g2 guess)
g2
(math-sqrt-float-iter a g2))))
;;; True if A and B differ only in the last digit of precision. [P F F]
(defun math-nearly-equal-float (a b)
(let ((ediff (- (nth 2 a) (nth 2 b))))
(cond ((= ediff 0) ;; Expanded out for speed
(setq ediff (math-add (Math-integer-neg (nth 1 a)) (nth 1 b)))
(or (eq ediff 0)
(and (not (consp ediff))
(< ediff 10)
(> ediff -10)
(= (math-numdigs (nth 1 a)) calc-internal-prec))))
((= ediff 1)
(setq ediff (math-add (Math-integer-neg (nth 1 b))
(math-scale-int (nth 1 a) 1)))
(and (not (consp ediff))
(< ediff 10)
(> ediff -10)
(= (math-numdigs (nth 1 b)) calc-internal-prec)))
((= ediff -1)
(setq ediff (math-add (Math-integer-neg (nth 1 a))
(math-scale-int (nth 1 b) 1)))
(and (not (consp ediff))
(< ediff 10)
(> ediff -10)
(= (math-numdigs (nth 1 a)) calc-internal-prec))))))
(defun math-nearly-equal (a b) ; [P N N] [Public]
(setq a (math-float a))
(setq b (math-float b))
(if (eq (car a) 'polar) (setq a (math-complex a)))
(if (eq (car b) 'polar) (setq b (math-complex b)))
(if (eq (car a) 'cplx)
(if (eq (car b) 'cplx)
(and (or (math-nearly-equal-float (nth 1 a) (nth 1 b))
(and (math-nearly-zerop-float (nth 1 a) (nth 2 a))
(math-nearly-zerop-float (nth 1 b) (nth 2 b))))
(or (math-nearly-equal-float (nth 2 a) (nth 2 b))
(and (math-nearly-zerop-float (nth 2 a) (nth 1 a))
(math-nearly-zerop-float (nth 2 b) (nth 1 b)))))
(and (math-nearly-equal-float (nth 1 a) b)
(math-nearly-zerop-float (nth 2 a) b)))
(if (eq (car b) 'cplx)
(and (math-nearly-equal-float a (nth 1 b))
(math-nearly-zerop-float a (nth 2 b)))
(math-nearly-equal-float a b))))
;;; True if A is nearly zero compared to B. [P F F]
(defun math-nearly-zerop-float (a b)
(or (eq (nth 1 a) 0)
(<= (+ (math-numdigs (nth 1 a)) (nth 2 a))
(1+ (- (+ (math-numdigs (nth 1 b)) (nth 2 b)) calc-internal-prec)))))
(defun math-nearly-zerop (a b) ; [P N R] [Public]
(setq a (math-float a))
(setq b (math-float b))
(if (eq (car a) 'cplx)
(and (math-nearly-zerop-float (nth 1 a) b)
(math-nearly-zerop-float (nth 2 a) b))
(if (eq (car a) 'polar)
(math-nearly-zerop-float (nth 1 a) b)
(math-nearly-zerop-float a b))))
;;; This implementation could be improved, accuracy-wise.
(defun math-hypot (a b)
(cond ((Math-zerop a) (math-abs b))
((Math-zerop b) (math-abs a))
((not (Math-scalarp a))
(if (math-infinitep a)
(if (math-infinitep b)
(if (equal a b)
a
'(var nan var-nan))
a)
(calc-record-why 'scalarp a)
(list 'calcFunc-hypot a b)))
((not (Math-scalarp b))
(if (math-infinitep b)
b
(calc-record-why 'scalarp b)
(list 'calcFunc-hypot a b)))
((and (Math-numberp a) (Math-numberp b))
(math-with-extra-prec 1
(math-sqrt (math-add (calcFunc-abssqr a) (calcFunc-abssqr b)))))
((eq (car-safe a) 'hms)
(if (eq (car-safe b) 'hms) ; this helps sdev's of hms forms
(math-to-hms (math-hypot (math-from-hms a 'deg)
(math-from-hms b 'deg)))
(math-to-hms (math-hypot (math-from-hms a 'deg) b))))
((eq (car-safe b) 'hms)
(math-to-hms (math-hypot a (math-from-hms b 'deg))))
(t nil)))
(defalias 'calcFunc-hypot 'math-hypot)
(defun calcFunc-sqr (x)
(math-pow x 2))
(defun math-nth-root (a n)
(cond ((= n 2) (math-sqrt a))
((Math-zerop a) a)
((Math-negp a) nil)
((Math-integerp a)
(let ((root (math-nth-root-integer a n)))
(if (car root)
(cdr root)
(and (not calc-symbolic-mode)
(math-nth-root-float (math-float a) n
(math-float (cdr root)))))))
((eq (car-safe a) 'frac)
(let* ((num-root (math-nth-root-integer (nth 1 a) n))
(den-root (math-nth-root-integer (nth 2 a) n)))
(if (and (car num-root) (car den-root))
(list 'frac (cdr num-root) (cdr den-root))
(and (not calc-symbolic-mode)
(math-nth-root-float
(math-float a) n
(math-div-float (math-float (cdr num-root))
(math-float (cdr den-root))))))))
((eq (car-safe a) 'float)
(and (not calc-symbolic-mode)
(math-nth-root-float a n)))
((eq (car-safe a) 'polar)
(let ((root (math-nth-root (nth 1 a) n)))
(and root (list 'polar root (math-div (nth 2 a) n)))))
(t nil)))
;; The variables math-nrf-n, math-nrf-nf and math-nrf-nfm1 are local
;; to math-nth-root-float, but are used by math-nth-root-float-iter,
;; which is called by math-nth-root-float.
(defvar math-nrf-n)
(defvar math-nrf-nf)
(defvar math-nrf-nfm1)
(defun math-nth-root-float (a math-nrf-n &optional guess)
(math-inexact-result)
(math-with-extra-prec 1
(let ((math-nrf-nf (math-float math-nrf-n))
(math-nrf-nfm1 (math-float (1- math-nrf-n))))
(math-nth-root-float-iter a (or guess
(math-make-float
1 (/ (+ (math-numdigs (nth 1 a))
(nth 2 a)
(/ math-nrf-n 2))
math-nrf-n)))))))
(defun math-nth-root-float-iter (a guess)
(math-working "root" guess)
(let ((g2 (math-div-float (math-add-float (math-mul math-nrf-nfm1 guess)
(math-div-float
a (math-ipow guess (1- math-nrf-n))))
math-nrf-nf)))
(if (math-nearly-equal-float g2 guess)
g2
(math-nth-root-float-iter a g2))))
;; The variable math-nri-n is local to math-nth-root-integer, but
;; is used by math-nth-root-int-iter, which is called by
;; math-nth-root-int.
(defvar math-nri-n)
(defun math-nth-root-integer (a math-nri-n &optional guess) ; [I I S]
(math-nth-root-int-iter a (or guess
(math-scale-int 1 (/ (+ (math-numdigs a)
(1- math-nri-n))
math-nri-n)))))
(defun math-nth-root-int-iter (a guess)
(math-working "root" guess)
(let* ((q (math-idivmod a (math-ipow guess (1- math-nri-n))))
(s (math-add (car q) (math-mul (1- math-nri-n) guess)))
(g2 (math-idivmod s math-nri-n)))
(if (Math-natnum-lessp (car g2) guess)
(math-nth-root-int-iter a (car g2))
(cons (and (equal (car g2) guess)
(eq (cdr q) 0)
(eq (cdr g2) 0))
guess))))
(defun calcFunc-nroot (x n)
(calcFunc-pow x (if (integerp n)
(math-make-frac 1 n)
(math-div 1 n))))
;;;; Transcendental functions.
;;; All of these functions are defined on the complex plane.
;;; (Branch cuts, etc. follow Steele's Common Lisp book.)
;;; Most functions increase calc-internal-prec by 2 digits, then round
;;; down afterward. "-raw" functions use the current precision, require
;;; their arguments to be in float (or complex float) format, and always
;;; work in radians (where applicable).
(defun math-to-radians (a) ; [N N]
(cond ((eq (car-safe a) 'hms)
(math-from-hms a 'rad))
((memq calc-angle-mode '(deg hms))
(math-mul a (math-pi-over-180)))
(t a)))
(defun math-from-radians (a) ; [N N]
(cond ((eq calc-angle-mode 'deg)
(if (math-constp a)
(math-div a (math-pi-over-180))
(list 'calcFunc-deg a)))
((eq calc-angle-mode 'hms)
(math-to-hms a 'rad))
(t a)))
(defun math-to-radians-2 (a &optional force-symbolic) ; [N N]
(cond ((eq (car-safe a) 'hms)
(math-from-hms a 'rad))
((memq calc-angle-mode '(deg hms))
(if (or calc-symbolic-mode force-symbolic)
(math-div (math-mul a '(var pi var-pi)) 180)
(math-mul a (math-pi-over-180))))
(t a)))
(defun math-from-radians-2 (a &optional force-symbolic) ; [N N]
(cond ((memq calc-angle-mode '(deg hms))
(if (or calc-symbolic-mode force-symbolic)
(math-div (math-mul 180 a) '(var pi var-pi))
(math-div a (math-pi-over-180))))
(t a)))
;;; Sine, cosine, and tangent.
(defun calcFunc-sin (x) ; [N N] [Public]
(cond ((and (integerp x)
(if (eq calc-angle-mode 'deg)
(= (% x 90) 0)
(= x 0)))
(aref [0 1 0 -1] (math-mod (/ x 90) 4)))
((Math-scalarp x)
(math-with-extra-prec 2
(math-sin-raw (math-to-radians (math-float x)))))
((eq (car x) 'sdev)
(if (math-constp x)
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float (nth 1 x))))
(xs (math-to-radians (math-float (nth 2 x))))
(sc (math-sin-cos-raw xx)))
(math-make-sdev (car sc) (math-mul xs (cdr sc)))))
(math-make-sdev (calcFunc-sin (nth 1 x))
(math-mul (nth 2 x) (calcFunc-cos (nth 1 x))))))
((and (eq (car x) 'intv) (math-intv-constp x))
(calcFunc-cos (math-sub x (math-quarter-circle nil))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'scalarp x)
(list 'calcFunc-sin x))))
(defun calcFunc-cos (x) ; [N N] [Public]
(cond ((and (integerp x)
(if (eq calc-angle-mode 'deg)
(= (% x 90) 0)
(= x 0)))
(aref [1 0 -1 0] (math-mod (/ x 90) 4)))
((Math-scalarp x)
(math-with-extra-prec 2
(math-cos-raw (math-to-radians (math-float x)))))
((eq (car x) 'sdev)
(if (math-constp x)
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float (nth 1 x))))
(xs (math-to-radians (math-float (nth 2 x))))
(sc (math-sin-cos-raw xx)))
(math-make-sdev (cdr sc) (math-mul xs (car sc)))))
(math-make-sdev (calcFunc-cos (nth 1 x))
(math-mul (nth 2 x) (calcFunc-sin (nth 1 x))))))
((and (eq (car x) 'intv) (math-intv-constp x))
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float x)))
(na (math-floor (math-div (nth 2 xx) (math-pi))))
(nb (math-floor (math-div (nth 3 xx) (math-pi))))
(span (math-sub nb na)))
(if (memq span '(0 1))
(let ((int (math-sort-intv (nth 1 x)
(math-cos-raw (nth 2 xx))
(math-cos-raw (nth 3 xx)))))
(if (eq span 1)
(if (math-evenp na)
(math-make-intv (logior (nth 1 x) 2)
-1
(nth 3 int))
(math-make-intv (logior (nth 1 x) 1)
(nth 2 int)
1))
int))
(list 'intv 3 -1 1)))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'scalarp x)
(list 'calcFunc-cos x))))
(defun calcFunc-sincos (x) ; [V N] [Public]
(if (Math-scalarp x)
(math-with-extra-prec 2
(let ((sc (math-sin-cos-raw (math-to-radians (math-float x)))))
(list 'vec (cdr sc) (car sc)))) ; the vector [cos, sin]
(list 'vec (calcFunc-sin x) (calcFunc-cos x))))
(defun calcFunc-tan (x) ; [N N] [Public]
(cond ((and (integerp x)
(if (eq calc-angle-mode 'deg)
(= (% x 180) 0)
(= x 0)))
0)
((Math-scalarp x)
(math-with-extra-prec 2
(math-tan-raw (math-to-radians (math-float x)))))
((eq (car x) 'sdev)
(if (math-constp x)
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float (nth 1 x))))
(xs (math-to-radians (math-float (nth 2 x))))
(sc (math-sin-cos-raw xx)))
(if (and (math-zerop (cdr sc)) (not calc-infinite-mode))
(progn
(calc-record-why "*Division by zero")
(list 'calcFunc-tan x))
(math-make-sdev (math-div-float (car sc) (cdr sc))
(math-div-float xs (math-sqr (cdr sc)))))))
(math-make-sdev (calcFunc-tan (nth 1 x))
(math-div (nth 2 x)
(math-sqr (calcFunc-cos (nth 1 x)))))))
((and (eq (car x) 'intv) (math-intv-constp x))
(or (math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float x)))
(na (math-floor (math-div (math-sub (nth 2 xx)
(math-pi-over-2))
(math-pi))))
(nb (math-floor (math-div (math-sub (nth 3 xx)
(math-pi-over-2))
(math-pi)))))
(and (equal na nb)
(math-sort-intv (nth 1 x)
(math-tan-raw (nth 2 xx))
(math-tan-raw (nth 3 xx))))))
'(intv 3 (neg (var inf var-inf)) (var inf var-inf))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'scalarp x)
(list 'calcFunc-tan x))))
(defun calcFunc-sec (x)
(cond ((and (integerp x)
(eq calc-angle-mode 'deg)
(= (% x 180) 0))
(if (= (% x 360) 0)
1
-1))
((and (integerp x)
(eq calc-angle-mode 'rad)
(= x 0))
1)
((Math-scalarp x)
(math-with-extra-prec 2
(math-sec-raw (math-to-radians (math-float x)))))
((eq (car x) 'sdev)
(if (math-constp x)
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float (nth 1 x))))
(xs (math-to-radians (math-float (nth 2 x))))
(sc (math-sin-cos-raw xx)))
(if (and (math-zerop (cdr sc))
(not calc-infinite-mode))
(progn
(calc-record-why "*Division by zero")
(list 'calcFunc-sec x))
(math-make-sdev (math-div-float '(float 1 0) (cdr sc))
(math-div-float
(math-mul xs (car sc))
(math-sqr (cdr sc)))))))
(math-make-sdev (calcFunc-sec (nth 1 x))
(math-div
(math-mul (nth 2 x)
(calcFunc-sin (nth 1 x)))
(math-sqr (calcFunc-cos (nth 1 x)))))))
((and (eq (car x) 'intv)
(math-intv-constp x))
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float x)))
(na (math-floor (math-div (math-sub (nth 2 xx)
(math-pi-over-2))
(math-pi))))
(nb (math-floor (math-div (math-sub (nth 3 xx)
(math-pi-over-2))
(math-pi))))
(naa (math-floor (math-div (nth 2 xx) (math-pi-over-2))))
(nbb (math-floor (math-div (nth 3 xx) (math-pi-over-2))))
(span (math-sub nbb naa)))
(if (not (equal na nb))
'(intv 3 (neg (var inf var-inf)) (var inf var-inf))
(let ((int (math-sort-intv (nth 1 x)
(math-sec-raw (nth 2 xx))
(math-sec-raw (nth 3 xx)))))
(if (eq span 1)
(if (math-evenp (math-div (math-add naa 1) 2))
(math-make-intv (logior (nth 1 int) 2)
1
(nth 3 int))
(math-make-intv (logior (nth 1 int) 1)
(nth 2 int)
-1))
int))))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'scalarp x)
(list 'calcFunc-sec x))))
(defun calcFunc-csc (x)
(cond ((and (integerp x)
(eq calc-angle-mode 'deg)
(= (% (- x 90) 180) 0))
(if (= (% (- x 90) 360) 0)
1
-1))
((Math-scalarp x)
(math-with-extra-prec 2
(math-csc-raw (math-to-radians (math-float x)))))
((eq (car x) 'sdev)
(if (math-constp x)
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float (nth 1 x))))
(xs (math-to-radians (math-float (nth 2 x))))
(sc (math-sin-cos-raw xx)))
(if (and (math-zerop (car sc))
(not calc-infinite-mode))
(progn
(calc-record-why "*Division by zero")
(list 'calcFunc-csc x))
(math-make-sdev (math-div-float '(float 1 0) (car sc))
(math-div-float
(math-mul xs (cdr sc))
(math-sqr (car sc)))))))
(math-make-sdev (calcFunc-csc (nth 1 x))
(math-div
(math-mul (nth 2 x)
(calcFunc-cos (nth 1 x)))
(math-sqr (calcFunc-sin (nth 1 x)))))))
((and (eq (car x) 'intv)
(math-intv-constp x))
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float x)))
(na (math-floor (math-div (nth 2 xx) (math-pi))))
(nb (math-floor (math-div (nth 3 xx) (math-pi))))
(naa (math-floor (math-div (nth 2 xx) (math-pi-over-2))))
(nbb (math-floor (math-div (nth 3 xx) (math-pi-over-2))))
(span (math-sub nbb naa)))
(if (not (equal na nb))
'(intv 3 (neg (var inf var-inf)) (var inf var-inf))
(let ((int (math-sort-intv (nth 1 x)
(math-csc-raw (nth 2 xx))
(math-csc-raw (nth 3 xx)))))
(if (eq span 1)
(if (math-evenp (math-div naa 2))
(math-make-intv (logior (nth 1 int) 2)
1
(nth 3 int))
(math-make-intv (logior (nth 1 int) 1)
(nth 2 int)
-1))
int))))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'scalarp x)
(list 'calcFunc-csc x))))
(defun calcFunc-cot (x) ; [N N] [Public]
(cond ((and (integerp x)
(if (eq calc-angle-mode 'deg)
(= (% (- x 90) 180) 0)
(= x 0)))
0)
((Math-scalarp x)
(math-with-extra-prec 2
(math-cot-raw (math-to-radians (math-float x)))))
((eq (car x) 'sdev)
(if (math-constp x)
(math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float (nth 1 x))))
(xs (math-to-radians (math-float (nth 2 x))))
(sc (math-sin-cos-raw xx)))
(if (and (math-zerop (car sc)) (not calc-infinite-mode))
(progn
(calc-record-why "*Division by zero")
(list 'calcFunc-cot x))
(math-make-sdev (math-div-float (cdr sc) (car sc))
(math-div-float xs (math-sqr (car sc)))))))
(math-make-sdev (calcFunc-cot (nth 1 x))
(math-div (nth 2 x)
(math-sqr (calcFunc-sin (nth 1 x)))))))
((and (eq (car x) 'intv) (math-intv-constp x))
(or (math-with-extra-prec 2
(let* ((xx (math-to-radians (math-float x)))
(na (math-floor (math-div (nth 2 xx) (math-pi))))
(nb (math-floor (math-div (nth 3 xx) (math-pi)))))
(and (equal na nb)
(math-sort-intv (nth 1 x)
(math-cot-raw (nth 2 xx))
(math-cot-raw (nth 3 xx))))))
'(intv 3 (neg (var inf var-inf)) (var inf var-inf))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'scalarp x)
(list 'calcFunc-cot x))))
(defun math-sin-raw (x &optional orgx) ; [N N]
(cond ((eq (car x) 'cplx)
(let* ((expx (math-exp-raw (nth 2 x)))
(expmx (math-div-float '(float 1 0) expx))
(sc (math-sin-cos-raw (nth 1 x))))
(list 'cplx
(math-mul-float (car sc)
(math-mul-float (math-add-float expx expmx)
'(float 5 -1)))
(math-mul-float (cdr sc)
(math-mul-float (math-sub-float expx expmx)
'(float 5 -1))))))
((eq (car x) 'polar)
(math-polar (math-sin-raw (math-complex x))))
((Math-integer-negp (nth 1 x))
(math-neg-float (math-sin-raw (math-neg-float x) (if orgx orgx x))))
((math-lessp-float '(float 7 0) x) ; avoid inf loops due to roundoff
(math-sin-raw (math-mod x (math-two-pi)) (if orgx orgx x)))
(t (math-sin-raw-2 x (if orgx orgx x)))))
(defun math-cos-raw (x) ; [N N]
(if (eq (car-safe x) 'polar)
(math-polar (math-cos-raw (math-complex x)))
(math-sin-raw (math-sub (math-pi-over-2) x) x)))
(defun math-sec-raw (x) ; [N N]
(cond ((eq (car x) 'cplx)
(let* ((x (math-mul x '(float 1 0)))
(expx (math-exp-raw (nth 2 x)))
(expmx (math-div-float '(float 1 0) expx))
(sh (math-mul-float (math-sub-float expx expmx) '(float 5 -1)))
(ch (math-mul-float (math-add-float expx expmx) '(float 5 -1)))
(sc (math-sin-cos-raw (nth 1 x)))
(d (math-add-float
(math-mul-float (math-sqr (car sc))
(math-sqr sh))
(math-mul-float (math-sqr (cdr sc))
(math-sqr ch)))))
(and (not (eq (nth 1 d) 0))
(list 'cplx
(math-div-float (math-mul-float (cdr sc) ch) d)
(math-div-float (math-mul-float (car sc) sh) d)))))
((eq (car x) 'polar)
(math-polar (math-sec-raw (math-complex x))))
(t
(let ((cs (math-cos-raw x)))
(if (eq cs 0)
(math-div 1 0)
(math-div-float '(float 1 0) cs))))))
(defun math-csc-raw (x) ; [N N]
(cond ((eq (car x) 'cplx)
(let* ((x (math-mul x '(float 1 0)))
(expx (math-exp-raw (nth 2 x)))
(expmx (math-div-float '(float 1 0) expx))
(sh (math-mul-float (math-sub-float expx expmx) '(float 5 -1)))
(ch (math-mul-float (math-add-float expx expmx) '(float 5 -1)))
(sc (math-sin-cos-raw (nth 1 x)))
(d (math-add-float
(math-mul-float (math-sqr (car sc))
(math-sqr ch))
(math-mul-float (math-sqr (cdr sc))
(math-sqr sh)))))
(and (not (eq (nth 1 d) 0))
(list 'cplx
(math-div-float (math-mul-float (car sc) ch) d)
(math-div-float (math-mul-float (cdr sc) sh) d)))))
((eq (car x) 'polar)
(math-polar (math-csc-raw (math-complex x))))
(t
(let ((sn (math-sin-raw x)))
(if (eq sn 0)
(math-div 1 0)
(math-div-float '(float 1 0) sn))))))
(defun math-cot-raw (x) ; [N N]
(cond ((eq (car x) 'cplx)
(let* ((x (math-mul x '(float 1 0)))
(expx (math-exp-raw (nth 2 x)))
(expmx (math-div-float '(float 1 0) expx))
(sh (math-mul-float (math-sub-float expx expmx) '(float 5 -1)))
(ch (math-mul-float (math-add-float expx expmx) '(float 5 -1)))
(sc (math-sin-cos-raw (nth 1 x)))
(d (math-add-float
(math-sqr (car sc))
(math-sqr sh))))
(and (not (eq (nth 1 d) 0))
(list 'cplx
(math-div-float
(math-mul-float (car sc) (cdr sc))
d)
(math-neg
(math-div-float
(math-mul-float sh ch)
d))))))
((eq (car x) 'polar)
(math-polar (math-cot-raw (math-complex x))))
(t
(let ((sc (math-sin-cos-raw x)))
(if (eq (nth 1 (car sc)) 0)
(math-div (cdr sc) 0)
(math-div-float (cdr sc) (car sc)))))))
;;; This could use a smarter method: Reduce x as in math-sin-raw, then
;;; compute either sin(x) or cos(x), whichever is smaller, and compute
;;; the other using the identity sin(x)^2 + cos(x)^2 = 1.
(defun math-sin-cos-raw (x) ; [F.F F] (result is (sin x . cos x))
(cons (math-sin-raw x) (math-cos-raw x)))
(defun math-tan-raw (x) ; [N N]
(cond ((eq (car x) 'cplx)
(let* ((x (math-mul x '(float 2 0)))
(expx (math-exp-raw (nth 2 x)))
(expmx (math-div-float '(float 1 0) expx))
(sc (math-sin-cos-raw (nth 1 x)))
(d (math-add-float (cdr sc)
(math-mul-float (math-add-float expx expmx)
'(float 5 -1)))))
(and (not (eq (nth 1 d) 0))
(list 'cplx
(math-div-float (car sc) d)
(math-div-float (math-mul-float (math-sub-float expx
expmx)
'(float 5 -1)) d)))))
((eq (car x) 'polar)
(math-polar (math-tan-raw (math-complex x))))
(t
(let ((sc (math-sin-cos-raw x)))
(if (eq (nth 1 (cdr sc)) 0)
(math-div (car sc) 0)
(math-div-float (car sc) (cdr sc)))))))
(defun math-sin-raw-2 (x orgx) ; This avoids poss of inf recursion. [F F]
(let ((xmpo2 (math-sub-float (math-pi-over-2) x)))
(cond ((Math-integer-negp (nth 1 xmpo2))
(math-neg-float (math-sin-raw-2 (math-sub-float x (math-pi))
orgx)))
((math-lessp-float (math-pi-over-4) x)
(math-cos-raw-2 xmpo2 orgx))
((math-lessp-float x (math-neg (math-pi-over-4)))
(math-neg (math-cos-raw-2 (math-add (math-pi-over-2) x) orgx)))
((math-with-extra-prec -1 (math-nearly-zerop-float x orgx))
'(float 0 0))
((math-use-emacs-fn 'sin x))
(calc-symbolic-mode (signal 'inexact-result nil))
(t (math-sin-series x 6 4 x (math-neg-float (math-sqr-float x)))))))
(defun math-cos-raw-2 (x orgx) ; [F F]
(cond ((math-with-extra-prec -1 (math-nearly-zerop-float x orgx))
'(float 1 0))
((math-use-emacs-fn 'cos x))
(calc-symbolic-mode (signal 'inexact-result nil))
(t (let ((xnegsqr (math-neg-float (math-sqr-float x))))
(math-sin-series
(math-add-float '(float 1 0)
(math-mul-float xnegsqr '(float 5 -1)))
24 5 xnegsqr xnegsqr)))))
(defun math-sin-series (sum nfac n x xnegsqr)
(math-working "sin" sum)
(let* ((nextx (math-mul-float x xnegsqr))
(nextsum (math-add-float sum (math-div-float nextx
(math-float nfac)))))
(if (math-nearly-equal-float sum nextsum)
sum
(math-sin-series nextsum (math-mul nfac (* n (1+ n)))
(+ n 2) nextx xnegsqr))))
;;; Inverse sine, cosine, tangent.
(defun calcFunc-arcsin (x) ; [N N] [Public]
(cond ((eq x 0) 0)
((and (eq x 1) (eq calc-angle-mode 'deg)) 90)
((and (eq x -1) (eq calc-angle-mode 'deg)) -90)
(calc-symbolic-mode (signal 'inexact-result nil))
((Math-numberp x)
(math-with-extra-prec 2
(math-from-radians (math-arcsin-raw (math-float x)))))
((eq (car x) 'sdev)
(math-make-sdev (calcFunc-arcsin (nth 1 x))
(math-from-radians
(math-div (nth 2 x)
(math-sqrt
(math-sub 1 (math-sqr (nth 1 x))))))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-arcsin (nth 2 x))
(calcFunc-arcsin (nth 3 x))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-arcsin x))))
(defun calcFunc-arccos (x) ; [N N] [Public]
(cond ((eq x 1) 0)
((and (eq x 0) (eq calc-angle-mode 'deg)) 90)
((and (eq x -1) (eq calc-angle-mode 'deg)) 180)
(calc-symbolic-mode (signal 'inexact-result nil))
((Math-numberp x)
(math-with-extra-prec 2
(math-from-radians (math-arccos-raw (math-float x)))))
((eq (car x) 'sdev)
(math-make-sdev (calcFunc-arccos (nth 1 x))
(math-from-radians
(math-div (nth 2 x)
(math-sqrt
(math-sub 1 (math-sqr (nth 1 x))))))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-arccos (nth 2 x))
(calcFunc-arccos (nth 3 x))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-arccos x))))
(defun calcFunc-arctan (x) ; [N N] [Public]
(cond ((eq x 0) 0)
((and (eq x 1) (eq calc-angle-mode 'deg)) 45)
((and (eq x -1) (eq calc-angle-mode 'deg)) -45)
((Math-numberp x)
(math-with-extra-prec 2
(math-from-radians (math-arctan-raw (math-float x)))))
((eq (car x) 'sdev)
(math-make-sdev (calcFunc-arctan (nth 1 x))
(math-from-radians
(math-div (nth 2 x)
(math-add 1 (math-sqr (nth 1 x)))))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-arctan (nth 2 x))
(calcFunc-arctan (nth 3 x))))
((equal x '(var inf var-inf))
(math-quarter-circle t))
((equal x '(neg (var inf var-inf)))
(math-neg (math-quarter-circle t)))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-arctan x))))
(defun math-arcsin-raw (x) ; [N N]
(let ((a (math-sqrt-raw (math-sub '(float 1 0) (math-sqr x)))))
(if (or (memq (car x) '(cplx polar))
(memq (car a) '(cplx polar)))
(math-with-extra-prec 2 ; use extra precision for difficult case
(math-mul '(cplx 0 -1)
(math-ln-raw (math-add (math-mul '(cplx 0 1) x) a))))
(math-arctan2-raw x a))))
(defun math-arccos-raw (x) ; [N N]
(math-sub (math-pi-over-2) (math-arcsin-raw x)))
(defun math-arctan-raw (x) ; [N N]
(cond ((memq (car x) '(cplx polar))
(math-with-extra-prec 2 ; extra-extra
(math-div (math-sub
(math-ln-raw (math-add 1 (math-mul '(cplx 0 1) x)))
(math-ln-raw (math-add 1 (math-mul '(cplx 0 -1) x))))
'(cplx 0 2))))
((Math-integer-negp (nth 1 x))
(math-neg-float (math-arctan-raw (math-neg-float x))))
((math-zerop x) x)
((math-use-emacs-fn 'atan x))
(calc-symbolic-mode (signal 'inexact-result nil))
((math-equal-int x 1) (math-pi-over-4))
((math-equal-int x -1) (math-neg (math-pi-over-4)))
((math-lessp-float '(float 414214 -6) x) ; if x > sqrt(2) - 1, reduce
(if (math-lessp-float '(float 1 0) x)
(math-sub-float (math-mul-float (math-pi) '(float 5 -1))
(math-arctan-raw (math-div-float '(float 1 0) x)))
(math-sub-float (math-mul-float (math-pi) '(float 25 -2))
(math-arctan-raw (math-div-float
(math-sub-float '(float 1 0) x)
(math-add-float '(float 1 0)
x))))))
(t (math-arctan-series x 3 x (math-neg-float (math-sqr-float x))))))
(defun math-arctan-series (sum n x xnegsqr)
(math-working "arctan" sum)
(let* ((nextx (math-mul-float x xnegsqr))
(nextsum (math-add-float sum (math-div-float nextx (math-float n)))))
(if (math-nearly-equal-float sum nextsum)
sum
(math-arctan-series nextsum (+ n 2) nextx xnegsqr))))
(defun calcFunc-arctan2 (y x) ; [F R R] [Public]
(if (Math-anglep y)
(if (Math-anglep x)
(math-with-extra-prec 2
(math-from-radians (math-arctan2-raw (math-float y)
(math-float x))))
(calc-record-why 'anglep x)
(list 'calcFunc-arctan2 y x))
(if (and (or (math-infinitep x) (math-anglep x))
(or (math-infinitep y) (math-anglep y)))
(progn
(if (math-posp x)
(setq x 1)
(if (math-negp x)
(setq x -1)
(or (math-zerop x)
(setq x nil))))
(if (math-posp y)
(setq y 1)
(if (math-negp y)
(setq y -1)
(or (math-zerop y)
(setq y nil))))
(if (and y x)
(calcFunc-arctan2 y x)
'(var nan var-nan)))
(calc-record-why 'anglep y)
(list 'calcFunc-arctan2 y x))))
(defun math-arctan2-raw (y x) ; [F R R]
(cond ((math-zerop y)
(if (math-negp x) (math-pi)
(if (or (math-floatp x) (math-floatp y)) '(float 0 0) 0)))
((math-zerop x)
(if (math-posp y)
(math-pi-over-2)
(math-neg (math-pi-over-2))))
((math-posp x)
(math-arctan-raw (math-div-float y x)))
((math-posp y)
(math-add-float (math-arctan-raw (math-div-float y x))
(math-pi)))
(t
(math-sub-float (math-arctan-raw (math-div-float y x))
(math-pi)))))
(defun calcFunc-arcsincos (x) ; [V N] [Public]
(if (and (Math-vectorp x)
(= (length x) 3))
(calcFunc-arctan2 (nth 2 x) (nth 1 x))
(math-reject-arg x "*Two-element vector expected")))
;;; Exponential function.
(defun calcFunc-exp (x) ; [N N] [Public]
(cond ((eq x 0) 1)
((and (memq x '(1 -1)) calc-symbolic-mode)
(if (eq x 1) '(var e var-e) (math-div 1 '(var e var-e))))
((Math-numberp x)
(math-with-extra-prec 2 (math-exp-raw (math-float x))))
((eq (car-safe x) 'sdev)
(let ((ex (calcFunc-exp (nth 1 x))))
(math-make-sdev ex (math-mul (nth 2 x) ex))))
((eq (car-safe x) 'intv)
(math-make-intv (nth 1 x) (calcFunc-exp (nth 2 x))
(calcFunc-exp (nth 3 x))))
((equal x '(var inf var-inf))
x)
((equal x '(neg (var inf var-inf)))
0)
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-exp x))))
(defun calcFunc-expm1 (x) ; [N N] [Public]
(cond ((eq x 0) 0)
((math-zerop x) '(float 0 0))
(calc-symbolic-mode (signal 'inexact-result nil))
((Math-numberp x)
(math-with-extra-prec 2
(let ((x (math-float x)))
(if (and (eq (car x) 'float)
(math-lessp-float x '(float 1 0))
(math-lessp-float '(float -1 0) x))
(math-exp-minus-1-raw x)
(math-add (math-exp-raw x) -1)))))
((eq (car-safe x) 'sdev)
(if (math-constp x)
(let ((ex (calcFunc-expm1 (nth 1 x))))
(math-make-sdev ex (math-mul (nth 2 x) (math-add ex 1))))
(math-make-sdev (calcFunc-expm1 (nth 1 x))
(math-mul (nth 2 x) (calcFunc-exp (nth 1 x))))))
((eq (car-safe x) 'intv)
(math-make-intv (nth 1 x)
(calcFunc-expm1 (nth 2 x))
(calcFunc-expm1 (nth 3 x))))
((equal x '(var inf var-inf))
x)
((equal x '(neg (var inf var-inf)))
-1)
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-expm1 x))))
(defun calcFunc-exp10 (x) ; [N N] [Public]
(if (eq x 0)
1
(math-pow '(float 1 1) x)))
(defun math-exp-raw (x) ; [N N]
(cond ((math-zerop x) '(float 1 0))
(calc-symbolic-mode (signal 'inexact-result nil))
((eq (car x) 'cplx)
(let ((expx (math-exp-raw (nth 1 x)))
(sc (math-sin-cos-raw (nth 2 x))))
(list 'cplx
(math-mul-float expx (cdr sc))
(math-mul-float expx (car sc)))))
((eq (car x) 'polar)
(let ((xc (math-complex x)))
(list 'polar
(math-exp-raw (nth 1 xc))
(math-from-radians (nth 2 xc)))))
((math-use-emacs-fn 'exp x))
((or (math-lessp-float '(float 5 -1) x)
(math-lessp-float x '(float -5 -1)))
(if (math-lessp-float '(float 921035 1) x)
(math-overflow)
(if (math-lessp-float x '(float -921035 1))
(math-underflow)))
(let* ((two-x (math-mul-float x '(float 2 0)))
(hint (math-scale-int (nth 1 two-x) (nth 2 two-x)))
(hfrac (math-sub-float x (math-mul-float (math-float hint)
'(float 5 -1)))))
(math-mul-float (math-ipow (math-sqrt-e) hint)
(math-add-float '(float 1 0)
(math-exp-minus-1-raw hfrac)))))
(t (math-add-float '(float 1 0) (math-exp-minus-1-raw x)))))
(defun math-exp-minus-1-raw (x) ; [F F]
(math-exp-series x 2 3 x x))
(defun math-exp-series (sum nfac n xpow x)
(math-working "exp" sum)
(let* ((nextx (math-mul-float xpow x))
(nextsum (math-add-float sum (math-div-float nextx
(math-float nfac)))))
(if (math-nearly-equal-float sum nextsum)
sum
(math-exp-series nextsum (math-mul nfac n) (1+ n) nextx x))))
;;; Logarithms.
(defun calcFunc-ln (x) ; [N N] [Public]
(cond ((math-zerop x)
(if calc-infinite-mode
'(neg (var inf var-inf))
(math-reject-arg x "*Logarithm of zero")))
((eq x 1) 0)
((Math-numberp x)
(math-with-extra-prec 2 (math-ln-raw (math-float x))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-ln (nth 1 x))
(math-div (nth 2 x) (nth 1 x))))
((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x))
(Math-zerop (nth 2 x))
(not (math-intv-constp x))))
(let ((calc-infinite-mode t))
(math-make-intv (nth 1 x) (calcFunc-ln (nth 2 x))
(calcFunc-ln (nth 3 x)))))
((equal x '(var e var-e))
1)
((and (eq (car-safe x) '^)
(equal (nth 1 x) '(var e var-e))
(math-known-realp (nth 2 x)))
(nth 2 x))
((math-infinitep x)
(if (equal x '(var nan var-nan))
x
'(var inf var-inf)))
(t (calc-record-why 'numberp x)
(list 'calcFunc-ln x))))
(defun calcFunc-log10 (x) ; [N N] [Public]
(cond ((math-equal-int x 1)
(if (math-floatp x) '(float 0 0) 0))
((and (Math-integerp x)
(math-posp x)
(let ((res (math-integer-log x 10)))
(and (car res)
(setq x (cdr res)))))
x)
((and (eq (car-safe x) 'frac)
(eq (nth 1 x) 1)
(let ((res (math-integer-log (nth 2 x) 10)))
(and (car res)
(setq x (- (cdr res))))))
x)
((math-zerop x)
(if calc-infinite-mode
'(neg (var inf var-inf))
(math-reject-arg x "*Logarithm of zero")))
(calc-symbolic-mode (signal 'inexact-result nil))
((Math-numberp x)
(math-with-extra-prec 2
(let ((xf (math-float x)))
(if (eq (nth 1 xf) 0)
(math-reject-arg x "*Logarithm of zero"))
(if (Math-integer-posp (nth 1 xf))
(if (eq (nth 1 xf) 1) ; log10(1*10^n) = n
(math-float (nth 2 xf))
(let ((xdigs (1- (math-numdigs (nth 1 xf)))))
(math-add-float
(math-div-float (math-ln-raw-2
(list 'float (nth 1 xf) (- xdigs)))
(math-ln-10))
(math-float (+ (nth 2 xf) xdigs)))))
(math-div (calcFunc-ln xf) (math-ln-10))))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-log10 (nth 1 x))
(math-div (nth 2 x)
(math-mul (nth 1 x) (math-ln-10)))))
((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x))
(not (math-intv-constp x))))
(math-make-intv (nth 1 x)
(calcFunc-log10 (nth 2 x))
(calcFunc-log10 (nth 3 x))))
((math-infinitep x)
(if (equal x '(var nan var-nan))
x
'(var inf var-inf)))
(t (calc-record-why 'numberp x)
(list 'calcFunc-log10 x))))
(defun calcFunc-log (x &optional b) ; [N N N] [Public]
(cond ((or (null b) (equal b '(var e var-e)))
(math-normalize (list 'calcFunc-ln x)))
((or (eq b 10) (equal b '(float 1 1)))
(math-normalize (list 'calcFunc-log10 x)))
((math-zerop x)
(if calc-infinite-mode
(math-div (calcFunc-ln x) (calcFunc-ln b))
(math-reject-arg x "*Logarithm of zero")))
((math-zerop b)
(if calc-infinite-mode
(math-div (calcFunc-ln x) (calcFunc-ln b))
(math-reject-arg b "*Logarithm of zero")))
((math-equal-int b 1)
(if calc-infinite-mode
(math-div (calcFunc-ln x) 0)
(math-reject-arg b "*Logarithm base one")))
((math-equal-int x 1)
(if (math-floatp b) '(float 0 0) 0))
((and (Math-ratp x) (Math-ratp b)
(math-posp x) (math-posp b)
(let* ((sign 1) (inv nil)
(xx (if (Math-lessp 1 x)
x
(setq sign -1)
(math-div 1 x)))
(bb (if (Math-lessp 1 b)
b
(setq sign (- sign))
(math-div 1 b)))
(res (if (Math-lessp xx bb)
(setq inv (math-integer-log bb xx))
(math-integer-log xx bb))))
(and (car res)
(setq x (if inv
(math-div 1 (* sign (cdr res)))
(* sign (cdr res)))))))
x)
(calc-symbolic-mode (signal 'inexact-result nil))
((and (Math-numberp x) (Math-numberp b))
(math-with-extra-prec 2
(math-div (math-ln-raw (math-float x))
(math-log-base-raw b))))
((and (eq (car-safe x) 'sdev)
(Math-numberp b))
(math-make-sdev (calcFunc-log (nth 1 x) b)
(math-div (nth 2 x)
(math-mul (nth 1 x)
(math-log-base-raw b)))))
((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x))
(not (math-intv-constp x)))
(math-realp b))
(math-make-intv (nth 1 x)
(calcFunc-log (nth 2 x) b)
(calcFunc-log (nth 3 x) b)))
((or (eq (car-safe x) 'intv) (eq (car-safe b) 'intv))
(math-div (calcFunc-ln x) (calcFunc-ln b)))
((or (math-infinitep x)
(math-infinitep b))
(math-div (calcFunc-ln x) (calcFunc-ln b)))
(t (if (Math-numberp b)
(calc-record-why 'numberp x)
(calc-record-why 'numberp b))
(list 'calcFunc-log x b))))
(defun calcFunc-alog (x &optional b)
(cond ((or (null b) (equal b '(var e var-e)))
(math-normalize (list 'calcFunc-exp x)))
(t (math-pow b x))))
(defun calcFunc-ilog (x b)
(if (and (math-natnump x) (not (eq x 0))
(math-natnump b) (not (eq b 0)))
(if (eq b 1)
(math-reject-arg x "*Logarithm base one")
(if (Math-natnum-lessp x b)
0
(cdr (math-integer-log x b))))
(math-floor (calcFunc-log x b))))
(defun math-integer-log (x b)
(let ((pows (list b))
(pow (math-sqr b))
next
sum n)
(while (not (Math-lessp x pow))
(setq pows (cons pow pows)
pow (math-sqr pow)))
(setq n (lsh 1 (1- (length pows)))
sum n
pow (car pows))
(while (and (setq pows (cdr pows))
(Math-lessp pow x))
(setq n (/ n 2)
next (math-mul pow (car pows)))
(or (Math-lessp x next)
(setq pow next
sum (+ sum n))))
(cons (equal pow x) sum)))
(defvar math-log-base-cache nil)
(defun math-log-base-raw (b) ; [N N]
(if (not (and (equal (car math-log-base-cache) b)
(eq (nth 1 math-log-base-cache) calc-internal-prec)))
(setq math-log-base-cache (list b calc-internal-prec
(math-ln-raw (math-float b)))))
(nth 2 math-log-base-cache))
(defun calcFunc-lnp1 (x) ; [N N] [Public]
(cond ((Math-equal-int x -1)
(if calc-infinite-mode
'(neg (var inf var-inf))
(math-reject-arg x "*Logarithm of zero")))
((eq x 0) 0)
((math-zerop x) '(float 0 0))
(calc-symbolic-mode (signal 'inexact-result nil))
((Math-numberp x)
(math-with-extra-prec 2
(let ((x (math-float x)))
(if (and (eq (car x) 'float)
(math-lessp-float x '(float 5 -1))
(math-lessp-float '(float -5 -1) x))
(math-ln-plus-1-raw x)
(math-ln-raw (math-add-float x '(float 1 0)))))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-lnp1 (nth 1 x))
(math-div (nth 2 x) (math-add (nth 1 x) 1))))
((and (eq (car-safe x) 'intv) (or (Math-posp (nth 2 x))
(not (math-intv-constp x))))
(math-make-intv (nth 1 x)
(calcFunc-lnp1 (nth 2 x))
(calcFunc-lnp1 (nth 3 x))))
((math-infinitep x)
(if (equal x '(var nan var-nan))
x
'(var inf var-inf)))
(t (calc-record-why 'numberp x)
(list 'calcFunc-lnp1 x))))
(defun math-ln-raw (x) ; [N N] --- must be float format!
(cond ((eq (car-safe x) 'cplx)
(list 'cplx
(math-mul-float (math-ln-raw
(math-add-float (math-sqr-float (nth 1 x))
(math-sqr-float (nth 2 x))))
'(float 5 -1))
(math-arctan2-raw (nth 2 x) (nth 1 x))))
((eq (car x) 'polar)
(math-polar (list 'cplx
(math-ln-raw (nth 1 x))
(math-to-radians (nth 2 x)))))
((Math-equal-int x 1)
'(float 0 0))
(calc-symbolic-mode (signal 'inexact-result nil))
((math-posp (nth 1 x)) ; positive and real
(cond
((math-use-emacs-fn 'log x))
(t
(let ((xdigs (1- (math-numdigs (nth 1 x)))))
(math-add-float (math-ln-raw-2 (list 'float (nth 1 x) (- xdigs)))
(math-mul-float (math-float (+ (nth 2 x) xdigs))
(math-ln-10)))))))
((math-zerop x)
(math-reject-arg x "*Logarithm of zero"))
((eq calc-complex-mode 'polar) ; negative and real
(math-polar
(list 'cplx ; negative and real
(math-ln-raw (math-neg-float x))
(math-pi))))
(t (list 'cplx ; negative and real
(math-ln-raw (math-neg-float x))
(math-pi)))))
(defun math-ln-raw-2 (x) ; [F F]
(cond ((math-lessp-float '(float 14 -1) x)
(math-add-float (math-ln-raw-2 (math-mul-float x '(float 5 -1)))
(math-ln-2)))
(t ; now .7 < x <= 1.4
(math-ln-raw-3 (math-div-float (math-sub-float x '(float 1 0))
(math-add-float x '(float 1 0)))))))
(defun math-ln-raw-3 (x) ; [F F]
(math-mul-float (math-ln-raw-series x 3 x (math-sqr-float x))
'(float 2 0)))
;;; Compute ln((1+x)/(1-x))
(defun math-ln-raw-series (sum n x xsqr)
(math-working "log" sum)
(let* ((nextx (math-mul-float x xsqr))
(nextsum (math-add-float sum (math-div-float nextx (math-float n)))))
(if (math-nearly-equal-float sum nextsum)
sum
(math-ln-raw-series nextsum (+ n 2) nextx xsqr))))
(defun math-ln-plus-1-raw (x)
(math-lnp1-series x 2 x (math-neg x)))
(defun math-lnp1-series (sum n xpow x)
(math-working "lnp1" sum)
(let* ((nextx (math-mul-float xpow x))
(nextsum (math-add-float sum (math-div-float nextx (math-float n)))))
(if (math-nearly-equal-float sum nextsum)
sum
(math-lnp1-series nextsum (1+ n) nextx x))))
(defconst math-approx-ln-10
(math-read-number-simple "2.302585092994045684018")
"An approximation for ln(10).")
(math-defcache math-ln-10 math-approx-ln-10
(math-ln-raw-2 '(float 1 1)))
(defconst math-approx-ln-2
(math-read-number-simple "0.693147180559945309417")
"An approximation for ln(2).")
(math-defcache math-ln-2 math-approx-ln-2
(math-ln-raw-3 (math-float '(frac 1 3))))
;;; Hyperbolic functions.
(defun calcFunc-sinh (x) ; [N N] [Public]
(cond ((eq x 0) 0)
(math-expand-formulas
(math-normalize
(list '/ (list '- (list 'calcFunc-exp x)
(list 'calcFunc-exp (list 'neg x))) 2)))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(let ((expx (math-exp-raw (math-float x))))
(math-mul (math-add expx (math-div -1 expx)) '(float 5 -1)))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-sinh (nth 1 x))
(math-mul (nth 2 x) (calcFunc-cosh (nth 1 x)))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-sinh (nth 2 x))
(calcFunc-sinh (nth 3 x))))
((or (equal x '(var inf var-inf))
(equal x '(neg (var inf var-inf)))
(equal x '(var nan var-nan)))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-sinh x))))
(put 'calcFunc-sinh 'math-expandable t)
(defun calcFunc-cosh (x) ; [N N] [Public]
(cond ((eq x 0) 1)
(math-expand-formulas
(math-normalize
(list '/ (list '+ (list 'calcFunc-exp x)
(list 'calcFunc-exp (list 'neg x))) 2)))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(let ((expx (math-exp-raw (math-float x))))
(math-mul (math-add expx (math-div 1 expx)) '(float 5 -1)))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-cosh (nth 1 x))
(math-mul (nth 2 x)
(calcFunc-sinh (nth 1 x)))))
((and (eq (car x) 'intv) (math-intv-constp x))
(setq x (math-abs x))
(math-sort-intv (nth 1 x)
(calcFunc-cosh (nth 2 x))
(calcFunc-cosh (nth 3 x))))
((or (equal x '(var inf var-inf))
(equal x '(neg (var inf var-inf)))
(equal x '(var nan var-nan)))
(math-abs x))
(t (calc-record-why 'numberp x)
(list 'calcFunc-cosh x))))
(put 'calcFunc-cosh 'math-expandable t)
(defun calcFunc-tanh (x) ; [N N] [Public]
(cond ((eq x 0) 0)
(math-expand-formulas
(math-normalize
(let ((expx (list 'calcFunc-exp x))
(expmx (list 'calcFunc-exp (list 'neg x))))
(math-normalize
(list '/ (list '- expx expmx) (list '+ expx expmx))))))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(let* ((expx (calcFunc-exp (math-float x)))
(expmx (math-div 1 expx)))
(math-div (math-sub expx expmx)
(math-add expx expmx)))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-tanh (nth 1 x))
(math-div (nth 2 x)
(math-sqr (calcFunc-cosh (nth 1 x))))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-tanh (nth 2 x))
(calcFunc-tanh (nth 3 x))))
((equal x '(var inf var-inf))
1)
((equal x '(neg (var inf var-inf)))
-1)
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-tanh x))))
(put 'calcFunc-tanh 'math-expandable t)
(defun calcFunc-sech (x) ; [N N] [Public]
(cond ((eq x 0) 1)
(math-expand-formulas
(math-normalize
(list '/ 2 (list '+ (list 'calcFunc-exp x)
(list 'calcFunc-exp (list 'neg x))))))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(let ((expx (math-exp-raw (math-float x))))
(math-div '(float 2 0) (math-add expx (math-div 1 expx))))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-sech (nth 1 x))
(math-mul (nth 2 x)
(math-mul (calcFunc-sech (nth 1 x))
(calcFunc-tanh (nth 1 x))))))
((and (eq (car x) 'intv) (math-intv-constp x))
(setq x (math-abs x))
(math-sort-intv (nth 1 x)
(calcFunc-sech (nth 2 x))
(calcFunc-sech (nth 3 x))))
((or (equal x '(var inf var-inf))
(equal x '(neg (var inf var-inf))))
0)
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-sech x))))
(put 'calcFunc-sech 'math-expandable t)
(defun calcFunc-csch (x) ; [N N] [Public]
(cond ((eq x 0) (math-div 1 0))
(math-expand-formulas
(math-normalize
(list '/ 2 (list '- (list 'calcFunc-exp x)
(list 'calcFunc-exp (list 'neg x))))))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(let ((expx (math-exp-raw (math-float x))))
(math-div '(float 2 0) (math-add expx (math-div -1 expx))))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-csch (nth 1 x))
(math-mul (nth 2 x)
(math-mul (calcFunc-csch (nth 1 x))
(calcFunc-coth (nth 1 x))))))
((eq (car x) 'intv)
(if (and (Math-negp (nth 2 x))
(Math-posp (nth 3 x)))
'(intv 3 (neg (var inf var-inf)) (var inf var-inf))
(math-sort-intv (nth 1 x)
(calcFunc-csch (nth 2 x))
(calcFunc-csch (nth 3 x)))))
((or (equal x '(var inf var-inf))
(equal x '(neg (var inf var-inf))))
0)
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-csch x))))
(put 'calcFunc-csch 'math-expandable t)
(defun calcFunc-coth (x) ; [N N] [Public]
(cond ((eq x 0) (math-div 1 0))
(math-expand-formulas
(math-normalize
(let ((expx (list 'calcFunc-exp x))
(expmx (list 'calcFunc-exp (list 'neg x))))
(math-normalize
(list '/ (list '+ expx expmx) (list '- expx expmx))))))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(let* ((expx (calcFunc-exp (math-float x)))
(expmx (math-div 1 expx)))
(math-div (math-add expx expmx)
(math-sub expx expmx)))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-coth (nth 1 x))
(math-div (nth 2 x)
(math-sqr (calcFunc-sinh (nth 1 x))))))
((eq (car x) 'intv)
(if (and (Math-negp (nth 2 x))
(Math-posp (nth 3 x)))
'(intv 3 (neg (var inf var-inf)) (var inf var-inf))
(math-sort-intv (nth 1 x)
(calcFunc-coth (nth 2 x))
(calcFunc-coth (nth 3 x)))))
((equal x '(var inf var-inf))
1)
((equal x '(neg (var inf var-inf)))
-1)
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-coth x))))
(put 'calcFunc-coth 'math-expandable t)
(defun calcFunc-arcsinh (x) ; [N N] [Public]
(cond ((eq x 0) 0)
(math-expand-formulas
(math-normalize
(list 'calcFunc-ln (list '+ x (list 'calcFunc-sqrt
(list '+ (list '^ x 2) 1))))))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(math-ln-raw (math-add x (math-sqrt-raw (math-add (math-sqr x)
'(float 1 0)))))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-arcsinh (nth 1 x))
(math-div (nth 2 x)
(math-sqrt
(math-add (math-sqr (nth 1 x)) 1)))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-arcsinh (nth 2 x))
(calcFunc-arcsinh (nth 3 x))))
((or (equal x '(var inf var-inf))
(equal x '(neg (var inf var-inf)))
(equal x '(var nan var-nan)))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-arcsinh x))))
(put 'calcFunc-arcsinh 'math-expandable t)
(defun calcFunc-arccosh (x) ; [N N] [Public]
(cond ((eq x 1) 0)
((and (eq x -1) calc-symbolic-mode)
'(var pi var-pi))
((and (eq x 0) calc-symbolic-mode)
(math-div (math-mul '(var pi var-pi) '(var i var-i)) 2))
(math-expand-formulas
(math-normalize
(list 'calcFunc-ln (list '+ x (list 'calcFunc-sqrt
(list '- (list '^ x 2) 1))))))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(if (Math-equal-int x -1)
(math-imaginary (math-pi))
(math-with-extra-prec 2
(if (or t ; need to do this even in the real case!
(memq (car-safe x) '(cplx polar)))
(let ((xp1 (math-add 1 x))) ; this gets the branch cuts right
(math-ln-raw
(math-add x (math-mul xp1
(math-sqrt-raw
(math-div (math-sub
x
'(float 1 0))
xp1))))))
(math-ln-raw
(math-add x (math-sqrt-raw (math-add (math-sqr x)
'(float -1 0)))))))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-arccosh (nth 1 x))
(math-div (nth 2 x)
(math-sqrt
(math-add (math-sqr (nth 1 x)) -1)))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-arccosh (nth 2 x))
(calcFunc-arccosh (nth 3 x))))
((or (equal x '(var inf var-inf))
(equal x '(neg (var inf var-inf)))
(equal x '(var nan var-nan)))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-arccosh x))))
(put 'calcFunc-arccosh 'math-expandable t)
(defun calcFunc-arctanh (x) ; [N N] [Public]
(cond ((eq x 0) 0)
((and (Math-equal-int x 1) calc-infinite-mode)
'(var inf var-inf))
((and (Math-equal-int x -1) calc-infinite-mode)
'(neg (var inf var-inf)))
(math-expand-formulas
(list '/ (list '-
(list 'calcFunc-ln (list '+ 1 x))
(list 'calcFunc-ln (list '- 1 x))) 2))
((Math-numberp x)
(if calc-symbolic-mode (signal 'inexact-result nil))
(math-with-extra-prec 2
(if (or (memq (car-safe x) '(cplx polar))
(Math-lessp 1 x))
(math-mul (math-sub (math-ln-raw (math-add '(float 1 0) x))
(math-ln-raw (math-sub '(float 1 0) x)))
'(float 5 -1))
(if (and (math-equal-int x 1) calc-infinite-mode)
'(var inf var-inf)
(if (and (math-equal-int x -1) calc-infinite-mode)
'(neg (var inf var-inf))
(math-mul (math-ln-raw (math-div (math-add '(float 1 0) x)
(math-sub 1 x)))
'(float 5 -1)))))))
((eq (car-safe x) 'sdev)
(math-make-sdev (calcFunc-arctanh (nth 1 x))
(math-div (nth 2 x)
(math-sub 1 (math-sqr (nth 1 x))))))
((eq (car x) 'intv)
(math-sort-intv (nth 1 x)
(calcFunc-arctanh (nth 2 x))
(calcFunc-arctanh (nth 3 x))))
((equal x '(var nan var-nan))
x)
(t (calc-record-why 'numberp x)
(list 'calcFunc-arctanh x))))
(put 'calcFunc-arctanh 'math-expandable t)
;;; Convert A from HMS or degrees to radians.
(defun calcFunc-rad (a) ; [R R] [Public]
(cond ((or (Math-numberp a)
(eq (car a) 'intv))
(math-with-extra-prec 2
(math-mul a (math-pi-over-180))))
((eq (car a) 'hms)
(math-from-hms a 'rad))
((eq (car a) 'sdev)
(math-make-sdev (calcFunc-rad (nth 1 a))
(calcFunc-rad (nth 2 a))))
(math-expand-formulas
(math-div (math-mul a '(var pi var-pi)) 180))
((math-infinitep a) a)
(t (list 'calcFunc-rad a))))
(put 'calcFunc-rad 'math-expandable t)
;;; Convert A from HMS or radians to degrees.
(defun calcFunc-deg (a) ; [R R] [Public]
(cond ((or (Math-numberp a)
(eq (car a) 'intv))
(math-with-extra-prec 2
(math-div a (math-pi-over-180))))
((eq (car a) 'hms)
(math-from-hms a 'deg))
((eq (car a) 'sdev)
(math-make-sdev (calcFunc-deg (nth 1 a))
(calcFunc-deg (nth 2 a))))
(math-expand-formulas
(math-div (math-mul 180 a) '(var pi var-pi)))
((math-infinitep a) a)
(t (list 'calcFunc-deg a))))
(put 'calcFunc-deg 'math-expandable t)
(provide 'calc-math)
;;; calc-math.el ends here
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