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;; Generic functions for generalized numbers
;; Liam Healy Tue Feb  9 1999 - 21:38
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;; Time-stamp: <2011-02-15 21:31:47EST generic.lisp>
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;; Copyright 2011 Liam M. Healy
;; Distributed under the terms of the GNU General Public License
;;
;; This program 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.
;;
;; This program 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 this program.  If not, see <http://www.gnu.org/licenses/>.

(in-package :antik)

;;; Generic functions and methods for arithmetic methods on 
;;; generalized numbers.
;;; This needs some work -
;;; * handle both sequences and scalars without commenting out stuff?
;;;    (see /i; check type then call-next-method?)

;;;;****************************************************************************
;;;; Convert numeric types
;;;;****************************************************************************

(defgeneric coerce (number like)
  (:documentation "Make number into an object similar to like, or
                   if like is a number type, into that type.")
  (:method (x like) (error 'coerce-undefined :object x :to like))
  (:method ((number number) (like number))
	   (cl:coerce number (type-of like)))
  (:method ((number symbol) (like number))
	   (error "Can't coerce symbol."))
  (:method (number (like symbol))
	   (if (eq like 't)
	       number
	     (if (subtypep like 'number)
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		 (cl:coerce number like)
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	       (error "Don't know how to coerce to ~a" like))))
  (:method ((number sequence) like)
	   (map (type-of number) (lambda (x) (coerce x like)) number))
  (:method (object (like null))
    (error 'coerce-nil :object object)))

(defgeneric gconstant (name like)
  (:method ((name (eql 'pi)) like)
	   (coerce pi like)))

;;; (with-converted-numbers ('double-float x y z) (fuzzle x y z))
(defmacro with-converted-numbers ((like &rest symbols) &body body)
  (let ((lets 
	 (mapcar (lambda (x) (list x `(coerce ,x ,like))) symbols)))
    ;; can't us pi here, error is:
    ;; Error: SYMBOL-MACROLET symbol PI is globally declared SPECIAL
    `(symbol-macrolet ((gpi (gconstant 'pi ,like)))
       (let ,lets
	 ,@body))))

;;;;****************************************************************************
;;;; Dyadic functions
;;;;****************************************************************************

;;; There is a general problem with using coerce in these functions;
;;; if the second argument is T, the first argument will be returned
;;; and the function will recurse infinitely.  
(defgeneric +i (a b)
  (:documentation "Addition of generalized or regular numbers.")
  (:method ((a number) (b number))
	   (cl:+ a b))
  (:method (a b) (funcall-dyadic '+i a b 'zerop))
  (:method ((a sequence) (b sequence))
	   (map (type-of a) #'+i a b))
  (:method ((a sequence) b)		; b is a scalar
	   (map (type-of a) (lambda (x) (+i x b)) a))
  (:method (a (b sequence))		; a is a scalar
	   (map (type-of b) (lambda (x) (+i a x)) b)))

(defun + (&rest args)
  "Addition of generalized or regular numbers with an
   arbitrary number of arguments."
  (reduce #'+i args))

(defgeneric -i (a b)
  (:documentation
   "Subtraction/negation of generalized or regular numbers, internal.
    Users call -.")
  (:method ((a number) (b number))
    (cl:- a b))
  (:method (a b)
    (funcall-dyadic
     '+i a b (lambda (x) (or (null x) (zerop x))) (lambda (x) (* -1 x))))
  (:method ((a sequence) (b sequence))
    (if (null b)
	(*i -1 a)
	(map (type-of a) #'-i a b)))
  (:method ((a sequence) b)		; b is a scalar
    (map (type-of a) (lambda (x) (-i x b)) a))
  (:method (a (b sequence))		; a is a scalar
    (if (null b)
	(*i -1 a)
	(map (type-of b) (lambda (x) (-i a x)) b))))

(defun - (&rest args)
  "Subtraction of generalized or regular numbers."
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  (if (single args) (-i (first args) nil) (reduce #'-i args)))
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(defgeneric *i (a b)
  (:documentation "Multiplication of generalized or regular numbers.")
  ;; Ordinary number multiplication
  (:method ((a number) (b number))
	   (cl:* a b))
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  (:method (a b) (funcall-dyadic '*i a b))
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  ;; Two sequences
  (:method ((a sequence) (b sequence))
	   (if (null b)
	       a			; null second arg - return first
	     (progn
	       (assert (eq (length a) (length b)) (a b)
		 "Sequence arguments are not of the same length.")
	       (map (type-of a) #'*i a b))))
  ;; Scalar multiplication of sequences
  (:method ((a sequence) b)		; b is a scalar
	   (map (type-of a) (lambda (x) (*i x b)) a))
  (:method (a (b sequence))		; a is a scalar
	   (if (null b)
	       a			; null second arg - return first
	     (map (type-of b) (lambda (x) (*i a x)) b))))

(defun * (&rest args)
  "Multiplication of generalized or regular numbers with an
   arbitrary number of arguments."
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  (if (single args) (first args) (reduce #'*i args)))
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(defgeneric /i (a b)
  (:documentation "Division of generalized or regular numbers, internal.
    Users call /.")
  (:method ((a number) (b number))
    (cl:/ a b))
  (:method (a b) (funcall-dyadic '/i a b 'null (lambda (x) (/ 1 x))))
  ;; Division can be done as reciprocal and multiplication, or vice versa
  ;; For each generalized number, one of these must be specified
  ;; as a specialized method.
  ;; see below for reciprocal (:method (a (b null)) (/i 1 a)) 
  (:method (a b) (*i a (/ b)))
  ;; Two sequences
  (:method ((a sequence) (b sequence))
    (progn
      (assert (eq (length a) (length b)) (a b)
	      "Sequence arguments are not of the same length.")
      (map (type-of a) #'/i a b))))

;; Scalar division of sequences
(defmethod /i ((a sequence) b)
  (map (type-of a) (lambda (x) (/i x b)) a))

(defmethod /i (a (b sequence))
  (map (type-of b) (lambda (x) (/i a x)) b))

(defun / (&rest args)
  "Division of generalized or regular numbers for an
   arbitrary number of arguments."
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  (if (single args) (/i 1 (first args)) (reduce #'/i args)))
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;;; Put these into with-generalized numbers?
(defmacro incf (ref &optional (delta 1))
  `(setf ,ref (+i ,ref ,delta)))

(defmacro decf (ref &optional (delta 1))
  `(setf ,ref (- ,ref ,delta)))

;;;;****************************************************************************
;;;; Monadic functions
;;;;****************************************************************************

(defun sqrt (num)
  "The square root of the generalized or regular number."
  (expt num 1/2))

(defgeneric sin (num)
  (:documentation "The sine of the generalized or regular number.")
  (:method ((num rational))
	   (cl:sin (coerce num *read-default-float-format*)))
  (:method ((num number))
	   (cl:sin num))
  (:method ((x sequence))
	   (map (type-of x) #'sin x)))

(defgeneric cos (num)
  (:documentation "The cosine of the generalized or regular number.")
  (:method ((num rational))
	   (cl:cos (coerce num *read-default-float-format*)))
  (:method ((num number))    
	   (cl:cos num))
  (:method ((x sequence))
	   (map (type-of x) #'cos x)))

(defgeneric tan (num)
  (:documentation "The tangent of the generalized or regular number.")
  (:method ((num rational))
	   (cl:tan (coerce num *read-default-float-format*)))
  (:method ((num number))
	   (cl:tan num))
  (:method ((x sequence))
	   (map (type-of x) #'tan x)))

(defparameter *radian* 1)
(defun make-radian (x)
  "From the number x, return the representation of an angle in radians."
  (* *radian* x))

(defgeneric asin (arg)
  (:documentation "The arcsine of the generalized or regular number.")
  (:method ((arg number))
    (when (and (not (complexp arg)) (> (abs arg) 1))
      (restart-case 
	  (error 'making-complex-number :operation 'asin :number arg)
	(accept () :report "Accept complex answer.")
	(truncate ()
	  :report "Truncate argument to ±1."
	  (return-from asin
	    (if (plusp arg) (cl:asin 1.0d0) (cl:asin -1.0d0))))
	(hyperbolic-function ()
	  :report "Use the inverse hyperbolic sine."
	  (return-from asin (cl:asinh arg)))))
    (make-radian (cl:asin arg))))

(defgeneric acos (arg)
  (:documentation "The arccosine of the generalized or regular number.")
  (:method ((arg number))
    (when (and (not (complexp arg)) (> (abs arg) 1))
      (restart-case 
	  (error 'making-complex-number :operation 'acos :number arg)
	(accept () :report "Accept complex answer.")
	(truncate ()
	  :report "Truncate argument to ±1."
	  (return-from acos
	    (if (plusp arg) (cl:acos 1.0d0) (cl:acos -1.0d0))))
	(hyperbolic-function ()
	  :report "Use the inverse hyperbolic cosine."
	  (return-from acos (cl:acosh arg)))))
    (make-radian (cl:acos arg))))

(defgeneric atan (num &optional den)
  (:documentation "The arctangent of the generalized or regular number.")
  (:method ((num number) &optional (den 1))
    (make-radian
     (cl:atan (if (typep num 'rational)
		  (coerce num *read-default-float-format*)
		  num)
	      (if (zerop num)
		  (signum den)
		  den)))))

(defgeneric expt (num exponent)
  (:documentation "Raise the number to the exponent.")
  (:method ((num number) (exponent number))
    ;; Most calculations we do are real,
    ;; so trap the production of complex numbers.
    ;; Presumably if the number is already complex,
    ;; we don't mind complex numbers.
    (when (and (not (complexp num)) (minusp num) (not (integerp exponent)))
      (restart-case 
	  (error 'making-complex-number :operation 'expt :number num)
	(accept () :report "Accept complex answer.")
	(absolute-value ()
	  :report "Take the absolute value of the argument and return a real."
	  (setq num (abs num)))))
    (cl:expt num exponent)))

(defgeneric exp (num)
  (:documentation "The natural exponent e^num of the generalized or regular number.")
  (:method ((num rational))
	   (cl:exp (coerce num *read-default-float-format*)))
  (:method ((num number))
	   (cl:exp num))
  (:method ((x sequence))
	   (map (type-of x) #'exp x)))

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(defgeneric log (num &optional base)
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  (:documentation "The natural logarithm of the generalized or regular number.")
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  (:method ((num number) &optional base)
    (if base
	(cl:log num base)
	(cl:log num)))
  (:method ((x sequence) &optional base)
    (if base
	(map (type-of x) (alexandria:rcurry 'log base) x)
	(map (type-of x) #'log x))))
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(defgeneric abs (num)
  (:documentation "The absolute value.")
  (:method ((num number))
	   (cl:abs num))
  (:method ((seq sequence))
	   (map (type-of seq) #'abs seq)))

;;;;****************************************************************************
;;;; Comparisons
;;;;****************************************************************************

(defgeneric numbcomp (x)
  (:documentation "Function for number comparison.  If this function is
   defined for a generalized number and returns an object for which all
   comparison functions such as g<, g>, etc. are defined,
   then all comparison functions will automatically be defined
   for that object.")
  (:method ((x number)) x))

(defgeneric >= (a b)
  (:documentation "Greater than or equal.")
  (:method ((a number) (b number))
	   (cl:>= a b))
  (:method (a b) (cl:>= (numbcomp a) (numbcomp b))))

(defgeneric > (a b)
  (:documentation "Greater than")
  (:method ((a number) (b number)) (cl:> a b))
  (:method (a b) (cl:> (numbcomp a) (numbcomp b))))

(defgeneric <= (a b)
  (:documentation "Less than or equal.")
  (:method ((a number) (b number))
	   (cl:<= a b))
  (:method (a b) (cl:<= (numbcomp a) (numbcomp b))))

(defgeneric < (a b)
  (:documentation "Less than.")
  (:method ((a number) (b number)) (cl:< a b))
  (:method (a b) (cl:< (numbcomp a) (numbcomp b))))

(defgeneric = (a b)
  (:documentation "Numeric equal")
  (:method ((a number) (b number))
	   (if (cl:= a b) a))
  (:method (a b) (funcall-dyadic '= (numbcomp a) (numbcomp b))))

(defgeneric plusp (a)
  (:documentation "Positive.")
  (:method ((a number))
	   (cl:plusp a))
  (:method (a) (cl:plusp (numbcomp a))))

(defgeneric minusp (a)
  (:documentation "Negative")
  (:method ((a number))
	   (cl:minusp a))
  (:method (a) (cl:minusp (numbcomp a))))

(defgeneric zerop (a)
  (:documentation "Zero.")
  (:method ((a number))
	   (cl:zerop a))
  (:method ((x sequence))
	   (every #'zerop x))
  (:method (a) (cl:zerop (numbcomp a))))

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(defgeneric floor (number &optional divisor)
  (:documentation "Greatest multiple of divisor less than number.")
  (:method ((number number) &optional (divisor 1))
	   (cl:floor number divisor)))
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(defgeneric round (number &optional divisor)
  (:documentation "Nearest multiple of divisor to number.")
  (:method ((number number) &optional (divisor 1))
	   (cl:round number divisor)))
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(defgeneric signum (a)
  (:documentation "Signum(a).")
  (:method ((a number)) (cl:signum a))
  (:method (a) (cl:signum (numbcomp a))))

(defun min (&rest args)
  (reduce (lambda (x y) (if (< x y) x y)) args))

(defun max (&rest args)
  (reduce (lambda (x y) (if (> x y) x y)) args))