;;;; This file contains macro-like source transformations which
;;;; convert uses of certain functions into the canonical form desired
;;;; within the compiler. ### and other IR1 transforms and stuff.

;;;; This software is part of the SBCL system. See the README file for
;;;; more information.
;;;;
;;;; This software is derived from the CMU CL system, which was
;;;; written at Carnegie Mellon University and released into the
;;;; public domain. The software is in the public domain and is
;;;; provided with absolutely no warranty. See the COPYING and CREDITS
;;;; files for more information.

(in-package "SB!C")

;;; Convert into an IF so that IF optimizations will eliminate redundant
;;; negations.
(define-source-transform not (x) `(if ,x nil t))
(define-source-transform null (x) `(if ,x nil t))

;;; ENDP is just NULL with a LIST assertion. The assertion will be
;;; optimized away when SAFETY optimization is low; hopefully that
;;; is consistent with ANSI's "should return an error".
(define-source-transform endp (x) `(null (the list ,x)))

;;; We turn IDENTITY into PROG1 so that it is obvious that it just
;;; returns the first value of its argument. Ditto for VALUES with one
;;; arg.
(define-source-transform identity (x) `(prog1 ,x))
(define-source-transform values (x) `(prog1 ,x))

;;; Bind the values and make a closure that returns them.
(define-source-transform constantly (value)
  (let ((rest (gensym "CONSTANTLY-REST-")))
    `(lambda (&rest ,rest)
       (declare (ignore ,rest))
       ,value)))

;;; If the function has a known number of arguments, then return a
;;; lambda with the appropriate fixed number of args. If the
;;; destination is a FUNCALL, then do the &REST APPLY thing, and let
;;; MV optimization figure things out.
(deftransform complement ((fun) * * :node node :when :both)
  "open code"
  (multiple-value-bind (min max)
      (fun-type-nargs (continuation-type fun))
    (cond
     ((and min (eql min max))
      (let ((dums (make-gensym-list min)))
	`#'(lambda ,dums (not (funcall fun ,@dums)))))
     ((let* ((cont (node-cont node))
	     (dest (continuation-dest cont)))
	(and (combination-p dest)
	     (eq (combination-fun dest) cont)))
      '#'(lambda (&rest args)
	   (not (apply fun args))))
     (t
      (give-up-ir1-transform
       "The function doesn't have a fixed argument count.")))))

;;;; list hackery

;;; Translate CxR into CAR/CDR combos.
(defun source-transform-cxr (form)
  (if (/= (length form) 2)
      (values nil t)
      (let ((name (symbol-name (car form))))
	(do ((i (- (length name) 2) (1- i))
	     (res (cadr form)
		  `(,(ecase (char name i)
		       (#\A 'car)
		       (#\D 'cdr))
		    ,res)))
	    ((zerop i) res)))))

;;; Make source transforms to turn CxR forms into combinations of CAR
;;; and CDR. ANSI specifies that everything up to 4 A/D operations is
;;; defined.
(/show0 "about to set CxR source transforms")
(loop for i of-type index from 2 upto 4 do
      ;; Iterate over BUF = all names CxR where x = an I-element
      ;; string of #\A or #\D characters.
      (let ((buf (make-string (+ 2 i))))
	(setf (aref buf 0) #\C
	      (aref buf (1+ i)) #\R)
	(dotimes (j (ash 2 i))
	  (declare (type index j))
	  (dotimes (k i)
	    (declare (type index k))
	    (setf (aref buf (1+ k))
		  (if (logbitp k j) #\A #\D)))
	  (setf (info :function :source-transform (intern buf))
		#'source-transform-cxr))))
(/show0 "done setting CxR source transforms")

;;; Turn FIRST..FOURTH and REST into the obvious synonym, assuming
;;; whatever is right for them is right for us. FIFTH..TENTH turn into
;;; Nth, which can be expanded into a CAR/CDR later on if policy
;;; favors it.
(define-source-transform first (x) `(car ,x))
(define-source-transform rest (x) `(cdr ,x))
(define-source-transform second (x) `(cadr ,x))
(define-source-transform third (x) `(caddr ,x))
(define-source-transform fourth (x) `(cadddr ,x))
(define-source-transform fifth (x) `(nth 4 ,x))
(define-source-transform sixth (x) `(nth 5 ,x))
(define-source-transform seventh (x) `(nth 6 ,x))
(define-source-transform eighth (x) `(nth 7 ,x))
(define-source-transform ninth (x) `(nth 8 ,x))
(define-source-transform tenth (x) `(nth 9 ,x))

;;; Translate RPLACx to LET and SETF.
(define-source-transform rplaca (x y)
  (once-only ((n-x x))
    `(progn
       (setf (car ,n-x) ,y)
       ,n-x)))
(define-source-transform rplacd (x y)
  (once-only ((n-x x))
    `(progn
       (setf (cdr ,n-x) ,y)
       ,n-x)))

(define-source-transform nth (n l) `(car (nthcdr ,n ,l)))

(defvar *default-nthcdr-open-code-limit* 6)
(defvar *extreme-nthcdr-open-code-limit* 20)

(deftransform nthcdr ((n l) (unsigned-byte t) * :node node)
  "convert NTHCDR to CAxxR"
  (unless (constant-continuation-p n)
    (give-up-ir1-transform))
  (let ((n (continuation-value n)))
    (when (> n
	     (if (policy node (and (= speed 3) (= space 0)))
		 *extreme-nthcdr-open-code-limit*
		 *default-nthcdr-open-code-limit*))
      (give-up-ir1-transform))

    (labels ((frob (n)
	       (if (zerop n)
		   'l
		   `(cdr ,(frob (1- n))))))
      (frob n))))

;;;; arithmetic and numerology

(define-source-transform plusp (x) `(> ,x 0))
(define-source-transform minusp (x) `(< ,x 0))
(define-source-transform zerop (x) `(= ,x 0))

(define-source-transform 1+ (x) `(+ ,x 1))
(define-source-transform 1- (x) `(- ,x 1))

(define-source-transform oddp (x) `(not (zerop (logand ,x 1))))
(define-source-transform evenp (x) `(zerop (logand ,x 1)))

;;; Note that all the integer division functions are available for
;;; inline expansion.

(macrolet ((deffrob (fun)
	     `(define-source-transform ,fun (x &optional (y nil y-p))
		(declare (ignore y))
		(if y-p
		    (values nil t)
		    `(,',fun ,x 1)))))
  (deffrob truncate)
  (deffrob round)
  #-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
  (deffrob floor)
  #-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
  (deffrob ceiling))

(define-source-transform lognand (x y) `(lognot (logand ,x ,y)))
(define-source-transform lognor (x y) `(lognot (logior ,x ,y)))
(define-source-transform logandc1 (x y) `(logand (lognot ,x) ,y))
(define-source-transform logandc2 (x y) `(logand ,x (lognot ,y)))
(define-source-transform logorc1 (x y) `(logior (lognot ,x) ,y))
(define-source-transform logorc2 (x y) `(logior ,x (lognot ,y)))
(define-source-transform logtest (x y) `(not (zerop (logand ,x ,y))))
(define-source-transform logbitp (index integer)
  `(not (zerop (logand (ash 1 ,index) ,integer))))
(define-source-transform byte (size position) `(cons ,size ,position))
(define-source-transform byte-size (spec) `(car ,spec))
(define-source-transform byte-position (spec) `(cdr ,spec))
(define-source-transform ldb-test (bytespec integer)
  `(not (zerop (mask-field ,bytespec ,integer))))

;;; With the ratio and complex accessors, we pick off the "identity"
;;; case, and use a primitive to handle the cell access case.
(define-source-transform numerator (num)
  (once-only ((n-num `(the rational ,num)))
    `(if (ratiop ,n-num)
	 (%numerator ,n-num)
	 ,n-num)))
(define-source-transform denominator (num)
  (once-only ((n-num `(the rational ,num)))
    `(if (ratiop ,n-num)
	 (%denominator ,n-num)
	 1)))

;;;; interval arithmetic for computing bounds
;;;;
;;;; This is a set of routines for operating on intervals. It
;;;; implements a simple interval arithmetic package. Although SBCL
;;;; has an interval type in NUMERIC-TYPE, we choose to use our own
;;;; for two reasons:
;;;;
;;;;   1. This package is simpler than NUMERIC-TYPE.
;;;;
;;;;   2. It makes debugging much easier because you can just strip
;;;;   out these routines and test them independently of SBCL. (This is a
;;;;   big win!)
;;;;
;;;; One disadvantage is a probable increase in consing because we
;;;; have to create these new interval structures even though
;;;; numeric-type has everything we want to know. Reason 2 wins for
;;;; now.

;;; The basic interval type. It can handle open and closed intervals.
;;; A bound is open if it is a list containing a number, just like
;;; Lisp says. NIL means unbounded.
(defstruct (interval (:constructor %make-interval)
		     (:copier nil))
  low high)

(defun make-interval (&key low high)
  (labels ((normalize-bound (val)
	     (cond ((and (floatp val)
			 (float-infinity-p val))
		    ;; Handle infinities.
		    nil)
		   ((or (numberp val)
			(eq val nil))
		    ;; Handle any closed bounds.
		    val)
		   ((listp val)
		    ;; We have an open bound. Normalize the numeric
		    ;; bound. If the normalized bound is still a number
		    ;; (not nil), keep the bound open. Otherwise, the
		    ;; bound is really unbounded, so drop the openness.
		    (let ((new-val (normalize-bound (first val))))
		      (when new-val
			;; The bound exists, so keep it open still.
			(list new-val))))
		   (t
		    (error "unknown bound type in MAKE-INTERVAL")))))
    (%make-interval :low (normalize-bound low)
		    :high (normalize-bound high))))

;;; Given a number X, create a form suitable as a bound for an
;;; interval. Make the bound open if OPEN-P is T. NIL remains NIL.
#!-sb-fluid (declaim (inline set-bound))
(defun set-bound (x open-p)
  (if (and x open-p) (list x) x))

;;; Apply the function F to a bound X. If X is an open bound, then
;;; the result will be open. IF X is NIL, the result is NIL.
(defun bound-func (f x)
  (and x
       (with-float-traps-masked (:underflow :overflow :inexact :divide-by-zero)
	 ;; With these traps masked, we might get things like infinity
	 ;; or negative infinity returned. Check for this and return
	 ;; NIL to indicate unbounded.
	 (let ((y (funcall f (type-bound-number x))))
	   (if (and (floatp y)
		    (float-infinity-p y))
	       nil
	       (set-bound (funcall f (type-bound-number x)) (consp x)))))))

;;; Apply a binary operator OP to two bounds X and Y. The result is
;;; NIL if either is NIL. Otherwise bound is computed and the result
;;; is open if either X or Y is open.
;;;
;;; FIXME: only used in this file, not needed in target runtime
(defmacro bound-binop (op x y)
  `(and ,x ,y
       (with-float-traps-masked (:underflow :overflow :inexact :divide-by-zero)
	 (set-bound (,op (type-bound-number ,x)
			 (type-bound-number ,y))
		    (or (consp ,x) (consp ,y))))))

;;; Convert a numeric-type object to an interval object.
(defun numeric-type->interval (x)
  (declare (type numeric-type x))
  (make-interval :low (numeric-type-low x)
		 :high (numeric-type-high x)))

(defun copy-interval-limit (limit)
  (if (numberp limit)
      limit
      (copy-list limit)))

(defun copy-interval (x)
  (declare (type interval x))
  (make-interval :low (copy-interval-limit (interval-low x))
		 :high (copy-interval-limit (interval-high x))))

;;; Given a point P contained in the interval X, split X into two
;;; interval at the point P. If CLOSE-LOWER is T, then the left
;;; interval contains P. If CLOSE-UPPER is T, the right interval
;;; contains P. You can specify both to be T or NIL.
(defun interval-split (p x &optional close-lower close-upper)
  (declare (type number p)
	   (type interval x))
  (list (make-interval :low (copy-interval-limit (interval-low x))
		       :high (if close-lower p (list p)))
	(make-interval :low (if close-upper (list p) p)
		       :high (copy-interval-limit (interval-high x)))))

;;; Return the closure of the interval. That is, convert open bounds
;;; to closed bounds.
(defun interval-closure (x)
  (declare (type interval x))
  (make-interval :low (type-bound-number (interval-low x))
		 :high (type-bound-number (interval-high x))))

(defun signed-zero->= (x y)
  (declare (real x y))
  (or (> x y)
      (and (= x y)
	   (>= (float-sign (float x))
	       (float-sign (float y))))))

;;; For an interval X, if X >= POINT, return '+. If X <= POINT, return
;;; '-. Otherwise return NIL.
#+nil
(defun interval-range-info (x &optional (point 0))
  (declare (type interval x))
  (let ((lo (interval-low x))
	(hi (interval-high x)))
    (cond ((and lo (signed-zero->= (type-bound-number lo) point))
	   '+)
	  ((and hi (signed-zero->= point (type-bound-number hi)))
	   '-)
	  (t
	   nil))))
(defun interval-range-info (x &optional (point 0))
  (declare (type interval x))
  (labels ((signed->= (x y)
	     (if (and (zerop x) (zerop y) (floatp x) (floatp y))
		 (>= (float-sign x) (float-sign y))
		 (>= x y))))
    (let ((lo (interval-low x))
	  (hi (interval-high x)))
      (cond ((and lo (signed->= (type-bound-number lo) point))
	     '+)
	    ((and hi (signed->= point (type-bound-number hi)))
	     '-)
	    (t
	     nil)))))

;;; Test to see whether the interval X is bounded. HOW determines the
;;; test, and should be either ABOVE, BELOW, or BOTH.
(defun interval-bounded-p (x how)
  (declare (type interval x))
  (ecase how
    ('above
     (interval-high x))
    ('below
     (interval-low x))
    ('both
     (and (interval-low x) (interval-high x)))))

;;; signed zero comparison functions. Use these functions if we need
;;; to distinguish between signed zeroes.
(defun signed-zero-< (x y)
  (declare (real x y))
  (or (< x y)
      (and (= x y)
	   (< (float-sign (float x))
	      (float-sign (float y))))))
(defun signed-zero-> (x y)
  (declare (real x y))
  (or (> x y)
      (and (= x y)
	   (> (float-sign (float x))
	      (float-sign (float y))))))
(defun signed-zero-= (x y)
  (declare (real x y))
  (and (= x y)
       (= (float-sign (float x))
	  (float-sign (float y)))))
(defun signed-zero-<= (x y)
  (declare (real x y))
  (or (< x y)
      (and (= x y)
	   (<= (float-sign (float x))
	       (float-sign (float y))))))

;;; See whether the interval X contains the number P, taking into
;;; account that the interval might not be closed.
(defun interval-contains-p (p x)
  (declare (type number p)
	   (type interval x))
  ;; Does the interval X contain the number P?  This would be a lot
  ;; easier if all intervals were closed!
  (let ((lo (interval-low x))
	(hi (interval-high x)))
    (cond ((and lo hi)
	   ;; The interval is bounded
	   (if (and (signed-zero-<= (type-bound-number lo) p)
		    (signed-zero-<= p (type-bound-number hi)))
	       ;; P is definitely in the closure of the interval.
	       ;; We just need to check the end points now.
	       (cond ((signed-zero-= p (type-bound-number lo))
		      (numberp lo))
		     ((signed-zero-= p (type-bound-number hi))
		      (numberp hi))
		     (t t))
	       nil))
	  (hi
	   ;; Interval with upper bound
	   (if (signed-zero-< p (type-bound-number hi))
	       t
	       (and (numberp hi) (signed-zero-= p hi))))
	  (lo
	   ;; Interval with lower bound
	   (if (signed-zero-> p (type-bound-number lo))
	       t
	       (and (numberp lo) (signed-zero-= p lo))))
	  (t
	   ;; Interval with no bounds
	   t))))

;;; Determine whether two intervals X and Y intersect. Return T if so.
;;; If CLOSED-INTERVALS-P is T, the treat the intervals as if they
;;; were closed. Otherwise the intervals are treated as they are.
;;;
;;; Thus if X = [0, 1) and Y = (1, 2), then they do not intersect
;;; because no element in X is in Y. However, if CLOSED-INTERVALS-P
;;; is T, then they do intersect because we use the closure of X = [0,
;;; 1] and Y = [1, 2] to determine intersection.
(defun interval-intersect-p (x y &optional closed-intervals-p)
  (declare (type interval x y))
  (multiple-value-bind (intersect diff)
      (interval-intersection/difference (if closed-intervals-p
					    (interval-closure x)
					    x)
					(if closed-intervals-p
					    (interval-closure y)
					    y))
    (declare (ignore diff))
    intersect))

;;; Are the two intervals adjacent?  That is, is there a number
;;; between the two intervals that is not an element of either
;;; interval?  If so, they are not adjacent. For example [0, 1) and
;;; [1, 2] are adjacent but [0, 1) and (1, 2] are not because 1 lies
;;; between both intervals.
(defun interval-adjacent-p (x y)
  (declare (type interval x y))
  (flet ((adjacent (lo hi)
	   ;; Check to see whether lo and hi are adjacent. If either is
	   ;; nil, they can't be adjacent.
	   (when (and lo hi (= (type-bound-number lo) (type-bound-number hi)))
	     ;; The bounds are equal. They are adjacent if one of
	     ;; them is closed (a number). If both are open (consp),
	     ;; then there is a number that lies between them.
	     (or (numberp lo) (numberp hi)))))
    (or (adjacent (interval-low y) (interval-high x))
	(adjacent (interval-low x) (interval-high y)))))

;;; Compute the intersection and difference between two intervals.
;;; Two values are returned: the intersection and the difference.
;;;
;;; Let the two intervals be X and Y, and let I and D be the two
;;; values returned by this function. Then I = X intersect Y. If I
;;; is NIL (the empty set), then D is X union Y, represented as the
;;; list of X and Y. If I is not the empty set, then D is (X union Y)
;;; - I, which is a list of two intervals.
;;;
;;; For example, let X = [1,5] and Y = [-1,3). Then I = [1,3) and D =
;;; [-1,1) union [3,5], which is returned as a list of two intervals.
(defun interval-intersection/difference (x y)
  (declare (type interval x y))
  (let ((x-lo (interval-low x))
	(x-hi (interval-high x))
	(y-lo (interval-low y))
	(y-hi (interval-high y)))
    (labels
	((opposite-bound (p)
	   ;; If p is an open bound, make it closed. If p is a closed
	   ;; bound, make it open.
	   (if (listp p)
	       (first p)
	       (list p)))
	 (test-number (p int)
	   ;; Test whether P is in the interval.
	   (when (interval-contains-p (type-bound-number p)
				      (interval-closure int))
	     (let ((lo (interval-low int))
		   (hi (interval-high int)))
	       ;; Check for endpoints.
	       (cond ((and lo (= (type-bound-number p) (type-bound-number lo)))
		      (not (and (consp p) (numberp lo))))
		     ((and hi (= (type-bound-number p) (type-bound-number hi)))
		      (not (and (numberp p) (consp hi))))
		     (t t)))))
	 (test-lower-bound (p int)
	   ;; P is a lower bound of an interval.
	   (if p
	       (test-number p int)
	       (not (interval-bounded-p int 'below))))
	 (test-upper-bound (p int)
	   ;; P is an upper bound of an interval.
	   (if p
	       (test-number p int)
	       (not (interval-bounded-p int 'above)))))
      (let ((x-lo-in-y (test-lower-bound x-lo y))
	    (x-hi-in-y (test-upper-bound x-hi y))
	    (y-lo-in-x (test-lower-bound y-lo x))
	    (y-hi-in-x (test-upper-bound y-hi x)))
	(cond ((or x-lo-in-y x-hi-in-y y-lo-in-x y-hi-in-x)
	       ;; Intervals intersect. Let's compute the intersection
	       ;; and the difference.
	       (multiple-value-bind (lo left-lo left-hi)
		   (cond (x-lo-in-y (values x-lo y-lo (opposite-bound x-lo)))
			 (y-lo-in-x (values y-lo x-lo (opposite-bound y-lo))))
		 (multiple-value-bind (hi right-lo right-hi)
		     (cond (x-hi-in-y
			    (values x-hi (opposite-bound x-hi) y-hi))
			   (y-hi-in-x
			    (values y-hi (opposite-bound y-hi) x-hi)))
		   (values (make-interval :low lo :high hi)
			   (list (make-interval :low left-lo
						:high left-hi)
				 (make-interval :low right-lo
						:high right-hi))))))
	      (t
	       (values nil (list x y))))))))

;;; If intervals X and Y intersect, return a new interval that is the
;;; union of the two. If they do not intersect, return NIL.
(defun interval-merge-pair (x y)
  (declare (type interval x y))
  ;; If x and y intersect or are adjacent, create the union.
  ;; Otherwise return nil
  (when (or (interval-intersect-p x y)
	    (interval-adjacent-p x y))
    (flet ((select-bound (x1 x2 min-op max-op)
	     (let ((x1-val (type-bound-number x1))
		   (x2-val (type-bound-number x2)))
	       (cond ((and x1 x2)
		      ;; Both bounds are finite. Select the right one.
		      (cond ((funcall min-op x1-val x2-val)
			     ;; x1 is definitely better.
			     x1)
			    ((funcall max-op x1-val x2-val)
			     ;; x2 is definitely better.
			     x2)
			    (t
			     ;; Bounds are equal. Select either
			     ;; value and make it open only if
			     ;; both were open.
			     (set-bound x1-val (and (consp x1) (consp x2))))))
		     (t
		      ;; At least one bound is not finite. The
		      ;; non-finite bound always wins.
		      nil)))))
      (let* ((x-lo (copy-interval-limit (interval-low x)))
	     (x-hi (copy-interval-limit (interval-high x)))
	     (y-lo (copy-interval-limit (interval-low y)))
	     (y-hi (copy-interval-limit (interval-high y))))
	(make-interval :low (select-bound x-lo y-lo #'< #'>)
		       :high (select-bound x-hi y-hi #'> #'<))))))

;;; basic arithmetic operations on intervals. We probably should do
;;; true interval arithmetic here, but it's complicated because we
;;; have float and integer types and bounds can be open or closed.

;;; the negative of an interval
(defun interval-neg (x)
  (declare (type interval x))
  (make-interval :low (bound-func #'- (interval-high x))
		 :high (bound-func #'- (interval-low x))))

;;; Add two intervals.
(defun interval-add (x y)
  (declare (type interval x y))
  (make-interval :low (bound-binop + (interval-low x) (interval-low y))
		 :high (bound-binop + (interval-high x) (interval-high y))))

;;; Subtract two intervals.
(defun interval-sub (x y)
  (declare (type interval x y))
  (make-interval :low (bound-binop - (interval-low x) (interval-high y))
		 :high (bound-binop - (interval-high x) (interval-low y))))

;;; Multiply two intervals.
(defun interval-mul (x y)
  (declare (type interval x y))
  (flet ((bound-mul (x y)
	   (cond ((or (null x) (null y))
		  ;; Multiply by infinity is infinity
		  nil)
		 ((or (and (numberp x) (zerop x))
		      (and (numberp y) (zerop y)))
		  ;; Multiply by closed zero is special. The result
		  ;; is always a closed bound. But don't replace this
		  ;; with zero; we want the multiplication to produce
		  ;; the correct signed zero, if needed.
		  (* (type-bound-number x) (type-bound-number y)))
		 ((or (and (floatp x) (float-infinity-p x))
		      (and (floatp y) (float-infinity-p y)))
		  ;; Infinity times anything is infinity
		  nil)
		 (t
		  ;; General multiply. The result is open if either is open.
		  (bound-binop * x y)))))
    (let ((x-range (interval-range-info x))
	  (y-range (interval-range-info y)))
      (cond ((null x-range)
	     ;; Split x into two and multiply each separately
	     (destructuring-bind (x- x+) (interval-split 0 x t t)
	       (interval-merge-pair (interval-mul x- y)
				    (interval-mul x+ y))))
	    ((null y-range)
	     ;; Split y into two and multiply each separately
	     (destructuring-bind (y- y+) (interval-split 0 y t t)
	       (interval-merge-pair (interval-mul x y-)
				    (interval-mul x y+))))
	    ((eq x-range '-)
	     (interval-neg (interval-mul (interval-neg x) y)))
	    ((eq y-range '-)
	     (interval-neg (interval-mul x (interval-neg y))))
	    ((and (eq x-range '+) (eq y-range '+))
	     ;; If we are here, X and Y are both positive.
	     (make-interval
	      :low (bound-mul (interval-low x) (interval-low y))
	      :high (bound-mul (interval-high x) (interval-high y))))
	    (t
	     (error "internal error in INTERVAL-MUL"))))))

;;; Divide two intervals.
(defun interval-div (top bot)
  (declare (type interval top bot))
  (flet ((bound-div (x y y-low-p)
	   ;; Compute x/y
	   (cond ((null y)
		  ;; Divide by infinity means result is 0. However,
		  ;; we need to watch out for the sign of the result,
		  ;; to correctly handle signed zeros. We also need
		  ;; to watch out for positive or negative infinity.
		  (if (floatp (type-bound-number x))
		      (if y-low-p
			  (- (float-sign (type-bound-number x) 0.0))
			  (float-sign (type-bound-number x) 0.0))
		      0))
		 ((zerop (type-bound-number y))
		  ;; Divide by zero means result is infinity
		  nil)
		 ((and (numberp x) (zerop x))
		  ;; Zero divided by anything is zero.
		  x)
		 (t
		  (bound-binop / x y)))))
    (let ((top-range (interval-range-info top))
	  (bot-range (interval-range-info bot)))
      (cond ((null bot-range)
	     ;; The denominator contains zero, so anything goes!
	     (make-interval :low nil :high nil))
	    ((eq bot-range '-)
	     ;; Denominator is negative so flip the sign, compute the
	     ;; result, and flip it back.
	     (interval-neg (interval-div top (interval-neg bot))))
	    ((null top-range)
	     ;; Split top into two positive and negative parts, and
	     ;; divide each separately
	     (destructuring-bind (top- top+) (interval-split 0 top t t)
	       (interval-merge-pair (interval-div top- bot)
				    (interval-div top+ bot))))
	    ((eq top-range '-)
	     ;; Top is negative so flip the sign, divide, and flip the
	     ;; sign of the result.
	     (interval-neg (interval-div (interval-neg top) bot)))
	    ((and (eq top-range '+) (eq bot-range '+))
	     ;; the easy case
	     (make-interval
	      :low (bound-div (interval-low top) (interval-high bot) t)
	      :high (bound-div (interval-high top) (interval-low bot) nil)))
	    (t
	     (error "internal error in INTERVAL-DIV"))))))

;;; Apply the function F to the interval X. If X = [a, b], then the
;;; result is [f(a), f(b)]. It is up to the user to make sure the
;;; result makes sense. It will if F is monotonic increasing (or
;;; non-decreasing).
(defun interval-func (f x)
  (declare (type interval x))
  (let ((lo (bound-func f (interval-low x)))
	(hi (bound-func f (interval-high x))))
    (make-interval :low lo :high hi)))

;;; Return T if X < Y. That is every number in the interval X is
;;; always less than any number in the interval Y.
(defun interval-< (x y)
  (declare (type interval x y))
  ;; X < Y only if X is bounded above, Y is bounded below, and they
  ;; don't overlap.
  (when (and (interval-bounded-p x 'above)
	     (interval-bounded-p y 'below))
    ;; Intervals are bounded in the appropriate way. Make sure they
    ;; don't overlap.
    (let ((left (interval-high x))
	  (right (interval-low y)))
      (cond ((> (type-bound-number left)
		(type-bound-number right))
	     ;; The intervals definitely overlap, so result is NIL.
	     nil)
	    ((< (type-bound-number left)
		(type-bound-number right))
	     ;; The intervals definitely don't touch, so result is T.
	     t)
	    (t
	     ;; Limits are equal. Check for open or closed bounds.
	     ;; Don't overlap if one or the other are open.
	     (or (consp left) (consp right)))))))

;;; Return T if X >= Y. That is, every number in the interval X is
;;; always greater than any number in the interval Y.
(defun interval->= (x y)
  (declare (type interval x y))
  ;; X >= Y if lower bound of X >= upper bound of Y
  (when (and (interval-bounded-p x 'below)
	     (interval-bounded-p y 'above))
    (>= (type-bound-number (interval-low x))
	(type-bound-number (interval-high y)))))

;;; Return an interval that is the absolute value of X. Thus, if
;;; X = [-1 10], the result is [0, 10].
(defun interval-abs (x)
  (declare (type interval x))
  (case (interval-range-info x)
    ('+
     (copy-interval x))
    ('-
     (interval-neg x))
    (t
     (destructuring-bind (x- x+) (interval-split 0 x t t)
       (interval-merge-pair (interval-neg x-) x+)))))

;;; Compute the square of an interval.
(defun interval-sqr (x)
  (declare (type interval x))
  (interval-func (lambda (x) (* x x))
		 (interval-abs x)))

;;;; numeric DERIVE-TYPE methods

;;; a utility for defining derive-type methods of integer operations. If
;;; the types of both X and Y are integer types, then we compute a new
;;; integer type with bounds determined Fun when applied to X and Y.
;;; Otherwise, we use Numeric-Contagion.
(defun derive-integer-type (x y fun)
  (declare (type continuation x y) (type function fun))
  (let ((x (continuation-type x))
	(y (continuation-type y)))
    (if (and (numeric-type-p x) (numeric-type-p y)
	     (eq (numeric-type-class x) 'integer)
	     (eq (numeric-type-class y) 'integer)
	     (eq (numeric-type-complexp x) :real)
	     (eq (numeric-type-complexp y) :real))
	(multiple-value-bind (low high) (funcall fun x y)
	  (make-numeric-type :class 'integer
			     :complexp :real
			     :low low
			     :high high))
	(numeric-contagion x y))))

;;; simple utility to flatten a list
(defun flatten-list (x)
  (labels ((flatten-helper (x r);; 'r' is the stuff to the 'right'.
	     (cond ((null x) r)
		   ((atom x)
		    (cons x r))
		   (t (flatten-helper (car x)
				      (flatten-helper (cdr x) r))))))
    (flatten-helper x nil)))

;;; Take some type of continuation and massage it so that we get a
;;; list of the constituent types. If ARG is *EMPTY-TYPE*, return NIL
;;; to indicate failure.
(defun prepare-arg-for-derive-type (arg)
  (flet ((listify (arg)
	   (typecase arg
	     (numeric-type
	      (list arg))
	     (union-type
	      (union-type-types arg))
	     (t
	      (list arg)))))
    (unless (eq arg *empty-type*)
      ;; Make sure all args are some type of numeric-type. For member
      ;; types, convert the list of members into a union of equivalent
      ;; single-element member-type's.
      (let ((new-args nil))
	(dolist (arg (listify arg))
	  (if (member-type-p arg)
	      ;; Run down the list of members and convert to a list of
	      ;; member types.
	      (dolist (member (member-type-members arg))
		(push (if (numberp member)
			  (make-member-type :members (list member))
			  *empty-type*)
		      new-args))
	      (push arg new-args)))
	(unless (member *empty-type* new-args)
	  new-args)))))

;;; Convert from the standard type convention for which -0.0 and 0.0
;;; are equal to an intermediate convention for which they are
;;; considered different which is more natural for some of the
;;; optimisers.
#!-negative-zero-is-not-zero
(defun convert-numeric-type (type)
  (declare (type numeric-type type))
  ;;; Only convert real float interval delimiters types.
  (if (eq (numeric-type-complexp type) :real)
      (let* ((lo (numeric-type-low type))
	     (lo-val (type-bound-number lo))
	     (lo-float-zero-p (and lo (floatp lo-val) (= lo-val 0.0)))
	     (hi (numeric-type-high type))
	     (hi-val (type-bound-number hi))
	     (hi-float-zero-p (and hi (floatp hi-val) (= hi-val 0.0))))
	(if (or lo-float-zero-p hi-float-zero-p)
	    (make-numeric-type
	     :class (numeric-type-class type)
	     :format (numeric-type-format type)
	     :complexp :real
	     :low (if lo-float-zero-p
		      (if (consp lo)
			  (list (float 0.0 lo-val))
			  (float -0.0 lo-val))
		      lo)
	     :high (if hi-float-zero-p
		       (if (consp hi)
			   (list (float -0.0 hi-val))
			   (float 0.0 hi-val))
		       hi))
	    type))
      ;; Not real float.
      type))

;;; Convert back from the intermediate convention for which -0.0 and
;;; 0.0 are considered different to the standard type convention for
;;; which and equal.
#!-negative-zero-is-not-zero
(defun convert-back-numeric-type (type)
  (declare (type numeric-type type))
  ;;; Only convert real float interval delimiters types.
  (if (eq (numeric-type-complexp type) :real)
      (let* ((lo (numeric-type-low type))
	     (lo-val (type-bound-number lo))
	     (lo-float-zero-p
	      (and lo (floatp lo-val) (= lo-val 0.0)
		   (float-sign lo-val)))
	     (hi (numeric-type-high type))
	     (hi-val (type-bound-number hi))
	     (hi-float-zero-p
	      (and hi (floatp hi-val) (= hi-val 0.0)
		   (float-sign hi-val))))
	(cond
	  ;; (float +0.0 +0.0) => (member 0.0)
	  ;; (float -0.0 -0.0) => (member -0.0)
	  ((and lo-float-zero-p hi-float-zero-p)
	   ;; shouldn't have exclusive bounds here..
	   (aver (and (not (consp lo)) (not (consp hi))))
	   (if (= lo-float-zero-p hi-float-zero-p)
	       ;; (float +0.0 +0.0) => (member 0.0)
	       ;; (float -0.0 -0.0) => (member -0.0)
	       (specifier-type `(member ,lo-val))
	       ;; (float -0.0 +0.0) => (float 0.0 0.0)
	       ;; (float +0.0 -0.0) => (float 0.0 0.0)
	       (make-numeric-type :class (numeric-type-class type)
				  :format (numeric-type-format type)
				  :complexp :real
				  :low hi-val
				  :high hi-val)))
	  (lo-float-zero-p
	   (cond
	     ;; (float -0.0 x) => (float 0.0 x)
	     ((and (not (consp lo)) (minusp lo-float-zero-p))
	      (make-numeric-type :class (numeric-type-class type)
				 :format (numeric-type-format type)
				 :complexp :real
				 :low (float 0.0 lo-val)
				 :high hi))
	     ;; (float (+0.0) x) => (float (0.0) x)
	     ((and (consp lo) (plusp lo-float-zero-p))
	      (make-numeric-type :class (numeric-type-class type)
				 :format (numeric-type-format type)
				 :complexp :real
				 :low (list (float 0.0 lo-val))
				 :high hi))
	     (t
	      ;; (float +0.0 x) => (or (member 0.0) (float (0.0) x))
	      ;; (float (-0.0) x) => (or (member 0.0) (float (0.0) x))
	      (list (make-member-type :members (list (float 0.0 lo-val)))
		    (make-numeric-type :class (numeric-type-class type)
				       :format (numeric-type-format type)
				       :complexp :real
				       :low (list (float 0.0 lo-val))
				       :high hi)))))
	  (hi-float-zero-p
	   (cond
	     ;; (float x +0.0) => (float x 0.0)
	     ((and (not (consp hi)) (plusp hi-float-zero-p))
	      (make-numeric-type :class (numeric-type-class type)
				 :format (numeric-type-format type)
				 :complexp :real
				 :low lo
				 :high (float 0.0 hi-val)))
	     ;; (float x (-0.0)) => (float x (0.0))
	     ((and (consp hi) (minusp hi-float-zero-p))
	      (make-numeric-type :class (numeric-type-class type)
				 :format (numeric-type-format type)
				 :complexp :real
				 :low lo
				 :high (list (float 0.0 hi-val))))
	     (t
	      ;; (float x (+0.0)) => (or (member -0.0) (float x (0.0)))
	      ;; (float x -0.0) => (or (member -0.0) (float x (0.0)))
	      (list (make-member-type :members (list (float -0.0 hi-val)))
		    (make-numeric-type :class (numeric-type-class type)
				       :format (numeric-type-format type)
				       :complexp :real
				       :low lo
				       :high (list (float 0.0 hi-val)))))))
	  (t
	   type)))
      ;; not real float
      type))

;;; Convert back a possible list of numeric types.
#!-negative-zero-is-not-zero
(defun convert-back-numeric-type-list (type-list)
  (typecase type-list
    (list
     (let ((results '()))
       (dolist (type type-list)
	 (if (numeric-type-p type)
	     (let ((result (convert-back-numeric-type type)))
	       (if (listp result)
		   (setf results (append results result))
		   (push result results)))
	     (push type results)))
       results))
    (numeric-type
     (convert-back-numeric-type type-list))
    (union-type
     (convert-back-numeric-type-list (union-type-types type-list)))
    (t
     type-list)))

;;; FIXME: MAKE-CANONICAL-UNION-TYPE and CONVERT-MEMBER-TYPE probably
;;; belong in the kernel's type logic, invoked always, instead of in
;;; the compiler, invoked only during some type optimizations.

;;; Take a list of types and return a canonical type specifier,
;;; combining any MEMBER types together. If both positive and negative
;;; MEMBER types are present they are converted to a float type.
;;; XXX This would be far simpler if the type-union methods could handle
;;; member/number unions.
(defun make-canonical-union-type (type-list)
  (let ((members '())
	(misc-types '()))
    (dolist (type type-list)
      (if (member-type-p type)
	  (setf members (union members (member-type-members type)))
	  (push type misc-types)))
    #!+long-float
    (when (null (set-difference '(-0l0 0l0) members))
      #!-negative-zero-is-not-zero
      (push (specifier-type '(long-float 0l0 0l0)) misc-types)
      #!+negative-zero-is-not-zero
      (push (specifier-type '(long-float -0l0 0l0)) misc-types)
      (setf members (set-difference members '(-0l0 0l0))))
    (when (null (set-difference '(-0d0 0d0) members))
      #!-negative-zero-is-not-zero
      (push (specifier-type '(double-float 0d0 0d0)) misc-types)
      #!+negative-zero-is-not-zero
      (push (specifier-type '(double-float -0d0 0d0)) misc-types)
      (setf members (set-difference members '(-0d0 0d0))))
    (when (null (set-difference '(-0f0 0f0) members))
      #!-negative-zero-is-not-zero
      (push (specifier-type '(single-float 0f0 0f0)) misc-types)
      #!+negative-zero-is-not-zero
      (push (specifier-type '(single-float -0f0 0f0)) misc-types)
      (setf members (set-difference members '(-0f0 0f0))))
    (if members
	(apply #'type-union (make-member-type :members members) misc-types)
	(apply #'type-union misc-types))))

;;; Convert a member type with a single member to a numeric type.
(defun convert-member-type (arg)
  (let* ((members (member-type-members arg))
	 (member (first members))
	 (member-type (type-of member)))
    (aver (not (rest members)))
    (specifier-type `(,(if (subtypep member-type 'integer)
			   'integer
			   member-type)
		      ,member ,member))))

;;; This is used in defoptimizers for computing the resulting type of
;;; a function.
;;;
;;; Given the continuation ARG, derive the resulting type using the
;;; DERIVE-FCN. DERIVE-FCN takes exactly one argument which is some
;;; "atomic" continuation type like numeric-type or member-type
;;; (containing just one element). It should return the resulting
;;; type, which can be a list of types.
;;;
;;; For the case of member types, if a member-fcn is given it is
;;; called to compute the result otherwise the member type is first
;;; converted to a numeric type and the derive-fcn is call.
(defun one-arg-derive-type (arg derive-fcn member-fcn
				&optional (convert-type t))
  (declare (type function derive-fcn)
	   (type (or null function) member-fcn)
	   #!+negative-zero-is-not-zero (ignore convert-type))
  (let ((arg-list (prepare-arg-for-derive-type (continuation-type arg))))
    (when arg-list
      (flet ((deriver (x)
	       (typecase x
		 (member-type
		  (if member-fcn
		      (with-float-traps-masked
			  (:underflow :overflow :divide-by-zero)
			(make-member-type
			 :members (list
				   (funcall member-fcn
					    (first (member-type-members x))))))
		      ;; Otherwise convert to a numeric type.
		      (let ((result-type-list
			     (funcall derive-fcn (convert-member-type x))))
			#!-negative-zero-is-not-zero
			(if convert-type
			    (convert-back-numeric-type-list result-type-list)
			    result-type-list)
			#!+negative-zero-is-not-zero
			result-type-list)))
		 (numeric-type
		  #!-negative-zero-is-not-zero
		  (if convert-type
		      (convert-back-numeric-type-list
		       (funcall derive-fcn (convert-numeric-type x)))
		      (funcall derive-fcn x))
		  #!+negative-zero-is-not-zero
		  (funcall derive-fcn x))
		 (t
		  *universal-type*))))
	;; Run down the list of args and derive the type of each one,
	;; saving all of the results in a list.
	(let ((results nil))
	  (dolist (arg arg-list)
	    (let ((result (deriver arg)))
	      (if (listp result)
		  (setf results (append results result))
		  (push result results))))
	  (if (rest results)
	      (make-canonical-union-type results)
	      (first results)))))))

;;; Same as ONE-ARG-DERIVE-TYPE, except we assume the function takes
;;; two arguments. DERIVE-FCN takes 3 args in this case: the two
;;; original args and a third which is T to indicate if the two args
;;; really represent the same continuation. This is useful for
;;; deriving the type of things like (* x x), which should always be
;;; positive. If we didn't do this, we wouldn't be able to tell.
(defun two-arg-derive-type (arg1 arg2 derive-fcn fcn
				 &optional (convert-type t))
  #!+negative-zero-is-not-zero
  (declare (ignore convert-type))
  (flet (#!-negative-zero-is-not-zero
	 (deriver (x y same-arg)
	   (cond ((and (member-type-p x) (member-type-p y))
		  (let* ((x (first (member-type-members x)))
			 (y (first (member-type-members y)))
			 (result (with-float-traps-masked
				     (:underflow :overflow :divide-by-zero
				      :invalid)
				   (funcall fcn x y))))
		    (cond ((null result))
			  ((and (floatp result) (float-nan-p result))
			   (make-numeric-type :class 'float
					      :format (type-of result)
					      :complexp :real))
			  (t
			   (make-member-type :members (list result))))))
		 ((and (member-type-p x) (numeric-type-p y))
		  (let* ((x (convert-member-type x))
			 (y (if convert-type (convert-numeric-type y) y))
			 (result (funcall derive-fcn x y same-arg)))
		    (if convert-type
			(convert-back-numeric-type-list result)
			result)))
		 ((and (numeric-type-p x) (member-type-p y))
		  (let* ((x (if convert-type (convert-numeric-type x) x))
			 (y (convert-member-type y))
			 (result (funcall derive-fcn x y same-arg)))
		    (if convert-type
			(convert-back-numeric-type-list result)
			result)))
		 ((and (numeric-type-p x) (numeric-type-p y))
		  (let* ((x (if convert-type (convert-numeric-type x) x))
			 (y (if convert-type (convert-numeric-type y) y))
			 (result (funcall derive-fcn x y same-arg)))
		    (if convert-type
			(convert-back-numeric-type-list result)
			result)))
		 (t
		  *universal-type*)))
	 #!+negative-zero-is-not-zero
	 (deriver (x y same-arg)
	   (cond ((and (member-type-p x) (member-type-p y))
		  (let* ((x (first (member-type-members x)))
			 (y (first (member-type-members y)))
			 (result (with-float-traps-masked
				     (:underflow :overflow :divide-by-zero)
				   (funcall fcn x y))))
		    (if result
			(make-member-type :members (list result)))))
		 ((and (member-type-p x) (numeric-type-p y))
		  (let ((x (convert-member-type x)))
		    (funcall derive-fcn x y same-arg)))
		 ((and (numeric-type-p x) (member-type-p y))
		  (let ((y (convert-member-type y)))
		    (funcall derive-fcn x y same-arg)))
		 ((and (numeric-type-p x) (numeric-type-p y))
		  (funcall derive-fcn x y same-arg))
		 (t
		  *universal-type*))))
    (let ((same-arg (same-leaf-ref-p arg1 arg2))
	  (a1 (prepare-arg-for-derive-type (continuation-type arg1)))
	  (a2 (prepare-arg-for-derive-type (continuation-type arg2))))
      (when (and a1 a2)
	(let ((results nil))
	  (if same-arg
	      ;; Since the args are the same continuation, just run
	      ;; down the lists.
	      (dolist (x a1)
		(let ((result (deriver x x same-arg)))
		  (if (listp result)
		      (setf results (append results result))
		      (push result results))))
	      ;; Try all pairwise combinations.
	      (dolist (x a1)
		(dolist (y a2)
		  (let ((result (or (deriver x y same-arg)
				    (numeric-contagion x y))))
		    (if (listp result)
			(setf results (append results result))
			(push result results))))))
	  (if (rest results)
	      (make-canonical-union-type results)
	      (first results)))))))

#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(progn
(defoptimizer (+ derive-type) ((x y))
  (derive-integer-type
   x y
   #'(lambda (x y)
       (flet ((frob (x y)
		(if (and x y)
		    (+ x y)
		    nil)))
	 (values (frob (numeric-type-low x) (numeric-type-low y))
		 (frob (numeric-type-high x) (numeric-type-high y)))))))

(defoptimizer (- derive-type) ((x y))
  (derive-integer-type
   x y
   #'(lambda (x y)
       (flet ((frob (x y)
		(if (and x y)
		    (- x y)
		    nil)))
	 (values (frob (numeric-type-low x) (numeric-type-high y))
		 (frob (numeric-type-high x) (numeric-type-low y)))))))

(defoptimizer (* derive-type) ((x y))
  (derive-integer-type
   x y
   #'(lambda (x y)
       (let ((x-low (numeric-type-low x))
	     (x-high (numeric-type-high x))
	     (y-low (numeric-type-low y))
	     (y-high (numeric-type-high y)))
	 (cond ((not (and x-low y-low))
		(values nil nil))
	       ((or (minusp x-low) (minusp y-low))
		(if (and x-high y-high)
		    (let ((max (* (max (abs x-low) (abs x-high))
				  (max (abs y-low) (abs y-high)))))
		      (values (- max) max))
		    (values nil nil)))
	       (t
		(values (* x-low y-low)
			(if (and x-high y-high)
			    (* x-high y-high)
			    nil))))))))

(defoptimizer (/ derive-type) ((x y))
  (numeric-contagion (continuation-type x) (continuation-type y)))

) ; PROGN

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(progn
(defun +-derive-type-aux (x y same-arg)
  (if (and (numeric-type-real-p x)
	   (numeric-type-real-p y))
      (let ((result
	     (if same-arg
		 (let ((x-int (numeric-type->interval x)))
		   (interval-add x-int x-int))
		 (interval-add (numeric-type->interval x)
			       (numeric-type->interval y))))
	    (result-type (numeric-contagion x y)))
	;; If the result type is a float, we need to be sure to coerce
	;; the bounds into the correct type.
	(when (eq (numeric-type-class result-type) 'float)
	  (setf result (interval-func
			#'(lambda (x)
			    (coerce x (or (numeric-type-format result-type)
					  'float)))
			result)))
	(make-numeric-type
	 :class (if (and (eq (numeric-type-class x) 'integer)
			 (eq (numeric-type-class y) 'integer))
		    ;; The sum of integers is always an integer.
		    'integer
		    (numeric-type-class result-type))
	 :format (numeric-type-format result-type)
	 :low (interval-low result)
	 :high (interval-high result)))
      ;; general contagion
      (numeric-contagion x y)))

(defoptimizer (+ derive-type) ((x y))
  (two-arg-derive-type x y #'+-derive-type-aux #'+))

(defun --derive-type-aux (x y same-arg)
  (if (and (numeric-type-real-p x)
	   (numeric-type-real-p y))
      (let ((result
	     ;; (- X X) is always 0.
	     (if same-arg
		 (make-interval :low 0 :high 0)
		 (interval-sub (numeric-type->interval x)
			       (numeric-type->interval y))))
	    (result-type (numeric-contagion x y)))
	;; If the result type is a float, we need to be sure to coerce
	;; the bounds into the correct type.
	(when (eq (numeric-type-class result-type) 'float)
	  (setf result (interval-func
			#'(lambda (x)
			    (coerce x (or (numeric-type-format result-type)
					  'float)))
			result)))
	(make-numeric-type
	 :class (if (and (eq (numeric-type-class x) 'integer)
			 (eq (numeric-type-class y) 'integer))
		    ;; The difference of integers is always an integer.
		    'integer
		    (numeric-type-class result-type))
	 :format (numeric-type-format result-type)
	 :low (interval-low result)
	 :high (interval-high result)))
      ;; general contagion
      (numeric-contagion x y)))

(defoptimizer (- derive-type) ((x y))
  (two-arg-derive-type x y #'--derive-type-aux #'-))

(defun *-derive-type-aux (x y same-arg)
  (if (and (numeric-type-real-p x)
	   (numeric-type-real-p y))
      (let ((result
	     ;; (* X X) is always positive, so take care to do it right.
	     (if same-arg
		 (interval-sqr (numeric-type->interval x))
		 (interval-mul (numeric-type->interval x)
			       (numeric-type->interval y))))
	    (result-type (numeric-contagion x y)))
	;; If the result type is a float, we need to be sure to coerce
	;; the bounds into the correct type.
	(when (eq (numeric-type-class result-type) 'float)
	  (setf result (interval-func
			#'(lambda (x)
			    (coerce x (or (numeric-type-format result-type)
					  'float)))
			result)))
	(make-numeric-type
	 :class (if (and (eq (numeric-type-class x) 'integer)
			 (eq (numeric-type-class y) 'integer))
		    ;; The product of integers is always an integer.
		    'integer
		    (numeric-type-class result-type))
	 :format (numeric-type-format result-type)
	 :low (interval-low result)
	 :high (interval-high result)))
      (numeric-contagion x y)))

(defoptimizer (* derive-type) ((x y))
  (two-arg-derive-type x y #'*-derive-type-aux #'*))

(defun /-derive-type-aux (x y same-arg)
  (if (and (numeric-type-real-p x)
	   (numeric-type-real-p y))
      (let ((result
	     ;; (/ X X) is always 1, except if X can contain 0. In
	     ;; that case, we shouldn't optimize the division away
	     ;; because we want 0/0 to signal an error.
	     (if (and same-arg
		      (not (interval-contains-p
			    0 (interval-closure (numeric-type->interval y)))))
		 (make-interval :low 1 :high 1)
		 (interval-div (numeric-type->interval x)
			       (numeric-type->interval y))))
	    (result-type (numeric-contagion x y)))
	;; If the result type is a float, we need to be sure to coerce
	;; the bounds into the correct type.
	(when (eq (numeric-type-class result-type) 'float)
	  (setf result (interval-func
			#'(lambda (x)
			    (coerce x (or (numeric-type-format result-type)
					  'float)))
			result)))
	(make-numeric-type :class (numeric-type-class result-type)
			   :format (numeric-type-format result-type)
			   :low (interval-low result)
			   :high (interval-high result)))
      (numeric-contagion x y)))

(defoptimizer (/ derive-type) ((x y))
  (two-arg-derive-type x y #'/-derive-type-aux #'/))

) ; PROGN


;;; KLUDGE: All this ASH optimization is suppressed under CMU CL
;;; because as of version 2.4.6 for Debian, CMU CL blows up on (ASH
;;; 1000000000 -100000000000) (i.e. ASH of two bignums yielding zero)
;;; and it's hard to avoid that calculation in here.
#-(and cmu sb-xc-host)
(progn

(defun ash-derive-type-aux (n-type shift same-arg)
  (declare (ignore same-arg))
  (flet ((ash-outer (n s)
	   (when (and (fixnump s)
		      (<= s 64)
		      (> s sb!vm:*target-most-negative-fixnum*))
	     (ash n s)))
         ;; KLUDGE: The bare 64's here should be related to
         ;; symbolic machine word size values somehow.

	 (ash-inner (n s)
	   (if (and (fixnump s)
		    (> s sb!vm:*target-most-negative-fixnum*))
             (ash n (min s 64))
             (if (minusp n) -1 0))))
    (or (and (csubtypep n-type (specifier-type 'integer))
	     (csubtypep shift (specifier-type 'integer))
	     (let ((n-low (numeric-type-low n-type))
		   (n-high (numeric-type-high n-type))
		   (s-low (numeric-type-low shift))
		   (s-high (numeric-type-high shift)))
	       (make-numeric-type :class 'integer  :complexp :real
				  :low (when n-low
					 (if (minusp n-low)
                                           (ash-outer n-low s-high)
                                           (ash-inner n-low s-low)))
				  :high (when n-high
					  (if (minusp n-high)
                                            (ash-inner n-high s-low)
                                            (ash-outer n-high s-high))))))
	*universal-type*)))

(defoptimizer (ash derive-type) ((n shift))
  (two-arg-derive-type n shift #'ash-derive-type-aux #'ash))
) ; PROGN

#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(macrolet ((frob (fun)
	     `#'(lambda (type type2)
		  (declare (ignore type2))
		  (let ((lo (numeric-type-low type))
			(hi (numeric-type-high type)))
		    (values (if hi (,fun hi) nil) (if lo (,fun lo) nil))))))

  (defoptimizer (%negate derive-type) ((num))
    (derive-integer-type num num (frob -))))

(defoptimizer (lognot derive-type) ((int))
  (derive-integer-type int int
		       (lambda (type type2)
			 (declare (ignore type2))
			 (let ((lo (numeric-type-low type))
			       (hi (numeric-type-high type)))
			   (values (if hi (lognot hi) nil)
				   (if lo (lognot lo) nil)
				   (numeric-type-class type)
				   (numeric-type-format type))))))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (%negate derive-type) ((num))
  (flet ((negate-bound (b)
           (and b
		(set-bound (- (type-bound-number b))
			   (consp b)))))
    (one-arg-derive-type num
			 (lambda (type)
			   (modified-numeric-type
			    type
			    :low (negate-bound (numeric-type-high type))
			    :high (negate-bound (numeric-type-low type))))
			 #'-)))

#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (abs derive-type) ((num))
  (let ((type (continuation-type num)))
    (if (and (numeric-type-p type)
	     (eq (numeric-type-class type) 'integer)
	     (eq (numeric-type-complexp type) :real))
	(let ((lo (numeric-type-low type))
	      (hi (numeric-type-high type)))
	  (make-numeric-type :class 'integer :complexp :real
			     :low (cond ((and hi (minusp hi))
					 (abs hi))
					(lo
					 (max 0 lo))
					(t
					 0))
			     :high (if (and hi lo)
				       (max (abs hi) (abs lo))
				       nil)))
	(numeric-contagion type type))))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defun abs-derive-type-aux (type)
  (cond ((eq (numeric-type-complexp type) :complex)
	 ;; The absolute value of a complex number is always a
	 ;; non-negative float.
	 (let* ((format (case (numeric-type-class type)
			  ((integer rational) 'single-float)
			  (t (numeric-type-format type))))
		(bound-format (or format 'float)))
	   (make-numeric-type :class 'float
			      :format format
			      :complexp :real
			      :low (coerce 0 bound-format)
			      :high nil)))
	(t
	 ;; The absolute value of a real number is a non-negative real
	 ;; of the same type.
	 (let* ((abs-bnd (interval-abs (numeric-type->interval type)))
		(class (numeric-type-class type))
		(format (numeric-type-format type))
		(bound-type (or format class 'real)))
	   (make-numeric-type
	    :class class
	    :format format
	    :complexp :real
	    :low (coerce-numeric-bound (interval-low abs-bnd) bound-type)
	    :high (coerce-numeric-bound
		   (interval-high abs-bnd) bound-type))))))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (abs derive-type) ((num))
  (one-arg-derive-type num #'abs-derive-type-aux #'abs))

#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (truncate derive-type) ((number divisor))
  (let ((number-type (continuation-type number))
	(divisor-type (continuation-type divisor))
	(integer-type (specifier-type 'integer)))
    (if (and (numeric-type-p number-type)
	     (csubtypep number-type integer-type)
	     (numeric-type-p divisor-type)
	     (csubtypep divisor-type integer-type))
	(let ((number-low (numeric-type-low number-type))
	      (number-high (numeric-type-high number-type))
	      (divisor-low (numeric-type-low divisor-type))
	      (divisor-high (numeric-type-high divisor-type)))
	  (values-specifier-type
	   `(values ,(integer-truncate-derive-type number-low number-high
						   divisor-low divisor-high)
		    ,(integer-rem-derive-type number-low number-high
					      divisor-low divisor-high))))
	*universal-type*)))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(progn

(defun rem-result-type (number-type divisor-type)
  ;; Figure out what the remainder type is. The remainder is an
  ;; integer if both args are integers; a rational if both args are
  ;; rational; and a float otherwise.
  (cond ((and (csubtypep number-type (specifier-type 'integer))
	      (csubtypep divisor-type (specifier-type 'integer)))
	 'integer)
	((and (csubtypep number-type (specifier-type 'rational))
	      (csubtypep divisor-type (specifier-type 'rational)))
	 'rational)
	((and (csubtypep number-type (specifier-type 'float))
	      (csubtypep divisor-type (specifier-type 'float)))
	 ;; Both are floats so the result is also a float, of
	 ;; the largest type.
	 (or (float-format-max (numeric-type-format number-type)
			       (numeric-type-format divisor-type))
	     'float))
	((and (csubtypep number-type (specifier-type 'float))
	      (csubtypep divisor-type (specifier-type 'rational)))
	 ;; One of the arguments is a float and the other is a
	 ;; rational. The remainder is a float of the same
	 ;; type.
	 (or (numeric-type-format number-type) 'float))
	((and (csubtypep divisor-type (specifier-type 'float))
	      (csubtypep number-type (specifier-type 'rational)))
	 ;; One of the arguments is a float and the other is a
	 ;; rational. The remainder is a float of the same
	 ;; type.
	 (or (numeric-type-format divisor-type) 'float))
	(t
	 ;; Some unhandled combination. This usually means both args
	 ;; are REAL so the result is a REAL.
	 'real)))

(defun truncate-derive-type-quot (number-type divisor-type)
  (let* ((rem-type (rem-result-type number-type divisor-type))
	 (number-interval (numeric-type->interval number-type))
	 (divisor-interval (numeric-type->interval divisor-type)))
    ;;(declare (type (member '(integer rational float)) rem-type))
    ;; We have real numbers now.
    (cond ((eq rem-type 'integer)
	   ;; Since the remainder type is INTEGER, both args are
	   ;; INTEGERs.
	   (let* ((res (integer-truncate-derive-type
			(interval-low number-interval)
			(interval-high number-interval)
			(interval-low divisor-interval)
			(interval-high divisor-interval))))
	     (specifier-type (if (listp res) res 'integer))))
	  (t
	   (let ((quot (truncate-quotient-bound
			(interval-div number-interval
				      divisor-interval))))
	     (specifier-type `(integer ,(or (interval-low quot) '*)
				       ,(or (interval-high quot) '*))))))))

(defun truncate-derive-type-rem (number-type divisor-type)
  (let* ((rem-type (rem-result-type number-type divisor-type))
	 (number-interval (numeric-type->interval number-type))
	 (divisor-interval (numeric-type->interval divisor-type))
	 (rem (truncate-rem-bound number-interval divisor-interval)))
    ;;(declare (type (member '(integer rational float)) rem-type))
    ;; We have real numbers now.
    (cond ((eq rem-type 'integer)
	   ;; Since the remainder type is INTEGER, both args are
	   ;; INTEGERs.
	   (specifier-type `(,rem-type ,(or (interval-low rem) '*)
				       ,(or (interval-high rem) '*))))
	  (t
	   (multiple-value-bind (class format)
	       (ecase rem-type
		 (integer
		  (values 'integer nil))
		 (rational
		  (values 'rational nil))
		 ((or single-float double-float #!+long-float long-float)
		  (values 'float rem-type))
		 (float
		  (values 'float nil))
		 (real
		  (values nil nil)))
	     (when (member rem-type '(float single-float double-float
				      	    #!+long-float long-float))
	       (setf rem (interval-func #'(lambda (x)
					    (coerce x rem-type))
					rem)))
	     (make-numeric-type :class class
				:format format
				:low (interval-low rem)
				:high (interval-high rem)))))))

(defun truncate-derive-type-quot-aux (num div same-arg)
  (declare (ignore same-arg))
  (if (and (numeric-type-real-p num)
	   (numeric-type-real-p div))
      (truncate-derive-type-quot num div)
      *empty-type*))

(defun truncate-derive-type-rem-aux (num div same-arg)
  (declare (ignore same-arg))
  (if (and (numeric-type-real-p num)
	   (numeric-type-real-p div))
      (truncate-derive-type-rem num div)
      *empty-type*))

(defoptimizer (truncate derive-type) ((number divisor))
  (let ((quot (two-arg-derive-type number divisor
				   #'truncate-derive-type-quot-aux #'truncate))
	(rem (two-arg-derive-type number divisor
				  #'truncate-derive-type-rem-aux #'rem)))
    (when (and quot rem)
      (make-values-type :required (list quot rem)))))

(defun ftruncate-derive-type-quot (number-type divisor-type)
  ;; The bounds are the same as for truncate. However, the first
  ;; result is a float of some type. We need to determine what that
  ;; type is. Basically it's the more contagious of the two types.
  (let ((q-type (truncate-derive-type-quot number-type divisor-type))
	(res-type (numeric-contagion number-type divisor-type)))
    (make-numeric-type :class 'float
		       :format (numeric-type-format res-type)
		       :low (numeric-type-low q-type)
		       :high (numeric-type-high q-type))))

(defun ftruncate-derive-type-quot-aux (n d same-arg)
  (declare (ignore same-arg))
  (if (and (numeric-type-real-p n)
	   (numeric-type-real-p d))
      (ftruncate-derive-type-quot n d)
      *empty-type*))

(defoptimizer (ftruncate derive-type) ((number divisor))
  (let ((quot
	 (two-arg-derive-type number divisor
			      #'ftruncate-derive-type-quot-aux #'ftruncate))
	(rem (two-arg-derive-type number divisor
				  #'truncate-derive-type-rem-aux #'rem)))
    (when (and quot rem)
      (make-values-type :required (list quot rem)))))

(defun %unary-truncate-derive-type-aux (number)
  (truncate-derive-type-quot number (specifier-type '(integer 1 1))))

(defoptimizer (%unary-truncate derive-type) ((number))
  (one-arg-derive-type number
		       #'%unary-truncate-derive-type-aux
		       #'%unary-truncate))

;;; Define optimizers for FLOOR and CEILING.
(macrolet
    ((frob-opt (name q-name r-name)
       (let ((q-aux (symbolicate q-name "-AUX"))
	     (r-aux (symbolicate r-name "-AUX")))
	 `(progn
	   ;; Compute type of quotient (first) result.
	   (defun ,q-aux (number-type divisor-type)
	     (let* ((number-interval
		     (numeric-type->interval number-type))
		    (divisor-interval
		     (numeric-type->interval divisor-type))
		    (quot (,q-name (interval-div number-interval
						 divisor-interval))))
	       (specifier-type `(integer ,(or (interval-low quot) '*)
					 ,(or (interval-high quot) '*)))))
	   ;; Compute type of remainder.
	   (defun ,r-aux (number-type divisor-type)
	     (let* ((divisor-interval
		     (numeric-type->interval divisor-type))
		    (rem (,r-name divisor-interval))
		    (result-type (rem-result-type number-type divisor-type)))
	       (multiple-value-bind (class format)
		   (ecase result-type
		     (integer
		      (values 'integer nil))
		     (rational
		      (values 'rational nil))
		     ((or single-float double-float #!+long-float long-float)
		      (values 'float result-type))
		     (float
		      (values 'float nil))
		     (real
		      (values nil nil)))
		 (when (member result-type '(float single-float double-float
					     #!+long-float long-float))
		   ;; Make sure that the limits on the interval have
		   ;; the right type.
		   (setf rem (interval-func (lambda (x)
					      (coerce x result-type))
					    rem)))
		 (make-numeric-type :class class
				    :format format
				    :low (interval-low rem)
				    :high (interval-high rem)))))
	   ;; the optimizer itself
	   (defoptimizer (,name derive-type) ((number divisor))
	     (flet ((derive-q (n d same-arg)
		      (declare (ignore same-arg))
		      (if (and (numeric-type-real-p n)
			       (numeric-type-real-p d))
			  (,q-aux n d)
			  *empty-type*))
		    (derive-r (n d same-arg)
		      (declare (ignore same-arg))
		      (if (and (numeric-type-real-p n)
			       (numeric-type-real-p d))
			  (,r-aux n d)
			  *empty-type*)))
	       (let ((quot (two-arg-derive-type
			    number divisor #'derive-q #',name))
		     (rem (two-arg-derive-type
			   number divisor #'derive-r #'mod)))
		 (when (and quot rem)
		   (make-values-type :required (list quot rem))))))))))

  ;; FIXME: DEF-FROB-OPT, not just FROB-OPT
  (frob-opt floor floor-quotient-bound floor-rem-bound)
  (frob-opt ceiling ceiling-quotient-bound ceiling-rem-bound))

;;; Define optimizers for FFLOOR and FCEILING
(macrolet
    ((frob-opt (name q-name r-name)
       (let ((q-aux (symbolicate "F" q-name "-AUX"))
	     (r-aux (symbolicate r-name "-AUX")))
	 `(progn
	   ;; Compute type of quotient (first) result.
	   (defun ,q-aux (number-type divisor-type)
	     (let* ((number-interval
		     (numeric-type->interval number-type))
		    (divisor-interval
		     (numeric-type->interval divisor-type))
		    (quot (,q-name (interval-div number-interval
						 divisor-interval)))
		    (res-type (numeric-contagion number-type divisor-type)))
	       (make-numeric-type
		:class (numeric-type-class res-type)
		:format (numeric-type-format res-type)
		:low  (interval-low quot)
		:high (interval-high quot))))

	   (defoptimizer (,name derive-type) ((number divisor))
	     (flet ((derive-q (n d same-arg)
		      (declare (ignore same-arg))
		      (if (and (numeric-type-real-p n)
			       (numeric-type-real-p d))
			  (,q-aux n d)
			  *empty-type*))
		    (derive-r (n d same-arg)
		      (declare (ignore same-arg))
		      (if (and (numeric-type-real-p n)
			       (numeric-type-real-p d))
			  (,r-aux n d)
			  *empty-type*)))
	       (let ((quot (two-arg-derive-type
			    number divisor #'derive-q #',name))
		     (rem (two-arg-derive-type
			   number divisor #'derive-r #'mod)))
		 (when (and quot rem)
		   (make-values-type :required (list quot rem))))))))))

  ;; FIXME: DEF-FROB-OPT, not just FROB-OPT
  (frob-opt ffloor floor-quotient-bound floor-rem-bound)
  (frob-opt fceiling ceiling-quotient-bound ceiling-rem-bound))

;;; functions to compute the bounds on the quotient and remainder for
;;; the FLOOR function
(defun floor-quotient-bound (quot)
  ;; Take the floor of the quotient and then massage it into what we
  ;; need.
  (let ((lo (interval-low quot))
	(hi (interval-high quot)))
    ;; Take the floor of the lower bound. The result is always a
    ;; closed lower bound.
    (setf lo (if lo
		 (floor (type-bound-number lo))
		 nil))
    ;; For the upper bound, we need to be careful.
    (setf hi
	  (cond ((consp hi)
		 ;; An open bound. We need to be careful here because
		 ;; the floor of '(10.0) is 9, but the floor of
		 ;; 10.0 is 10.
		 (multiple-value-bind (q r) (floor (first hi))
		   (if (zerop r)
		       (1- q)
		       q)))
		(hi
		 ;; A closed bound, so the answer is obvious.
		 (floor hi))
		(t
		 hi)))
    (make-interval :low lo :high hi)))
(defun floor-rem-bound (div)
  ;; The remainder depends only on the divisor. Try to get the
  ;; correct sign for the remainder if we can.
  (case (interval-range-info div)
    (+
     ;; The divisor is always positive.
     (let ((rem (interval-abs div)))
       (setf (interval-low rem) 0)
       (when (and (numberp (interval-high rem))
		  (not (zerop (interval-high rem))))
	 ;; The remainder never contains the upper bound. However,
	 ;; watch out for the case where the high limit is zero!
	 (setf (interval-high rem) (list (interval-high rem))))
       rem))
    (-
     ;; The divisor is always negative.
     (let ((rem (interval-neg (interval-abs div))))
       (setf (interval-high rem) 0)
       (when (numberp (interval-low rem))
	 ;; The remainder never contains the lower bound.
	 (setf (interval-low rem) (list (interval-low rem))))
       rem))
    (otherwise
     ;; The divisor can be positive or negative. All bets off. The
     ;; magnitude of remainder is the maximum value of the divisor.
     (let ((limit (type-bound-number (interval-high (interval-abs div)))))
       ;; The bound never reaches the limit, so make the interval open.
       (make-interval :low (if limit
			       (list (- limit))
			       limit)
		      :high (list limit))))))
#| Test cases
(floor-quotient-bound (make-interval :low 0.3 :high 10.3))
=> #S(INTERVAL :LOW 0 :HIGH 10)
(floor-quotient-bound (make-interval :low 0.3 :high '(10.3)))
=> #S(INTERVAL :LOW 0 :HIGH 10)
(floor-quotient-bound (make-interval :low 0.3 :high 10))
=> #S(INTERVAL :LOW 0 :HIGH 10)
(floor-quotient-bound (make-interval :low 0.3 :high '(10)))
=> #S(INTERVAL :LOW 0 :HIGH 9)
(floor-quotient-bound (make-interval :low '(0.3) :high 10.3))
=> #S(INTERVAL :LOW 0 :HIGH 10)
(floor-quotient-bound (make-interval :low '(0.0) :high 10.3))
=> #S(INTERVAL :LOW 0 :HIGH 10)
(floor-quotient-bound (make-interval :low '(-1.3) :high 10.3))
=> #S(INTERVAL :LOW -2 :HIGH 10)
(floor-quotient-bound (make-interval :low '(-1.0) :high 10.3))
=> #S(INTERVAL :LOW -1 :HIGH 10)
(floor-quotient-bound (make-interval :low -1.0 :high 10.3))
=> #S(INTERVAL :LOW -1 :HIGH 10)

(floor-rem-bound (make-interval :low 0.3 :high 10.3))
=> #S(INTERVAL :LOW 0 :HIGH '(10.3))
(floor-rem-bound (make-interval :low 0.3 :high '(10.3)))
=> #S(INTERVAL :LOW 0 :HIGH '(10.3))
(floor-rem-bound (make-interval :low -10 :high -2.3))
#S(INTERVAL :LOW (-10) :HIGH 0)
(floor-rem-bound (make-interval :low 0.3 :high 10))
=> #S(INTERVAL :LOW 0 :HIGH '(10))
(floor-rem-bound (make-interval :low '(-1.3) :high 10.3))
=> #S(INTERVAL :LOW '(-10.3) :HIGH '(10.3))
(floor-rem-bound (make-interval :low '(-20.3) :high 10.3))
=> #S(INTERVAL :LOW (-20.3) :HIGH (20.3))
|#

;;; same functions for CEILING
(defun ceiling-quotient-bound (quot)
  ;; Take the ceiling of the quotient and then massage it into what we
  ;; need.
  (let ((lo (interval-low quot))
	(hi (interval-high quot)))
    ;; Take the ceiling of the upper bound. The result is always a
    ;; closed upper bound.
    (setf hi (if hi
		 (ceiling (type-bound-number hi))
		 nil))
    ;; For the lower bound, we need to be careful.
    (setf lo
	  (cond ((consp lo)
		 ;; An open bound. We need to be careful here because
		 ;; the ceiling of '(10.0) is 11, but the ceiling of
		 ;; 10.0 is 10.
		 (multiple-value-bind (q r) (ceiling (first lo))
		   (if (zerop r)
		       (1+ q)
		       q)))
		(lo
		 ;; A closed bound, so the answer is obvious.
		 (ceiling lo))
		(t
		 lo)))
    (make-interval :low lo :high hi)))
(defun ceiling-rem-bound (div)
  ;; The remainder depends only on the divisor. Try to get the
  ;; correct sign for the remainder if we can.
  (case (interval-range-info div)
    (+
     ;; Divisor is always positive. The remainder is negative.
     (let ((rem (interval-neg (interval-abs div))))
       (setf (interval-high rem) 0)
       (when (and (numberp (interval-low rem))
		  (not (zerop (interval-low rem))))
	 ;; The remainder never contains the upper bound. However,
	 ;; watch out for the case when the upper bound is zero!
	 (setf (interval-low rem) (list (interval-low rem))))
       rem))
    (-
     ;; Divisor is always negative. The remainder is positive
     (let ((rem (interval-abs div)))
       (setf (interval-low rem) 0)
       (when (numberp (interval-high rem))
	 ;; The remainder never contains the lower bound.
	 (setf (interval-high rem) (list (interval-high rem))))
       rem))
    (otherwise
     ;; The divisor can be positive or negative. All bets off. The
     ;; magnitude of remainder is the maximum value of the divisor.
     (let ((limit (type-bound-number (interval-high (interval-abs div)))))
       ;; The bound never reaches the limit, so make the interval open.
       (make-interval :low (if limit
			       (list (- limit))
			       limit)
		      :high (list limit))))))

#| Test cases
(ceiling-quotient-bound (make-interval :low 0.3 :high 10.3))
=> #S(INTERVAL :LOW 1 :HIGH 11)
(ceiling-quotient-bound (make-interval :low 0.3 :high '(10.3)))
=> #S(INTERVAL :LOW 1 :HIGH 11)
(ceiling-quotient-bound (make-interval :low 0.3 :high 10))
=> #S(INTERVAL :LOW 1 :HIGH 10)
(ceiling-quotient-bound (make-interval :low 0.3 :high '(10)))
=> #S(INTERVAL :LOW 1 :HIGH 10)
(ceiling-quotient-bound (make-interval :low '(0.3) :high 10.3))
=> #S(INTERVAL :LOW 1 :HIGH 11)
(ceiling-quotient-bound (make-interval :low '(0.0) :high 10.3))
=> #S(INTERVAL :LOW 1 :HIGH 11)
(ceiling-quotient-bound (make-interval :low '(-1.3) :high 10.3))
=> #S(INTERVAL :LOW -1 :HIGH 11)
(ceiling-quotient-bound (make-interval :low '(-1.0) :high 10.3))
=> #S(INTERVAL :LOW 0 :HIGH 11)
(ceiling-quotient-bound (make-interval :low -1.0 :high 10.3))
=> #S(INTERVAL :LOW -1 :HIGH 11)

(ceiling-rem-bound (make-interval :low 0.3 :high 10.3))
=> #S(INTERVAL :LOW (-10.3) :HIGH 0)
(ceiling-rem-bound (make-interval :low 0.3 :high '(10.3)))
=> #S(INTERVAL :LOW 0 :HIGH '(10.3))
(ceiling-rem-bound (make-interval :low -10 :high -2.3))
=> #S(INTERVAL :LOW 0 :HIGH (10))
(ceiling-rem-bound (make-interval :low 0.3 :high 10))
=> #S(INTERVAL :LOW (-10) :HIGH 0)
(ceiling-rem-bound (make-interval :low '(-1.3) :high 10.3))
=> #S(INTERVAL :LOW (-10.3) :HIGH (10.3))
(ceiling-rem-bound (make-interval :low '(-20.3) :high 10.3))
=> #S(INTERVAL :LOW (-20.3) :HIGH (20.3))
|#

(defun truncate-quotient-bound (quot)
  ;; For positive quotients, truncate is exactly like floor. For
  ;; negative quotients, truncate is exactly like ceiling. Otherwise,
  ;; it's the union of the two pieces.
  (case (interval-range-info quot)
    (+
     ;; just like FLOOR
     (floor-quotient-bound quot))
    (-
     ;; just like CEILING
     (ceiling-quotient-bound quot))
    (otherwise
     ;; Split the interval into positive and negative pieces, compute
     ;; the result for each piece and put them back together.
     (destructuring-bind (neg pos) (interval-split 0 quot t t)
       (interval-merge-pair (ceiling-quotient-bound neg)
			    (floor-quotient-bound pos))))))

(defun truncate-rem-bound (num div)
  ;; This is significantly more complicated than FLOOR or CEILING. We
  ;; need both the number and the divisor to determine the range. The
  ;; basic idea is to split the ranges of NUM and DEN into positive
  ;; and negative pieces and deal with each of the four possibilities
  ;; in turn.
  (case (interval-range-info num)
    (+
     (case (interval-range-info div)
       (+
	(floor-rem-bound div))
       (-
	(ceiling-rem-bound div))
       (otherwise
	(destructuring-bind (neg pos) (interval-split 0 div t t)
	  (interval-merge-pair (truncate-rem-bound num neg)
			       (truncate-rem-bound num pos))))))
    (-
     (case (interval-range-info div)
       (+
	(ceiling-rem-bound div))
       (-
	(floor-rem-bound div))
       (otherwise
	(destructuring-bind (neg pos) (interval-split 0 div t t)
	  (interval-merge-pair (truncate-rem-bound num neg)
			       (truncate-rem-bound num pos))))))
    (otherwise
     (destructuring-bind (neg pos) (interval-split 0 num t t)
       (interval-merge-pair (truncate-rem-bound neg div)
			    (truncate-rem-bound pos div))))))
) ; PROGN

;;; Derive useful information about the range. Returns three values:
;;; - '+ if its positive, '- negative, or nil if it overlaps 0.
;;; - The abs of the minimal value (i.e. closest to 0) in the range.
;;; - The abs of the maximal value if there is one, or nil if it is
;;;   unbounded.
(defun numeric-range-info (low high)
  (cond ((and low (not (minusp low)))
	 (values '+ low high))
	((and high (not (plusp high)))
	 (values '- (- high) (if low (- low) nil)))
	(t
	 (values nil 0 (and low high (max (- low) high))))))

(defun integer-truncate-derive-type
       (number-low number-high divisor-low divisor-high)
  ;; The result cannot be larger in magnitude than the number, but the
  ;; sign might change. If we can determine the sign of either the
  ;; number or the divisor, we can eliminate some of the cases.
  (multiple-value-bind (number-sign number-min number-max)
      (numeric-range-info number-low number-high)
    (multiple-value-bind (divisor-sign divisor-min divisor-max)
	(numeric-range-info divisor-low divisor-high)
      (when (and divisor-max (zerop divisor-max))
	;; We've got a problem: guaranteed division by zero.
	(return-from integer-truncate-derive-type t))
      (when (zerop divisor-min)
	;; We'll assume that they aren't going to divide by zero.
	(incf divisor-min))
      (cond ((and number-sign divisor-sign)
	     ;; We know the sign of both.
	     (if (eq number-sign divisor-sign)
		 ;; Same sign, so the result will be positive.
		 `(integer ,(if divisor-max
				(truncate number-min divisor-max)
				0)
			   ,(if number-max
				(truncate number-max divisor-min)
				'*))
		 ;; Different signs, the result will be negative.
		 `(integer ,(if number-max
				(- (truncate number-max divisor-min))
				'*)
			   ,(if divisor-max
				(- (truncate number-min divisor-max))
				0))))
	    ((eq divisor-sign '+)
	     ;; The divisor is positive. Therefore, the number will just
	     ;; become closer to zero.
	     `(integer ,(if number-low
			    (truncate number-low divisor-min)
			    '*)
		       ,(if number-high
			    (truncate number-high divisor-min)
			    '*)))
	    ((eq divisor-sign '-)
	     ;; The divisor is negative. Therefore, the absolute value of
	     ;; the number will become closer to zero, but the sign will also
	     ;; change.
	     `(integer ,(if number-high
			    (- (truncate number-high divisor-min))
			    '*)
		       ,(if number-low
			    (- (truncate number-low divisor-min))
			    '*)))
	    ;; The divisor could be either positive or negative.
	    (number-max
	     ;; The number we are dividing has a bound. Divide that by the
	     ;; smallest posible divisor.
	     (let ((bound (truncate number-max divisor-min)))
	       `(integer ,(- bound) ,bound)))
	    (t
	     ;; The number we are dividing is unbounded, so we can't tell
	     ;; anything about the result.
	     `integer)))))

#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defun integer-rem-derive-type
       (number-low number-high divisor-low divisor-high)
  (if (and divisor-low divisor-high)
      ;; We know the range of the divisor, and the remainder must be
      ;; smaller than the divisor. We can tell the sign of the
      ;; remainer if we know the sign of the number.
      (let ((divisor-max (1- (max (abs divisor-low) (abs divisor-high)))))
	`(integer ,(if (or (null number-low)
			   (minusp number-low))
		       (- divisor-max)
		       0)
		  ,(if (or (null number-high)
			   (plusp number-high))
		       divisor-max
		       0)))
      ;; The divisor is potentially either very positive or very
      ;; negative. Therefore, the remainer is unbounded, but we might
      ;; be able to tell something about the sign from the number.
      `(integer ,(if (and number-low (not (minusp number-low)))
		     ;; The number we are dividing is positive.
		     ;; Therefore, the remainder must be positive.
		     0
		     '*)
		,(if (and number-high (not (plusp number-high)))
		     ;; The number we are dividing is negative.
		     ;; Therefore, the remainder must be negative.
		     0
		     '*))))

#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (random derive-type) ((bound &optional state))
  (let ((type (continuation-type bound)))
    (when (numeric-type-p type)
      (let ((class (numeric-type-class type))
	    (high (numeric-type-high type))
	    (format (numeric-type-format type)))
	(make-numeric-type
	 :class class
	 :format format
	 :low (coerce 0 (or format class 'real))
	 :high (cond ((not high) nil)
		     ((eq class 'integer) (max (1- high) 0))
		     ((or (consp high) (zerop high)) high)
		     (t `(,high))))))))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defun random-derive-type-aux (type)
  (let ((class (numeric-type-class type))
	(high (numeric-type-high type))
	(format (numeric-type-format type)))
    (make-numeric-type
	 :class class
	 :format format
	 :low (coerce 0 (or format class 'real))
	 :high (cond ((not high) nil)
		     ((eq class 'integer) (max (1- high) 0))
		     ((or (consp high) (zerop high)) high)
		     (t `(,high))))))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (random derive-type) ((bound &optional state))
  (one-arg-derive-type bound #'random-derive-type-aux nil))

;;;; DERIVE-TYPE methods for LOGAND, LOGIOR, and friends

;;; Return the maximum number of bits an integer of the supplied type
;;; can take up, or NIL if it is unbounded. The second (third) value
;;; is T if the integer can be positive (negative) and NIL if not.
;;; Zero counts as positive.
(defun integer-type-length (type)
  (if (numeric-type-p type)
      (let ((min (numeric-type-low type))
	    (max (numeric-type-high type)))
	(values (and min max (max (integer-length min) (integer-length max)))
		(or (null max) (not (minusp max)))
		(or (null min) (minusp min))))
      (values nil t t)))

(defun logand-derive-type-aux (x y &optional same-leaf)
  (declare (ignore same-leaf))
  (multiple-value-bind (x-len x-pos x-neg) (integer-type-length x)
    (declare (ignore x-pos))
    (multiple-value-bind (y-len y-pos y-neg) (integer-type-length  y)
      (declare (ignore y-pos))
      (if (not x-neg)
	  ;; X must be positive.
	  (if (not y-neg)
	      ;; They must both be positive.
	      (cond ((or (null x-len) (null y-len))
		     (specifier-type 'unsigned-byte))
		    ((or (zerop x-len) (zerop y-len))
		     (specifier-type '(integer 0 0)))
		    (t
		     (specifier-type `(unsigned-byte ,(min x-len y-len)))))
	      ;; X is positive, but Y might be negative.
	      (cond ((null x-len)
		     (specifier-type 'unsigned-byte))
		    ((zerop x-len)
		     (specifier-type '(integer 0 0)))
		    (t
		     (specifier-type `(unsigned-byte ,x-len)))))
	  ;; X might be negative.
	  (if (not y-neg)
	      ;; Y must be positive.
	      (cond ((null y-len)
		     (specifier-type 'unsigned-byte))
		    ((zerop y-len)
		     (specifier-type '(integer 0 0)))
		    (t
		     (specifier-type
		      `(unsigned-byte ,y-len))))
	      ;; Either might be negative.
	      (if (and x-len y-len)
		  ;; The result is bounded.
		  (specifier-type `(signed-byte ,(1+ (max x-len y-len))))
		  ;; We can't tell squat about the result.
		  (specifier-type 'integer)))))))

(defun logior-derive-type-aux (x y &optional same-leaf)
  (declare (ignore same-leaf))
  (multiple-value-bind (x-len x-pos x-neg) (integer-type-length x)
    (multiple-value-bind (y-len y-pos y-neg) (integer-type-length y)
      (cond
       ((and (not x-neg) (not y-neg))
	;; Both are positive.
	(if (and x-len y-len (zerop x-len) (zerop y-len))
	    (specifier-type '(integer 0 0))
	    (specifier-type `(unsigned-byte ,(if (and x-len y-len)
					     (max x-len y-len)
					     '*)))))
       ((not x-pos)
	;; X must be negative.
	(if (not y-pos)
	    ;; Both are negative. The result is going to be negative
	    ;; and be the same length or shorter than the smaller.
	    (if (and x-len y-len)
		;; It's bounded.
		(specifier-type `(integer ,(ash -1 (min x-len y-len)) -1))
		;; It's unbounded.
		(specifier-type '(integer * -1)))
	    ;; X is negative, but we don't know about Y. The result
	    ;; will be negative, but no more negative than X.
	    (specifier-type
	     `(integer ,(or (numeric-type-low x) '*)
		       -1))))
       (t
	;; X might be either positive or negative.
	(if (not y-pos)
	    ;; But Y is negative. The result will be negative.
	    (specifier-type
	     `(integer ,(or (numeric-type-low y) '*)
		       -1))
	    ;; We don't know squat about either. It won't get any bigger.
	    (if (and x-len y-len)
		;; Bounded.
		(specifier-type `(signed-byte ,(1+ (max x-len y-len))))
		;; Unbounded.
		(specifier-type 'integer))))))))

(defun logxor-derive-type-aux (x y &optional same-leaf)
  (declare (ignore same-leaf))
  (multiple-value-bind (x-len x-pos x-neg) (integer-type-length x)
    (multiple-value-bind (y-len y-pos y-neg) (integer-type-length y)
      (cond
       ((or (and (not x-neg) (not y-neg))
	    (and (not x-pos) (not y-pos)))
	;; Either both are negative or both are positive. The result
	;; will be positive, and as long as the longer.
	(if (and x-len y-len (zerop x-len) (zerop y-len))
	    (specifier-type '(integer 0 0))
	    (specifier-type `(unsigned-byte ,(if (and x-len y-len)
					     (max x-len y-len)
					     '*)))))
       ((or (and (not x-pos) (not y-neg))
	    (and (not y-neg) (not y-pos)))
	;; Either X is negative and Y is positive of vice-versa. The
	;; result will be negative.
	(specifier-type `(integer ,(if (and x-len y-len)
				       (ash -1 (max x-len y-len))
				       '*)
				  -1)))
       ;; We can't tell what the sign of the result is going to be.
       ;; All we know is that we don't create new bits.
       ((and x-len y-len)
	(specifier-type `(signed-byte ,(1+ (max x-len y-len)))))
       (t
	(specifier-type 'integer))))))

(macrolet ((deffrob (logfcn)
	     (let ((fcn-aux (symbolicate logfcn "-DERIVE-TYPE-AUX")))
	     `(defoptimizer (,logfcn derive-type) ((x y))
		(two-arg-derive-type x y #',fcn-aux #',logfcn)))))
  (deffrob logand)
  (deffrob logior)
  (deffrob logxor))

;;;; miscellaneous derive-type methods

(defoptimizer (integer-length derive-type) ((x))
  (let ((x-type (continuation-type x)))
    (when (and (numeric-type-p x-type)
               (csubtypep x-type (specifier-type 'integer)))
      ;; If the X is of type (INTEGER LO HI), then the INTEGER-LENGTH
      ;; of X is (INTEGER (MIN lo hi) (MAX lo hi), basically.  Be
      ;; careful about LO or HI being NIL, though.  Also, if 0 is
      ;; contained in X, the lower bound is obviously 0.
      (flet ((null-or-min (a b)
               (and a b (min (integer-length a)
                             (integer-length b))))
             (null-or-max (a b)
               (and a b (max (integer-length a)
                             (integer-length b)))))
        (let* ((min (numeric-type-low x-type))
               (max (numeric-type-high x-type))
               (min-len (null-or-min min max))
               (max-len (null-or-max min max)))
          (when (ctypep 0 x-type)
            (setf min-len 0))
          (specifier-type `(integer ,(or min-len '*) ,(or max-len '*))))))))

(defoptimizer (code-char derive-type) ((code))
  (specifier-type 'base-char))

(defoptimizer (values derive-type) ((&rest values))
  (values-specifier-type
   `(values ,@(mapcar (lambda (x)
			(type-specifier (continuation-type x)))
		      values))))

;;;; byte operations
;;;;
;;;; We try to turn byte operations into simple logical operations.
;;;; First, we convert byte specifiers into separate size and position
;;;; arguments passed to internal %FOO functions. We then attempt to
;;;; transform the %FOO functions into boolean operations when the
;;;; size and position are constant and the operands are fixnums.

(macrolet (;; Evaluate body with SIZE-VAR and POS-VAR bound to
	   ;; expressions that evaluate to the SIZE and POSITION of
	   ;; the byte-specifier form SPEC. We may wrap a let around
	   ;; the result of the body to bind some variables.
	   ;;
	   ;; If the spec is a BYTE form, then bind the vars to the
	   ;; subforms. otherwise, evaluate SPEC and use the BYTE-SIZE
	   ;; and BYTE-POSITION. The goal of this transformation is to
	   ;; avoid consing up byte specifiers and then immediately
	   ;; throwing them away.
	   (with-byte-specifier ((size-var pos-var spec) &body body)
	     (once-only ((spec `(macroexpand ,spec))
			 (temp '(gensym)))
			`(if (and (consp ,spec)
				  (eq (car ,spec) 'byte)
				  (= (length ,spec) 3))
			(let ((,size-var (second ,spec))
			      (,pos-var (third ,spec)))
			  ,@body)
			(let ((,size-var `(byte-size ,,temp))
			      (,pos-var `(byte-position ,,temp)))
			  `(let ((,,temp ,,spec))
			     ,,@body))))))

  (define-source-transform ldb (spec int)
    (with-byte-specifier (size pos spec)
      `(%ldb ,size ,pos ,int)))

  (define-source-transform dpb (newbyte spec int)
    (with-byte-specifier (size pos spec)
      `(%dpb ,newbyte ,size ,pos ,int)))

  (define-source-transform mask-field (spec int)
    (with-byte-specifier (size pos spec)
      `(%mask-field ,size ,pos ,int)))

  (define-source-transform deposit-field (newbyte spec int)
    (with-byte-specifier (size pos spec)
      `(%deposit-field ,newbyte ,size ,pos ,int))))

(defoptimizer (%ldb derive-type) ((size posn num))
  (let ((size (continuation-type size)))
    (if (and (numeric-type-p size)
	     (csubtypep size (specifier-type 'integer)))
	(let ((size-high (numeric-type-high size)))
	  (if (and size-high (<= size-high sb!vm:n-word-bits))
	      (specifier-type `(unsigned-byte ,size-high))
	      (specifier-type 'unsigned-byte)))
	*universal-type*)))

(defoptimizer (%mask-field derive-type) ((size posn num))
  (let ((size (continuation-type size))
	(posn (continuation-type posn)))
    (if (and (numeric-type-p size)
	     (csubtypep size (specifier-type 'integer))
	     (numeric-type-p posn)
	     (csubtypep posn (specifier-type 'integer)))
	(let ((size-high (numeric-type-high size))
	      (posn-high (numeric-type-high posn)))
	  (if (and size-high posn-high
		   (<= (+ size-high posn-high) sb!vm:n-word-bits))
	      (specifier-type `(unsigned-byte ,(+ size-high posn-high)))
	      (specifier-type 'unsigned-byte)))
	*universal-type*)))

(defoptimizer (%dpb derive-type) ((newbyte size posn int))
  (let ((size (continuation-type size))
	(posn (continuation-type posn))
	(int (continuation-type int)))
    (if (and (numeric-type-p size)
	     (csubtypep size (specifier-type 'integer))
	     (numeric-type-p posn)
	     (csubtypep posn (specifier-type 'integer))
	     (numeric-type-p int)
	     (csubtypep int (specifier-type 'integer)))
	(let ((size-high (numeric-type-high size))
	      (posn-high (numeric-type-high posn))
	      (high (numeric-type-high int))
	      (low (numeric-type-low int)))
	  (if (and size-high posn-high high low
		   (<= (+ size-high posn-high) sb!vm:n-word-bits))
	      (specifier-type
	       (list (if (minusp low) 'signed-byte 'unsigned-byte)
		     (max (integer-length high)
			  (integer-length low)
			  (+ size-high posn-high))))
	      *universal-type*))
	*universal-type*)))

(defoptimizer (%deposit-field derive-type) ((newbyte size posn int))
  (let ((size (continuation-type size))
	(posn (continuation-type posn))
	(int (continuation-type int)))
    (if (and (numeric-type-p size)
	     (csubtypep size (specifier-type 'integer))
	     (numeric-type-p posn)
	     (csubtypep posn (specifier-type 'integer))
	     (numeric-type-p int)
	     (csubtypep int (specifier-type 'integer)))
	(let ((size-high (numeric-type-high size))
	      (posn-high (numeric-type-high posn))
	      (high (numeric-type-high int))
	      (low (numeric-type-low int)))
	  (if (and size-high posn-high high low
		   (<= (+ size-high posn-high) sb!vm:n-word-bits))
	      (specifier-type
	       (list (if (minusp low) 'signed-byte 'unsigned-byte)
		     (max (integer-length high)
			  (integer-length low)
			  (+ size-high posn-high))))
	      *universal-type*))
	*universal-type*)))

(deftransform %ldb ((size posn int)
		    (fixnum fixnum integer)
		    (unsigned-byte #.sb!vm:n-word-bits))
  "convert to inline logical operations"
  `(logand (ash int (- posn))
	   (ash ,(1- (ash 1 sb!vm:n-word-bits))
		(- size ,sb!vm:n-word-bits))))

(deftransform %mask-field ((size posn int)
			   (fixnum fixnum integer)
			   (unsigned-byte #.sb!vm:n-word-bits))
  "convert to inline logical operations"
  `(logand int
	   (ash (ash ,(1- (ash 1 sb!vm:n-word-bits))
		     (- size ,sb!vm:n-word-bits))
		posn)))

;;; Note: for %DPB and %DEPOSIT-FIELD, we can't use
;;;   (OR (SIGNED-BYTE N) (UNSIGNED-BYTE N))
;;; as the result type, as that would allow result types that cover
;;; the range -2^(n-1) .. 1-2^n, instead of allowing result types of
;;; (UNSIGNED-BYTE N) and result types of (SIGNED-BYTE N).

(deftransform %dpb ((new size posn int)
		    *
		    (unsigned-byte #.sb!vm:n-word-bits))
  "convert to inline logical operations"
  `(let ((mask (ldb (byte size 0) -1)))
     (logior (ash (logand new mask) posn)
	     (logand int (lognot (ash mask posn))))))

(deftransform %dpb ((new size posn int)
		    *
		    (signed-byte #.sb!vm:n-word-bits))
  "convert to inline logical operations"
  `(let ((mask (ldb (byte size 0) -1)))
     (logior (ash (logand new mask) posn)
	     (logand int (lognot (ash mask posn))))))

(deftransform %deposit-field ((new size posn int)
			      *
			      (unsigned-byte #.sb!vm:n-word-bits))
  "convert to inline logical operations"
  `(let ((mask (ash (ldb (byte size 0) -1) posn)))
     (logior (logand new mask)
	     (logand int (lognot mask)))))

(deftransform %deposit-field ((new size posn int)
			      *
			      (signed-byte #.sb!vm:n-word-bits))
  "convert to inline logical operations"
  `(let ((mask (ash (ldb (byte size 0) -1) posn)))
     (logior (logand new mask)
	     (logand int (lognot mask)))))

;;; miscellanous numeric transforms

;;; If a constant appears as the first arg, swap the args.
(deftransform commutative-arg-swap ((x y) * * :defun-only t :node node)
  (if (and (constant-continuation-p x)
	   (not (constant-continuation-p y)))
      `(,(continuation-fun-name (basic-combination-fun node))
	y
	,(continuation-value x))
      (give-up-ir1-transform)))

(dolist (x '(= char= + * logior logand logxor))
  (%deftransform x '(function * *) #'commutative-arg-swap
		 "place constant arg last"))

;;; Handle the case of a constant BOOLE-CODE.
(deftransform boole ((op x y) * * :when :both)
  "convert to inline logical operations"
  (unless (constant-continuation-p op)
    (give-up-ir1-transform "BOOLE code is not a constant."))
  (let ((control (continuation-value op)))
    (case control
      (#.boole-clr 0)
      (#.boole-set -1)
      (#.boole-1 'x)
      (#.boole-2 'y)
      (#.boole-c1 '(lognot x))
      (#.boole-c2 '(lognot y))
      (#.boole-and '(logand x y))
      (#.boole-ior '(logior x y))
      (#.boole-xor '(logxor x y))
      (#.boole-eqv '(logeqv x y))
      (#.boole-nand '(lognand x y))
      (#.boole-nor '(lognor x y))
      (#.boole-andc1 '(logandc1 x y))
      (#.boole-andc2 '(logandc2 x y))
      (#.boole-orc1 '(logorc1 x y))
      (#.boole-orc2 '(logorc2 x y))
      (t
       (abort-ir1-transform "~S is an illegal control arg to BOOLE."
			    control)))))

;;;; converting special case multiply/divide to shifts

;;; If arg is a constant power of two, turn * into a shift.
(deftransform * ((x y) (integer integer) * :when :both)
  "convert x*2^k to shift"
  (unless (constant-continuation-p y)
    (give-up-ir1-transform))
  (let* ((y (continuation-value y))
	 (y-abs (abs y))
	 (len (1- (integer-length y-abs))))
    (unless (= y-abs (ash 1 len))
      (give-up-ir1-transform))
    (if (minusp y)
	`(- (ash x ,len))
	`(ash x ,len))))

;;; If both arguments and the result are (UNSIGNED-BYTE 32), try to
;;; come up with a ``better'' multiplication using multiplier
;;; recoding. There are two different ways the multiplier can be
;;; recoded. The more obvious is to shift X by the correct amount for
;;; each bit set in Y and to sum the results. But if there is a string
;;; of bits that are all set, you can add X shifted by one more then
;;; the bit position of the first set bit and subtract X shifted by
;;; the bit position of the last set bit. We can't use this second
;;; method when the high order bit is bit 31 because shifting by 32
;;; doesn't work too well.
(deftransform * ((x y)
		 ((unsigned-byte 32) (unsigned-byte 32))
		 (unsigned-byte 32))
  "recode as shift and add"
  (unless (constant-continuation-p y)
    (give-up-ir1-transform))
  (let ((y (continuation-value y))
	(result nil)
	(first-one nil))
    (labels ((tub32 (x) `(truly-the (unsigned-byte 32) ,x))
	     (add (next-factor)
	       (setf result
		     (tub32
		      (if result
			  `(+ ,result ,(tub32 next-factor))
			  next-factor)))))
      (declare (inline add))
      (dotimes (bitpos 32)
	(if first-one
	    (when (not (logbitp bitpos y))
	      (add (if (= (1+ first-one) bitpos)
		       ;; There is only a single bit in the string.
		       `(ash x ,first-one)
		       ;; There are at least two.
		       `(- ,(tub32 `(ash x ,bitpos))
			   ,(tub32 `(ash x ,first-one)))))
	      (setf first-one nil))
	    (when (logbitp bitpos y)
	      (setf first-one bitpos))))
      (when first-one
	(cond ((= first-one 31))
	      ((= first-one 30)
	       (add '(ash x 30)))
	      (t
	       (add `(- ,(tub32 '(ash x 31)) ,(tub32 `(ash x ,first-one))))))
	(add '(ash x 31))))
    (or result 0)))

;;; If arg is a constant power of two, turn FLOOR into a shift and
;;; mask. If CEILING, add in (1- (ABS Y)) and then do FLOOR.
(flet ((frob (y ceil-p)
	 (unless (constant-continuation-p y)
	   (give-up-ir1-transform))
	 (let* ((y (continuation-value y))
		(y-abs (abs y))
		(len (1- (integer-length y-abs))))
	   (unless (= y-abs (ash 1 len))
	     (give-up-ir1-transform))
	   (let ((shift (- len))
		 (mask (1- y-abs)))
	     `(let ,(when ceil-p `((x (+ x ,(1- y-abs)))))
		,(if (minusp y)
		     `(values (ash (- x) ,shift)
			      (- (logand (- x) ,mask)))
		     `(values (ash x ,shift)
			      (logand x ,mask))))))))
  (deftransform floor ((x y) (integer integer) *)
    "convert division by 2^k to shift"
    (frob y nil))
  (deftransform ceiling ((x y) (integer integer) *)
    "convert division by 2^k to shift"
    (frob y t)))

;;; Do the same for MOD.
(deftransform mod ((x y) (integer integer) * :when :both)
  "convert remainder mod 2^k to LOGAND"
  (unless (constant-continuation-p y)
    (give-up-ir1-transform))
  (let* ((y (continuation-value y))
	 (y-abs (abs y))
	 (len (1- (integer-length y-abs))))
    (unless (= y-abs (ash 1 len))
      (give-up-ir1-transform))
    (let ((mask (1- y-abs)))
      (if (minusp y)
	  `(- (logand (- x) ,mask))
	  `(logand x ,mask)))))

;;; If arg is a constant power of two, turn TRUNCATE into a shift and mask.
(deftransform truncate ((x y) (integer integer))
  "convert division by 2^k to shift"
  (unless (constant-continuation-p y)
    (give-up-ir1-transform))
  (let* ((y (continuation-value y))
	 (y-abs (abs y))
	 (len (1- (integer-length y-abs))))
    (unless (= y-abs (ash 1 len))
      (give-up-ir1-transform))
    (let* ((shift (- len))
	   (mask (1- y-abs)))
      `(if (minusp x)
	   (values ,(if (minusp y)
			`(ash (- x) ,shift)
			`(- (ash (- x) ,shift)))
		   (- (logand (- x) ,mask)))
	   (values ,(if (minusp y)
			`(- (ash (- x) ,shift))
			`(ash x ,shift))
		   (logand x ,mask))))))

;;; And the same for REM.
(deftransform rem ((x y) (integer integer) * :when :both)
  "convert remainder mod 2^k to LOGAND"
  (unless (constant-continuation-p y)
    (give-up-ir1-transform))
  (let* ((y (continuation-value y))
	 (y-abs (abs y))
	 (len (1- (integer-length y-abs))))
    (unless (= y-abs (ash 1 len))
      (give-up-ir1-transform))
    (let ((mask (1- y-abs)))
      `(if (minusp x)
	   (- (logand (- x) ,mask))
	   (logand x ,mask)))))

;;;; arithmetic and logical identity operation elimination

;;; Flush calls to various arith functions that convert to the
;;; identity function or a constant.
(macrolet ((def-frob (name identity result)
             `(deftransform ,name ((x y) (* (constant-arg (member ,identity)))
                                    * :when :both)
                "fold identity operations"
                ',result)))
  (def-frob ash 0 x)
  (def-frob logand -1 x)
  (def-frob logand 0 0)
  (def-frob logior 0 x)
  (def-frob logior -1 -1)
  (def-frob logxor -1 (lognot x))
  (def-frob logxor 0 x))

;;; These are restricted to rationals, because (- 0 0.0) is 0.0, not -0.0, and
;;; (* 0 -4.0) is -0.0.
(deftransform - ((x y) ((constant-arg (member 0)) rational) *
		 :when :both)
  "convert (- 0 x) to negate"
  '(%negate y))
(deftransform * ((x y) (rational (constant-arg (member 0))) *
		 :when :both)
  "convert (* x 0) to 0"
  0)

;;; Return T if in an arithmetic op including continuations X and Y,
;;; the result type is not affected by the type of X. That is, Y is at
;;; least as contagious as X.
#+nil
(defun not-more-contagious (x y)
  (declare (type continuation x y))
  (let ((x (continuation-type x))
	(y (continuation-type y)))
    (values (type= (numeric-contagion x y)
		   (numeric-contagion y y)))))
;;; Patched version by Raymond Toy. dtc: Should be safer although it
;;; XXX needs more work as valid transforms are missed; some cases are
;;; specific to particular transform functions so the use of this
;;; function may need a re-think.
(defun not-more-contagious (x y)
  (declare (type continuation x y))
  (flet ((simple-numeric-type (num)
	   (and (numeric-type-p num)
		;; Return non-NIL if NUM is integer, rational, or a float
		;; of some type (but not FLOAT)
		(case (numeric-type-class num)
		  ((integer rational)
		   t)
		  (float
		   (numeric-type-format num))
		  (t
		   nil)))))
    (let ((x (continuation-type x))
	  (y (continuation-type y)))
      (if (and (simple-numeric-type x)
	       (simple-numeric-type y))
	  (values (type= (numeric-contagion x y)
			 (numeric-contagion y y)))))))

;;; Fold (+ x 0).
;;;
;;; If y is not constant, not zerop, or is contagious, or a positive
;;; float +0.0 then give up.
(deftransform + ((x y) (t (constant-arg t)) * :when :both)
  "fold zero arg"
  (let ((val (continuation-value y)))
    (unless (and (zerop val)
		 (not (and (floatp val) (plusp (float-sign val))))
		 (not-more-contagious y x))
      (give-up-ir1-transform)))
  'x)

;;; Fold (- x 0).
;;;
;;; If y is not constant, not zerop, or is contagious, or a negative
;;; float -0.0 then give up.
(deftransform - ((x y) (t (constant-arg t)) * :when :both)
  "fold zero arg"
  (let ((val (continuation-value y)))
    (unless (and (zerop val)
		 (not (and (floatp val) (minusp (float-sign val))))
		 (not-more-contagious y x))
      (give-up-ir1-transform)))
  'x)

;;; Fold (OP x +/-1)
(macrolet ((def-frob (name result minus-result)
             `(deftransform ,name ((x y) (t (constant-arg real))
                                    * :when :both)
                "fold identity operations"
                (let ((val (continuation-value y)))
                  (unless (and (= (abs val) 1)
                               (not-more-contagious y x))
                    (give-up-ir1-transform))
                  (if (minusp val) ',minus-result ',result)))))
  (def-frob * x (%negate x))
  (def-frob / x (%negate x))
  (def-frob expt x (/ 1 x)))

;;; Fold (expt x n) into multiplications for small integral values of
;;; N; convert (expt x 1/2) to sqrt.
(deftransform expt ((x y) (t (constant-arg real)) *)
  "recode as multiplication or sqrt"
  (let ((val (continuation-value y)))
    ;; If Y would cause the result to be promoted to the same type as
    ;; Y, we give up. If not, then the result will be the same type
    ;; as X, so we can replace the exponentiation with simple
    ;; multiplication and division for small integral powers.
    (unless (not-more-contagious y x)
      (give-up-ir1-transform))
    (cond ((zerop val) '(float 1 x))
	  ((= val 2) '(* x x))
	  ((= val -2) '(/ (* x x)))
	  ((= val 3) '(* x x x))
	  ((= val -3) '(/ (* x x x)))
	  ((= val 1/2) '(sqrt x))
	  ((= val -1/2) '(/ (sqrt x)))
	  (t (give-up-ir1-transform)))))

;;; KLUDGE: Shouldn't (/ 0.0 0.0), etc. cause exceptions in these
;;; transformations?
;;; Perhaps we should have to prove that the denominator is nonzero before
;;; doing them?  -- WHN 19990917
(macrolet ((def-frob (name)
             `(deftransform ,name ((x y) ((constant-arg (integer 0 0)) integer)
                                   * :when :both)
                "fold zero arg"
                0)))
  (def-frob ash)
  (def-frob /))

(macrolet ((def-frob (name)
             `(deftransform ,name ((x y) ((constant-arg (integer 0 0)) integer)
                                   * :when :both)
                "fold zero arg"
                '(values 0 0))))
  (def-frob truncate)
  (def-frob round)
  (def-frob floor)
  (def-frob ceiling))


;;;; character operations

(deftransform char-equal ((a b) (base-char base-char))
  "open code"
  '(let* ((ac (char-code a))
	  (bc (char-code b))
	  (sum (logxor ac bc)))
     (or (zerop sum)
	 (when (eql sum #x20)
	   (let ((sum (+ ac bc)))
	     (and (> sum 161) (< sum 213)))))))

(deftransform char-upcase ((x) (base-char))
  "open code"
  '(let ((n-code (char-code x)))
     (if (and (> n-code #o140)	; Octal 141 is #\a.
	      (< n-code #o173))	; Octal 172 is #\z.
	 (code-char (logxor #x20 n-code))
	 x)))

(deftransform char-downcase ((x) (base-char))
  "open code"
  '(let ((n-code (char-code x)))
     (if (and (> n-code 64)	; 65 is #\A.
	      (< n-code 91))	; 90 is #\Z.
	 (code-char (logxor #x20 n-code))
	 x)))

;;;; equality predicate transforms

;;; Return true if X and Y are continuations whose only use is a
;;; reference to the same leaf, and the value of the leaf cannot
;;; change.
(defun same-leaf-ref-p (x y)
  (declare (type continuation x y))
  (let ((x-use (continuation-use x))
	(y-use (continuation-use y)))
    (and (ref-p x-use)
	 (ref-p y-use)
	 (eq (ref-leaf x-use) (ref-leaf y-use))
	 (constant-reference-p x-use))))

;;; If X and Y are the same leaf, then the result is true. Otherwise,
;;; if there is no intersection between the types of the arguments,
;;; then the result is definitely false.
(deftransform simple-equality-transform ((x y) * *
					 :defun-only t
					 :when :both)
  (cond ((same-leaf-ref-p x y)
	 t)
	((not (types-equal-or-intersect (continuation-type x)
					(continuation-type y)))
	 nil)
	(t
	 (give-up-ir1-transform))))

(macrolet ((def-frob (x)
             `(%deftransform ',x '(function * *) #'simple-equality-transform)))
  (def-frob eq)
  (def-frob char=)
  (def-frob equal))

;;; This is similar to SIMPLE-EQUALITY-PREDICATE, except that we also
;;; try to convert to a type-specific predicate or EQ:
;;; -- If both args are characters, convert to CHAR=. This is better than
;;;    just converting to EQ, since CHAR= may have special compilation
;;;    strategies for non-standard representations, etc.
;;; -- If either arg is definitely not a number, then we can compare
;;;    with EQ.
;;; -- Otherwise, we try to put the arg we know more about second. If X
;;;    is constant then we put it second. If X is a subtype of Y, we put
;;;    it second. These rules make it easier for the back end to match
;;;    these interesting cases.
;;; -- If Y is a fixnum, then we quietly pass because the back end can
;;;    handle that case, otherwise give an efficiency note.
(deftransform eql ((x y) * * :when :both)
  "convert to simpler equality predicate"
  (let ((x-type (continuation-type x))
	(y-type (continuation-type y))
	(char-type (specifier-type 'character))
	(number-type (specifier-type 'number)))
    (cond ((same-leaf-ref-p x y)
	   t)
	  ((not (types-equal-or-intersect x-type y-type))
	   nil)
	  ((and (csubtypep x-type char-type)
		(csubtypep y-type char-type))
	   '(char= x y))
	  ((or (not (types-equal-or-intersect x-type number-type))
	       (not (types-equal-or-intersect y-type number-type)))
	   '(eq x y))
	  ((and (not (constant-continuation-p y))
		(or (constant-continuation-p x)
		    (and (csubtypep x-type y-type)
			 (not (csubtypep y-type x-type)))))
	   '(eql y x))
	  (t
	   (give-up-ir1-transform)))))

;;; Convert to EQL if both args are rational and complexp is specified
;;; and the same for both.
(deftransform = ((x y) * * :when :both)
  "open code"
  (let ((x-type (continuation-type x))
	(y-type (continuation-type y)))
    (if (and (csubtypep x-type (specifier-type 'number))
	     (csubtypep y-type (specifier-type 'number)))
	(cond ((or (and (csubtypep x-type (specifier-type 'float))
			(csubtypep y-type (specifier-type 'float)))
		   (and (csubtypep x-type (specifier-type '(complex float)))
			(csubtypep y-type (specifier-type '(complex float)))))
	       ;; They are both floats. Leave as = so that -0.0 is
	       ;; handled correctly.
	       (give-up-ir1-transform))
	      ((or (and (csubtypep x-type (specifier-type 'rational))
			(csubtypep y-type (specifier-type 'rational)))
		   (and (csubtypep x-type
				   (specifier-type '(complex rational)))
			(csubtypep y-type
				   (specifier-type '(complex rational)))))
	       ;; They are both rationals and complexp is the same.
	       ;; Convert to EQL.
	       '(eql x y))
	      (t
	       (give-up-ir1-transform
		"The operands might not be the same type.")))
	(give-up-ir1-transform
	 "The operands might not be the same type."))))

;;; If CONT's type is a numeric type, then return the type, otherwise
;;; GIVE-UP-IR1-TRANSFORM.
(defun numeric-type-or-lose (cont)
  (declare (type continuation cont))
  (let ((res (continuation-type cont)))
    (unless (numeric-type-p res) (give-up-ir1-transform))
    res))

;;; See whether we can statically determine (< X Y) using type
;;; information. If X's high bound is < Y's low, then X < Y.
;;; Similarly, if X's low is >= to Y's high, the X >= Y (so return
;;; NIL). If not, at least make sure any constant arg is second.
;;;
;;; FIXME: Why should constant argument be second? It would be nice to
;;; find out and explain.
#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defun ir1-transform-< (x y first second inverse)
  (if (same-leaf-ref-p x y)
      nil
      (let* ((x-type (numeric-type-or-lose x))
	     (x-lo (numeric-type-low x-type))
	     (x-hi (numeric-type-high x-type))
	     (y-type (numeric-type-or-lose y))
	     (y-lo (numeric-type-low y-type))
	     (y-hi (numeric-type-high y-type)))
	(cond ((and x-hi y-lo (< x-hi y-lo))
	       t)
	      ((and y-hi x-lo (>= x-lo y-hi))
	       nil)
	      ((and (constant-continuation-p first)
		    (not (constant-continuation-p second)))
	       `(,inverse y x))
	      (t
	       (give-up-ir1-transform))))))
#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defun ir1-transform-< (x y first second inverse)
  (if (same-leaf-ref-p x y)
      nil
      (let ((xi (numeric-type->interval (numeric-type-or-lose x)))
	    (yi (numeric-type->interval (numeric-type-or-lose y))))
	(cond ((interval-< xi yi)
	       t)
	      ((interval->= xi yi)
	       nil)
	      ((and (constant-continuation-p first)
		    (not (constant-continuation-p second)))
	       `(,inverse y x))
	      (t
	       (give-up-ir1-transform))))))

(deftransform < ((x y) (integer integer) * :when :both)
  (ir1-transform-< x y x y '>))

(deftransform > ((x y) (integer integer) * :when :both)
  (ir1-transform-< y x x y '<))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(deftransform < ((x y) (float float) * :when :both)
  (ir1-transform-< x y x y '>))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(deftransform > ((x y) (float float) * :when :both)
  (ir1-transform-< y x x y '<))

;;;; converting N-arg comparisons
;;;;
;;;; We convert calls to N-arg comparison functions such as < into
;;;; two-arg calls. This transformation is enabled for all such
;;;; comparisons in this file. If any of these predicates are not
;;;; open-coded, then the transformation should be removed at some
;;;; point to avoid pessimization.

;;; This function is used for source transformation of N-arg
;;; comparison functions other than inequality. We deal both with
;;; converting to two-arg calls and inverting the sense of the test,
;;; if necessary. If the call has two args, then we pass or return a
;;; negated test as appropriate. If it is a degenerate one-arg call,
;;; then we transform to code that returns true. Otherwise, we bind
;;; all the arguments and expand into a bunch of IFs.
(declaim (ftype (function (symbol list boolean) *) multi-compare))
(defun multi-compare (predicate args not-p)
  (let ((nargs (length args)))
    (cond ((< nargs 1) (values nil t))
	  ((= nargs 1) `(progn ,@args t))
	  ((= nargs 2)
	   (if not-p
	       `(if (,predicate ,(first args) ,(second args)) nil t)
	       (values nil t)))
	  (t
	   (do* ((i (1- nargs) (1- i))
		 (last nil current)
		 (current (gensym) (gensym))
		 (vars (list current) (cons current vars))
		 (result t (if not-p
			       `(if (,predicate ,current ,last)
				    nil ,result)
			       `(if (,predicate ,current ,last)
				    ,result nil))))
	       ((zerop i)
		`((lambda ,vars ,result) . ,args)))))))

(define-source-transform = (&rest args) (multi-compare '= args nil))
(define-source-transform < (&rest args) (multi-compare '< args nil))
(define-source-transform > (&rest args) (multi-compare '> args nil))
(define-source-transform <= (&rest args) (multi-compare '> args t))
(define-source-transform >= (&rest args) (multi-compare '< args t))

(define-source-transform char= (&rest args) (multi-compare 'char= args nil))
(define-source-transform char< (&rest args) (multi-compare 'char< args nil))
(define-source-transform char> (&rest args) (multi-compare 'char> args nil))
(define-source-transform char<= (&rest args) (multi-compare 'char> args t))
(define-source-transform char>= (&rest args) (multi-compare 'char< args t))

(define-source-transform char-equal (&rest args)
  (multi-compare 'char-equal args nil))
(define-source-transform char-lessp (&rest args)
  (multi-compare 'char-lessp args nil))
(define-source-transform char-greaterp (&rest args)
  (multi-compare 'char-greaterp args nil))
(define-source-transform char-not-greaterp (&rest args)
  (multi-compare 'char-greaterp args t))
(define-source-transform char-not-lessp (&rest args)
  (multi-compare 'char-lessp args t))

;;; This function does source transformation of N-arg inequality
;;; functions such as /=. This is similar to Multi-Compare in the <3
;;; arg cases. If there are more than two args, then we expand into
;;; the appropriate n^2 comparisons only when speed is important.
(declaim (ftype (function (symbol list) *) multi-not-equal))
(defun multi-not-equal (predicate args)
  (let ((nargs (length args)))
    (cond ((< nargs 1) (values nil t))
	  ((= nargs 1) `(progn ,@args t))
	  ((= nargs 2)
	   `(if (,predicate ,(first args) ,(second args)) nil t))
	  ((not (policy *lexenv*
			(and (>= speed space)
			     (>= speed compilation-speed))))
	   (values nil t))
	  (t
	   (let ((vars (make-gensym-list nargs)))
	     (do ((var vars next)
		  (next (cdr vars) (cdr next))
		  (result t))
		 ((null next)
		  `((lambda ,vars ,result) . ,args))
	       (let ((v1 (first var)))
		 (dolist (v2 next)
		   (setq result `(if (,predicate ,v1 ,v2) nil ,result))))))))))

(define-source-transform /= (&rest args) (multi-not-equal '= args))
(define-source-transform char/= (&rest args) (multi-not-equal 'char= args))
(define-source-transform char-not-equal (&rest args)
  (multi-not-equal 'char-equal args))

;;; Expand MAX and MIN into the obvious comparisons.
(define-source-transform max (arg &rest more-args)
  (if (null more-args)
      `(values ,arg)
      (once-only ((arg1 arg)
		  (arg2 `(max ,@more-args)))
	`(if (> ,arg1 ,arg2)
	     ,arg1 ,arg2))))
(define-source-transform min (arg &rest more-args)
  (if (null more-args)
      `(values ,arg)
      (once-only ((arg1 arg)
		  (arg2 `(min ,@more-args)))
	`(if (< ,arg1 ,arg2)
	     ,arg1 ,arg2))))

;;;; converting N-arg arithmetic functions
;;;;
;;;; N-arg arithmetic and logic functions are associated into two-arg
;;;; versions, and degenerate cases are flushed.

;;; Left-associate FIRST-ARG and MORE-ARGS using FUNCTION.
(declaim (ftype (function (symbol t list) list) associate-args))
(defun associate-args (function first-arg more-args)
  (let ((next (rest more-args))
	(arg (first more-args)))
    (if (null next)
	`(,function ,first-arg ,arg)
	(associate-args function `(,function ,first-arg ,arg) next))))

;;; Do source transformations for transitive functions such as +.
;;; One-arg cases are replaced with the arg and zero arg cases with
;;; the identity. If LEAF-FUN is true, then replace two-arg calls with
;;; a call to that function.
(defun source-transform-transitive (fun args identity &optional leaf-fun)
  (declare (symbol fun leaf-fun) (list args))
  (case (length args)
    (0 identity)
    (1 `(values ,(first args)))
    (2 (if leaf-fun
	   `(,leaf-fun ,(first args) ,(second args))
	   (values nil t)))
    (t
     (associate-args fun (first args) (rest args)))))

(define-source-transform + (&rest args)
  (source-transform-transitive '+ args 0))
(define-source-transform * (&rest args)
  (source-transform-transitive '* args 1))
(define-source-transform logior (&rest args)
  (source-transform-transitive 'logior args 0))
(define-source-transform logxor (&rest args)
  (source-transform-transitive 'logxor args 0))
(define-source-transform logand (&rest args)
  (source-transform-transitive 'logand args -1))

(define-source-transform logeqv (&rest args)
  (if (evenp (length args))
      `(lognot (logxor ,@args))
      `(logxor ,@args)))

;;; Note: we can't use SOURCE-TRANSFORM-TRANSITIVE for GCD and LCM
;;; because when they are given one argument, they return its absolute
;;; value.

(define-source-transform gcd (&rest args)
  (case (length args)
    (0 0)
    (1 `(abs (the integer ,(first args))))
    (2 (values nil t))
    (t (associate-args 'gcd (first args) (rest args)))))

(define-source-transform lcm (&rest args)
  (case (length args)
    (0 1)
    (1 `(abs (the integer ,(first args))))
    (2 (values nil t))
    (t (associate-args 'lcm (first args) (rest args)))))

;;; Do source transformations for intransitive n-arg functions such as
;;; /. With one arg, we form the inverse. With two args we pass.
;;; Otherwise we associate into two-arg calls.
(declaim (ftype (function (symbol list t) list) source-transform-intransitive))
(defun source-transform-intransitive (function args inverse)
  (case (length args)
    ((0 2) (values nil t))
    (1 `(,@inverse ,(first args)))
    (t (associate-args function (first args) (rest args)))))

(define-source-transform - (&rest args)
  (source-transform-intransitive '- args '(%negate)))
(define-source-transform / (&rest args)
  (source-transform-intransitive '/ args '(/ 1)))

;;;; transforming APPLY

;;; We convert APPLY into MULTIPLE-VALUE-CALL so that the compiler
;;; only needs to understand one kind of variable-argument call. It is
;;; more efficient to convert APPLY to MV-CALL than MV-CALL to APPLY.
(define-source-transform apply (fun arg &rest more-args)
  (let ((args (cons arg more-args)))
    `(multiple-value-call ,fun
       ,@(mapcar (lambda (x)
		   `(values ,x))
		 (butlast args))
       (values-list ,(car (last args))))))

;;;; transforming FORMAT
;;;;
;;;; If the control string is a compile-time constant, then replace it
;;;; with a use of the FORMATTER macro so that the control string is
;;;; ``compiled.'' Furthermore, if the destination is either a stream
;;;; or T and the control string is a function (i.e. FORMATTER), then
;;;; convert the call to FORMAT to just a FUNCALL of that function.

(deftransform format ((dest control &rest args) (t simple-string &rest t) *
		      :policy (> speed space))
  (unless (constant-continuation-p control)
    (give-up-ir1-transform "The control string is not a constant."))
  (let ((arg-names (make-gensym-list (length args))))
    `(lambda (dest control ,@arg-names)
       (declare (ignore control))
       (format dest (formatter ,(continuation-value control)) ,@arg-names))))

(deftransform format ((stream control &rest args) (stream function &rest t) *
		      :policy (> speed space))
  (let ((arg-names (make-gensym-list (length args))))
    `(lambda (stream control ,@arg-names)
       (funcall control stream ,@arg-names)
       nil)))

(deftransform format ((tee control &rest args) ((member t) function &rest t) *
		      :policy (> speed space))
  (let ((arg-names (make-gensym-list (length args))))
    `(lambda (tee control ,@arg-names)
       (declare (ignore tee))
       (funcall control *standard-output* ,@arg-names)
       nil)))

(defoptimizer (coerce derive-type) ((value type))
  (let ((value-type (continuation-type value))
        (type-type (continuation-type type)))
    (labels
        ((good-cons-type-p (cons-type)
           ;; Make sure the cons-type we're looking at is something
           ;; we're prepared to handle which is basically something
           ;; that array-element-type can return.
           (or (and (member-type-p cons-type)
                    (null (rest (member-type-members cons-type)))
                    (null (first (member-type-members cons-type))))
               (let ((car-type (cons-type-car-type cons-type)))
                 (and (member-type-p car-type)
                      (null (rest (member-type-members car-type)))
                      (or (symbolp (first (member-type-members car-type)))
                          (numberp (first (member-type-members car-type)))
                          (and (listp (first (member-type-members car-type)))
                               (numberp (first (first (member-type-members
                                                       car-type))))))
                      (good-cons-type-p (cons-type-cdr-type cons-type))))))
         (unconsify-type (good-cons-type)
           ;; Convert the "printed" respresentation of a cons
           ;; specifier into a type specifier.  That is, the specifier
           ;; (cons (eql signed-byte) (cons (eql 16) null)) is
           ;; converted to (signed-byte 16).
           (cond ((or (null good-cons-type)
                      (eq good-cons-type 'null))
                   nil)
                 ((and (eq (first good-cons-type) 'cons)
                       (eq (first (second good-cons-type)) 'member))
                   `(,(second (second good-cons-type))
                     ,@(unconsify-type (caddr good-cons-type))))))
         (coerceable-p (c-type)
           ;; Can the value be coerced to the given type?  Coerce is
           ;; complicated, so we don't handle every possible case
           ;; here---just the most common and easiest cases:
           ;;
           ;; o Any real can be coerced to a float type.
           ;; o Any number can be coerced to a complex single/double-float.
           ;; o An integer can be coerced to an integer.
           (let ((coerced-type c-type))
             (or (and (subtypep coerced-type 'float)
                      (csubtypep value-type (specifier-type 'real)))
                 (and (subtypep coerced-type
                                '(or (complex single-float)
                                  (complex double-float)))
                      (csubtypep value-type (specifier-type 'number)))
                 (and (subtypep coerced-type 'integer)
                      (csubtypep value-type (specifier-type 'integer))))))
         (process-types (type)
           ;; FIXME:
           ;; This needs some work because we should be able to derive
           ;; the resulting type better than just the type arg of
           ;; coerce.  That is, if x is (integer 10 20), the (coerce x
           ;; 'double-float) should say (double-float 10d0 20d0)
           ;; instead of just double-float.
           (cond ((member-type-p type)
                   (let ((members (member-type-members type)))
                     (if (every #'coerceable-p members)
                       (specifier-type `(or ,@members))
                       *universal-type*)))
                 ((and (cons-type-p type)
                       (good-cons-type-p type))
                   (let ((c-type (unconsify-type (type-specifier type))))
                     (if (coerceable-p c-type)
                       (specifier-type c-type)
                       *universal-type*)))
                 (t
                   *universal-type*))))
      (cond ((union-type-p type-type)
              (apply #'type-union (mapcar #'process-types
                                          (union-type-types type-type))))
            ((or (member-type-p type-type)
                 (cons-type-p type-type))
              (process-types type-type))
            (t
              *universal-type*)))))

(defoptimizer (array-element-type derive-type) ((array))
  (let* ((array-type (continuation-type array)))
    (labels ((consify (list)
              (if (endp list)
                  '(eql nil)
                  `(cons (eql ,(car list)) ,(consify (rest list)))))
            (get-element-type (a)
              (let ((element-type
		     (type-specifier (array-type-specialized-element-type a))))
                (cond ((eq element-type '*)
                       (specifier-type 'type-specifier))
		      ((symbolp element-type)
                       (make-member-type :members (list element-type)))
                      ((consp element-type)
                       (specifier-type (consify element-type)))
                      (t
                       (error "can't understand type ~S~%" element-type))))))
      (cond ((array-type-p array-type)
	     (get-element-type array-type))
	    ((union-type-p array-type)             
             (apply #'type-union
                    (mapcar #'get-element-type (union-type-types array-type))))
	    (t
	     *universal-type*)))))

;;;; debuggers' little helpers

;;; for debugging when transforms are behaving mysteriously,
;;; e.g. when debugging a problem with an ASH transform
;;;   (defun foo (&optional s)
;;;     (sb-c::/report-continuation s "S outside WHEN")
;;;     (when (and (integerp s) (> s 3))
;;;       (sb-c::/report-continuation s "S inside WHEN")
;;;       (let ((bound (ash 1 (1- s))))
;;;         (sb-c::/report-continuation bound "BOUND")
;;;         (let ((x (- bound))
;;;     	  (y (1- bound)))
;;;   	      (sb-c::/report-continuation x "X")
;;;           (sb-c::/report-continuation x "Y"))
;;;         `(integer ,(- bound) ,(1- bound)))))
;;; (The DEFTRANSFORM doesn't do anything but report at compile time,
;;; and the function doesn't do anything at all.)
#!+sb-show
(progn
  (defknown /report-continuation (t t) null)
  (deftransform /report-continuation ((x message) (t t))
    (format t "~%/in /REPORT-CONTINUATION~%")
    (format t "/(CONTINUATION-TYPE X)=~S~%" (continuation-type x))
    (when (constant-continuation-p x)
      (format t "/(CONTINUATION-VALUE X)=~S~%" (continuation-value x)))
    (format t "/MESSAGE=~S~%" (continuation-value message))
    (give-up-ir1-transform "not a real transform"))
  (defun /report-continuation (&rest rest)
    (declare (ignore rest))))