;;;; This file contains floating-point-specific transforms, and may be
;;;; somewhat implementation-dependent in its assumptions of what the
;;;; formats are.

;;;; 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")

;;;; coercions

(defknown %single-float (real) single-float
    (movable foldable))
(defknown %double-float (real) double-float
    (movable foldable))

(deftransform float ((n f) (* single-float) *)
  '(%single-float n))

(deftransform float ((n f) (* double-float) *)
  '(%double-float n))

(deftransform float ((n) *)
  '(if (floatp n)
       n
       (%single-float n)))

(deftransform %single-float ((n) (single-float) *)
  'n)

(deftransform %double-float ((n) (double-float) *)
  'n)

;;; RANDOM
(macrolet ((frob (fun type)
             `(deftransform random ((num &optional state)
                                    (,type &optional *) *)
                "Use inline float operations."
                '(,fun num (or state *random-state*)))))
  (frob %random-single-float single-float)
  (frob %random-double-float double-float))

;;; Return an expression to generate an integer of N-BITS many random
;;; bits, using the minimal number of random chunks possible.
(defun generate-random-expr-for-power-of-2 (n-bits state)
  (declare (type (integer 1 #.sb!vm:n-word-bits) n-bits))
  (multiple-value-bind (n-chunk-bits chunk-expr)
      (cond ((<= n-bits n-random-chunk-bits)
             (values n-random-chunk-bits `(random-chunk ,state)))
            ((<= n-bits (* 2 n-random-chunk-bits))
             (values (* 2 n-random-chunk-bits) `(big-random-chunk ,state)))
            (t
             (error "Unexpectedly small N-RANDOM-CHUNK-BITS")))
    (if (< n-bits n-chunk-bits)
        `(logand ,(1- (ash 1 n-bits)) ,chunk-expr)
        chunk-expr)))

;;; This transform for compile-time constant word-sized integers
;;; generates an accept-reject loop to achieve equidistribution of the
;;; returned values. Several optimizations are done: If NUM is a power
;;; of two no loop is needed. If the random chunk size is half the word
;;; size only one chunk is used where sufficient. For values of NUM
;;; where it is possible and results in faster code, the rejection
;;; probability is reduced by accepting all values below the largest
;;; multiple of the limit that fits into one or two chunks and and doing
;;; a division to get the random value into the desired range.
(deftransform random ((num &optional state)
                      ((constant-arg (integer 1 #.(expt 2 sb!vm:n-word-bits)))
                       &optional *)
                      *
                      :policy (and (> speed compilation-speed)
                                   (> speed space)))
  "optimize to inlined RANDOM-CHUNK operations"
  (let ((num (lvar-value num)))
    (if (= num 1)
        0
        (flet ((chunk-n-bits-and-expr (n-bits)
                 (cond ((<= n-bits n-random-chunk-bits)
                        (values n-random-chunk-bits
                                '(random-chunk (or state *random-state*))))
                       ((<= n-bits (* 2 n-random-chunk-bits))
                        (values (* 2 n-random-chunk-bits)
                                '(big-random-chunk (or state *random-state*))))
                       (t
                        (error "Unexpectedly small N-RANDOM-CHUNK-BITS")))))
          (if (zerop (logand num (1- num)))
              ;; NUM is a power of 2.
              (let ((n-bits (integer-length (1- num))))
                (multiple-value-bind (n-chunk-bits chunk-expr)
                    (chunk-n-bits-and-expr n-bits)
                  (if (< n-bits n-chunk-bits)
                      `(logand ,(1- (ash 1 n-bits)) ,chunk-expr)
                      chunk-expr)))
              ;; Generate an accept-reject loop.
              (let ((n-bits (integer-length num)))
                (multiple-value-bind (n-chunk-bits chunk-expr)
                    (chunk-n-bits-and-expr n-bits)
                  (if (or (> (* num 3) (expt 2 n-chunk-bits))
                          (logbitp (- n-bits 2) num))
                      ;; Division can't help as the quotient is below 3,
                      ;; or is too costly as the rejection probability
                      ;; without it is already small (namely at most 1/4
                      ;; with the given test, which is experimentally a
                      ;; reasonable threshold and cheap to test for).
                      `(loop
                         (let ((bits ,(generate-random-expr-for-power-of-2
                                       n-bits '(or state *random-state*))))
                           (when (< bits num)
                             (return bits))))
                      (let ((d (truncate (expt 2 n-chunk-bits) num)))
                        `(loop
                           (let ((bits ,chunk-expr))
                             (when (< bits ,(* num d))
                               (return (values (truncate bits ,d)))))))))))))))


;;;; float accessors

(defknown make-single-float ((signed-byte 32)) single-float
  (movable flushable))

(defknown make-double-float ((signed-byte 32) (unsigned-byte 32)) double-float
  (movable flushable))

#-sb-xc-host
(deftransform make-single-float ((bits)
                                 ((signed-byte 32)))
  "Conditional constant folding"
  (unless (constant-lvar-p bits)
    (give-up-ir1-transform))
  (let* ((bits  (lvar-value bits))
         (float (make-single-float bits)))
    (when (float-nan-p float)
      (give-up-ir1-transform))
    float))

#-sb-xc-host
(deftransform make-double-float ((hi lo)
                                 ((signed-byte 32) (unsigned-byte 32)))
  "Conditional constant folding"
  (unless (and (constant-lvar-p hi)
               (constant-lvar-p lo))
    (give-up-ir1-transform))
  (let* ((hi    (lvar-value hi))
         (lo    (lvar-value lo))
         (float (make-double-float hi lo)))
    (when (float-nan-p float)
      (give-up-ir1-transform))
    float))

(defknown single-float-bits (single-float) (signed-byte 32)
  (movable foldable flushable))

(defknown double-float-high-bits (double-float) (signed-byte 32)
  (movable foldable flushable))

(defknown double-float-low-bits (double-float) (unsigned-byte 32)
  (movable foldable flushable))

(deftransform float-sign ((float &optional float2)
                          (single-float &optional single-float) *)
  (if float2
      (let ((temp (gensym)))
        `(let ((,temp (abs float2)))
          (if (minusp (single-float-bits float)) (- ,temp) ,temp)))
      '(if (minusp (single-float-bits float)) -1f0 1f0)))

(deftransform float-sign ((float &optional float2)
                          (double-float &optional double-float) *)
  (if float2
      (let ((temp (gensym)))
        `(let ((,temp (abs float2)))
          (if (minusp (double-float-high-bits float)) (- ,temp) ,temp)))
      '(if (minusp (double-float-high-bits float)) -1d0 1d0)))

;;;; DECODE-FLOAT, INTEGER-DECODE-FLOAT, and SCALE-FLOAT

(defknown decode-single-float (single-float)
  (values single-float single-float-exponent (single-float -1f0 1f0))
  (movable foldable flushable))

(defknown decode-double-float (double-float)
  (values double-float double-float-exponent (double-float -1d0 1d0))
  (movable foldable flushable))

(defknown integer-decode-single-float (single-float)
  (values single-float-significand single-float-int-exponent (integer -1 1))
  (movable foldable flushable))

(defknown integer-decode-double-float (double-float)
  (values double-float-significand double-float-int-exponent (integer -1 1))
  (movable foldable flushable))

(defknown scale-single-float (single-float integer) single-float
  (movable foldable flushable))

(defknown scale-double-float (double-float integer) double-float
  (movable foldable flushable))

(deftransform decode-float ((x) (single-float) *)
  '(decode-single-float x))

(deftransform decode-float ((x) (double-float) *)
  '(decode-double-float x))

(deftransform integer-decode-float ((x) (single-float) *)
  '(integer-decode-single-float x))

(deftransform integer-decode-float ((x) (double-float) *)
  '(integer-decode-double-float x))

(deftransform scale-float ((f ex) (single-float *) *)
  (if (and #!+x86 t #!-x86 nil
           (csubtypep (lvar-type ex)
                      (specifier-type '(signed-byte 32))))
      '(coerce (%scalbn (coerce f 'double-float) ex) 'single-float)
      '(scale-single-float f ex)))

(deftransform scale-float ((f ex) (double-float *) *)
  (if (and #!+x86 t #!-x86 nil
           (csubtypep (lvar-type ex)
                      (specifier-type '(signed-byte 32))))
      '(%scalbn f ex)
      '(scale-double-float f ex)))

;;; What is the CROSS-FLOAT-INFINITY-KLUDGE?
;;;
;;; SBCL's own implementation of floating point supports floating
;;; point infinities. Some of the old CMU CL :PROPAGATE-FLOAT-TYPE and
;;; :PROPAGATE-FUN-TYPE code, like the DEFOPTIMIZERs below, uses this
;;; floating point support. Thus, we have to avoid running it on the
;;; cross-compilation host, since we're not guaranteed that the
;;; cross-compilation host will support floating point infinities.
;;;
;;; If we wanted to live dangerously, we could conditionalize the code
;;; with #+(OR SBCL SB-XC) instead. That way, if the cross-compilation
;;; host happened to be SBCL, we'd be able to run the infinity-using
;;; code. Pro:
;;;   * SBCL itself gets built with more complete optimization.
;;; Con:
;;;   * You get a different SBCL depending on what your cross-compilation
;;;     host is.
;;; So far the pros and cons seem seem to be mostly academic, since
;;; AFAIK (WHN 2001-08-28) the propagate-foo-type optimizations aren't
;;; actually important in compiling SBCL itself. If this changes, then
;;; we have to decide:
;;;   * Go for simplicity, leaving things as they are.
;;;   * Go for performance at the expense of conceptual clarity,
;;;     using #+(OR SBCL SB-XC) and otherwise leaving the build
;;;     process as is.
;;;   * Go for performance at the expense of build time, using
;;;     #+(OR SBCL SB-XC) and also making SBCL do not just
;;;     make-host-1.sh and make-host-2.sh, but a third step
;;;     make-host-3.sh where it builds itself under itself. (Such a
;;;     3-step build process could also help with other things, e.g.
;;;     using specialized arrays to represent debug information.)
;;;   * Rewrite the code so that it doesn't depend on unportable
;;;     floating point infinities.

;;; optimizers for SCALE-FLOAT. If the float has bounds, new bounds
;;; are computed for the result, if possible.
#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(progn

(defun scale-float-derive-type-aux (f ex same-arg)
  (declare (ignore same-arg))
  (flet ((scale-bound (x n)
           ;; We need to be a bit careful here and catch any overflows
           ;; that might occur. We can ignore underflows which become
           ;; zeros.
           (set-bound
            (handler-case
             (scale-float (type-bound-number x) n)
             (floating-point-overflow ()
                nil))
            (consp x))))
    (when (and (numeric-type-p f) (numeric-type-p ex))
      (let ((f-lo (numeric-type-low f))
            (f-hi (numeric-type-high f))
            (ex-lo (numeric-type-low ex))
            (ex-hi (numeric-type-high ex))
            (new-lo nil)
            (new-hi nil))
        (when f-hi
          (if (< (float-sign (type-bound-number f-hi)) 0.0)
              (when ex-lo
                (setf new-hi (scale-bound f-hi ex-lo)))
              (when ex-hi
                (setf new-hi (scale-bound f-hi ex-hi)))))
        (when f-lo
          (if (< (float-sign (type-bound-number f-lo)) 0.0)
              (when ex-hi
                (setf new-lo (scale-bound f-lo ex-hi)))
              (when ex-lo
                (setf new-lo (scale-bound f-lo ex-lo)))))
        (make-numeric-type :class (numeric-type-class f)
                           :format (numeric-type-format f)
                           :complexp :real
                           :low new-lo
                           :high new-hi)))))
(defoptimizer (scale-single-float derive-type) ((f ex))
  (two-arg-derive-type f ex #'scale-float-derive-type-aux
                       #'scale-single-float t))
(defoptimizer (scale-double-float derive-type) ((f ex))
  (two-arg-derive-type f ex #'scale-float-derive-type-aux
                       #'scale-double-float t))

;;; DEFOPTIMIZERs for %SINGLE-FLOAT and %DOUBLE-FLOAT. This makes the
;;; FLOAT function return the correct ranges if the input has some
;;; defined range. Quite useful if we want to convert some type of
;;; bounded integer into a float.
(macrolet
    ((frob (fun type most-negative most-positive)
       (let ((aux-name (symbolicate fun "-DERIVE-TYPE-AUX")))
         `(progn
            (defun ,aux-name (num)
              ;; When converting a number to a float, the limits are
              ;; the same.
              (let* ((lo (bound-func (lambda (x)
                                       (if (< x ,most-negative)
                                           ,most-negative
                                           (coerce x ',type)))
                                     (numeric-type-low num)
                                     nil))
                     (hi (bound-func (lambda (x)
                                       (if (< ,most-positive x )
                                           ,most-positive
                                           (coerce x ',type)))
                                     (numeric-type-high num)
                                     nil)))
                (specifier-type `(,',type ,(or lo '*) ,(or hi '*)))))

            (defoptimizer (,fun derive-type) ((num))
              (handler-case
                  (one-arg-derive-type num #',aux-name #',fun)
                (type-error ()
                  nil)))))))
  (frob %single-float single-float
        most-negative-single-float most-positive-single-float)
  (frob %double-float double-float
        most-negative-double-float most-positive-double-float))
) ; PROGN

;;;; float contagion

(defun safe-ctype-for-single-coercion-p (x)
  ;; See comment in SAFE-SINGLE-COERCION-P -- this deals with the same
  ;; problem, but in the context of evaluated and compiled (+ <int> <single>)
  ;; giving different result if we fail to check for this.
  (or (not (csubtypep x (specifier-type 'integer)))
      #!+x86
      (csubtypep x (specifier-type `(integer ,most-negative-exactly-single-float-fixnum
                                             ,most-positive-exactly-single-float-fixnum)))
      #!-x86
      (csubtypep x (specifier-type 'fixnum))))

;;; Do some stuff to recognize when the loser is doing mixed float and
;;; rational arithmetic, or different float types, and fix it up. If
;;; we don't, he won't even get so much as an efficiency note.
(deftransform float-contagion-arg1 ((x y) * * :defun-only t :node node)
  (if (or (not (types-equal-or-intersect (lvar-type y) (specifier-type 'single-float)))
          (safe-ctype-for-single-coercion-p (lvar-type x)))
      `(,(lvar-fun-name (basic-combination-fun node))
         (float x y) y)
      (give-up-ir1-transform)))
(deftransform float-contagion-arg2 ((x y) * * :defun-only t :node node)
  (if (or (not (types-equal-or-intersect (lvar-type x) (specifier-type 'single-float)))
          (safe-ctype-for-single-coercion-p (lvar-type y)))
      `(,(lvar-fun-name (basic-combination-fun node))
         x (float y x))
      (give-up-ir1-transform)))

(dolist (x '(+ * / -))
  (%deftransform x '(function (rational float) *) #'float-contagion-arg1)
  (%deftransform x '(function (float rational) *) #'float-contagion-arg2))

(dolist (x '(= < > + * / -))
  (%deftransform x '(function (single-float double-float) *)
                 #'float-contagion-arg1)
  (%deftransform x '(function (double-float single-float) *)
                 #'float-contagion-arg2))

(macrolet ((def (type &rest args)
             `(deftransform * ((x y) (,type (constant-arg (member ,@args))) *
                               ;; Beware the SNaN!
                               :policy (zerop float-accuracy))
                "optimize multiplication by one"
                (let ((y (lvar-value y)))
                  (if (minusp y)
                      '(%negate x)
                      'x)))))
  (def single-float 1.0 -1.0)
  (def double-float 1.0d0 -1.0d0))

;;; Return the reciprocal of X if it can be represented exactly, NIL otherwise.
(defun maybe-exact-reciprocal (x)
  (unless (zerop x)
    (handler-case
        (multiple-value-bind (significand exponent sign)
            (integer-decode-float x)
          ;; only powers of 2 can be inverted exactly
          (unless (zerop (logand significand (1- significand)))
            (return-from maybe-exact-reciprocal nil))
          (let ((expected   (/ sign significand (expt 2 exponent)))
                (reciprocal (/ x)))
            (multiple-value-bind (significand exponent sign)
                (integer-decode-float reciprocal)
              ;; Denorms can't be inverted safely.
              (and (eql expected (* sign significand (expt 2 exponent)))
                   reciprocal))))
      (error () (return-from maybe-exact-reciprocal nil)))))

;;; Replace constant division by multiplication with exact reciprocal,
;;; if one exists.
(macrolet ((def (type)
             `(deftransform / ((x y) (,type (constant-arg ,type)) *
                               :node node)
                "convert to multiplication by reciprocal"
                (let ((n (lvar-value y)))
                  (if (policy node (zerop float-accuracy))
                      `(* x ,(/ n))
                      (let ((r (maybe-exact-reciprocal n)))
                        (if r
                            `(* x ,r)
                            (give-up-ir1-transform
                             "~S does not have an exact reciprocal"
                             n))))))))
  (def single-float)
  (def double-float))

;;; Optimize addition and subtraction of zero
(macrolet ((def (op type &rest args)
             `(deftransform ,op ((x y) (,type (constant-arg (member ,@args))) *
                                 ;; Beware the SNaN!
                                 :policy (zerop float-accuracy))
                'x)))
  ;; No signed zeros, thanks.
  (def + single-float 0 0.0)
  (def - single-float 0 0.0)
  (def + double-float 0 0.0 0.0d0)
  (def - double-float 0 0.0 0.0d0))

;;; On most platforms (+ x x) is faster than (* x 2)
(macrolet ((def (type &rest args)
             `(deftransform * ((x y) (,type (constant-arg (member ,@args))))
                '(+ x x))))
  (def single-float 2 2.0)
  (def double-float 2 2.0 2.0d0))

;;; Prevent ZEROP, PLUSP, and MINUSP from losing horribly. We can't in
;;; general float rational args to comparison, since Common Lisp
;;; semantics says we are supposed to compare as rationals, but we can
;;; do it for any rational that has a precise representation as a
;;; float (such as 0).
(macrolet ((frob (op)
             `(deftransform ,op ((x y) (float rational) *)
                "open-code FLOAT to RATIONAL comparison"
                (unless (constant-lvar-p y)
                  (give-up-ir1-transform
                   "The RATIONAL value isn't known at compile time."))
                (let ((val (lvar-value y)))
                  (unless (eql (rational (float val)) val)
                    (give-up-ir1-transform
                     "~S doesn't have a precise float representation."
                     val)))
                `(,',op x (float y x)))))
  (frob <)
  (frob >)
  (frob =))

;;;; irrational derive-type methods

;;; Derive the result to be float for argument types in the
;;; appropriate domain.
#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(dolist (stuff '((asin (real -1.0 1.0))
                 (acos (real -1.0 1.0))
                 (acosh (real 1.0))
                 (atanh (real -1.0 1.0))
                 (sqrt (real 0.0))))
  (destructuring-bind (name type) stuff
    (let ((type (specifier-type type)))
      (setf (fun-info-derive-type (fun-info-or-lose name))
            (lambda (call)
              (declare (type combination call))
              (when (csubtypep (lvar-type
                                (first (combination-args call)))
                               type)
                (specifier-type 'float)))))))

#+sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (log derive-type) ((x &optional y))
  (when (and (csubtypep (lvar-type x)
                        (specifier-type '(real 0.0)))
             (or (null y)
                 (csubtypep (lvar-type y)
                            (specifier-type '(real 0.0)))))
    (specifier-type 'float)))

;;;; irrational transforms

(defknown (%tan %sinh %asinh %atanh %log %logb %log10 %tan-quick)
          (double-float) double-float
  (movable foldable flushable))

(defknown (%sin %cos %tanh %sin-quick %cos-quick)
  (double-float) (double-float -1.0d0 1.0d0)
  (movable foldable flushable))

(defknown (%asin %atan)
  (double-float)
  (double-float #.(coerce (- (/ pi 2)) 'double-float)
                #.(coerce (/ pi 2) 'double-float))
  (movable foldable flushable))

(defknown (%acos)
  (double-float) (double-float 0.0d0 #.(coerce pi 'double-float))
  (movable foldable flushable))

(defknown (%cosh)
  (double-float) (double-float 1.0d0)
  (movable foldable flushable))

(defknown (%acosh %exp %sqrt)
  (double-float) (double-float 0.0d0)
  (movable foldable flushable))

(defknown %expm1
  (double-float) (double-float -1d0)
  (movable foldable flushable))

(defknown (%hypot)
  (double-float double-float) (double-float 0d0)
  (movable foldable flushable))

(defknown (%pow)
  (double-float double-float) double-float
  (movable foldable flushable))

(defknown (%atan2)
  (double-float double-float)
  (double-float #.(coerce (- pi) 'double-float)
                #.(coerce pi 'double-float))
  (movable foldable flushable))

(defknown (%scalb)
  (double-float double-float) double-float
  (movable foldable flushable))

(defknown (%scalbn)
  (double-float (signed-byte 32)) double-float
  (movable foldable flushable))

(defknown (%log1p)
  (double-float) double-float
  (movable foldable flushable))

(macrolet ((def (name prim rtype)
             `(progn
               (deftransform ,name ((x) (single-float) ,rtype)
                 `(coerce (,',prim (coerce x 'double-float)) 'single-float))
               (deftransform ,name ((x) (double-float) ,rtype)
                 `(,',prim x)))))
  (def exp %exp *)
  (def log %log float)
  (def sqrt %sqrt float)
  (def asin %asin float)
  (def acos %acos float)
  (def atan %atan *)
  (def sinh %sinh *)
  (def cosh %cosh *)
  (def tanh %tanh *)
  (def asinh %asinh *)
  (def acosh %acosh float)
  (def atanh %atanh float))

;;; The argument range is limited on the x86 FP trig. functions. A
;;; post-test can detect a failure (and load a suitable result), but
;;; this test is avoided if possible.
(macrolet ((def (name prim prim-quick)
             (declare (ignorable prim-quick))
             `(progn
                (deftransform ,name ((x) (single-float) *)
                  #!+x86 (cond ((csubtypep (lvar-type x)
                                           (specifier-type '(single-float
                                                             (#.(- (expt 2f0 63)))
                                                             (#.(expt 2f0 63)))))
                                `(coerce (,',prim-quick (coerce x 'double-float))
                                  'single-float))
                               (t
                                (compiler-notify
                                 "unable to avoid inline argument range check~@
                                  because the argument range (~S) was not within 2^63"
                                 (type-specifier (lvar-type x)))
                                `(coerce (,',prim (coerce x 'double-float)) 'single-float)))
                  #!-x86 `(coerce (,',prim (coerce x 'double-float)) 'single-float))
               (deftransform ,name ((x) (double-float) *)
                 #!+x86 (cond ((csubtypep (lvar-type x)
                                          (specifier-type '(double-float
                                                            (#.(- (expt 2d0 63)))
                                                            (#.(expt 2d0 63)))))
                               `(,',prim-quick x))
                              (t
                               (compiler-notify
                                "unable to avoid inline argument range check~@
                                 because the argument range (~S) was not within 2^63"
                                (type-specifier (lvar-type x)))
                               `(,',prim x)))
                 #!-x86 `(,',prim x)))))
  (def sin %sin %sin-quick)
  (def cos %cos %cos-quick)
  (def tan %tan %tan-quick))

(deftransform atan ((x y) (single-float single-float) *)
  `(coerce (%atan2 (coerce x 'double-float) (coerce y 'double-float))
    'single-float))
(deftransform atan ((x y) (double-float double-float) *)
  `(%atan2 x y))

(deftransform expt ((x y) ((single-float 0f0) single-float) *)
  `(coerce (%pow (coerce x 'double-float) (coerce y 'double-float))
    'single-float))
(deftransform expt ((x y) ((double-float 0d0) double-float) *)
  `(%pow x y))
(deftransform expt ((x y) ((single-float 0f0) (signed-byte 32)) *)
  `(coerce (%pow (coerce x 'double-float) (coerce y 'double-float))
    'single-float))
(deftransform expt ((x y) ((double-float 0d0) (signed-byte 32)) *)
  `(%pow x (coerce y 'double-float)))

;;; ANSI says log with base zero returns zero.
(deftransform log ((x y) (float float) float)
  '(if (zerop y) y (/ (log x) (log y))))

;;; Handle some simple transformations.

(deftransform abs ((x) ((complex double-float)) double-float)
  '(%hypot (realpart x) (imagpart x)))

(deftransform abs ((x) ((complex single-float)) single-float)
  '(coerce (%hypot (coerce (realpart x) 'double-float)
                   (coerce (imagpart x) 'double-float))
          'single-float))

(deftransform phase ((x) ((complex double-float)) double-float)
  '(%atan2 (imagpart x) (realpart x)))

(deftransform phase ((x) ((complex single-float)) single-float)
  '(coerce (%atan2 (coerce (imagpart x) 'double-float)
                   (coerce (realpart x) 'double-float))
          'single-float))

(deftransform phase ((x) ((float)) float)
  '(if (minusp (float-sign x))
       (float pi x)
       (float 0 x)))

;;; The number is of type REAL.
(defun numeric-type-real-p (type)
  (and (numeric-type-p type)
       (eq (numeric-type-complexp type) :real)))

;;; Coerce a numeric type bound to the given type while handling
;;; exclusive bounds.
(defun coerce-numeric-bound (bound type)
  (when bound
    (if (consp bound)
        (list (coerce (car bound) type))
        (coerce bound type))))

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

;;;; optimizers for elementary functions
;;;;
;;;; These optimizers compute the output range of the elementary
;;;; function, based on the domain of the input.

;;; Generate a specifier for a complex type specialized to the same
;;; type as the argument.
(defun complex-float-type (arg)
  (declare (type numeric-type arg))
  (let* ((format (case (numeric-type-class arg)
                   ((integer rational) 'single-float)
                   (t (numeric-type-format arg))))
         (float-type (or format 'float)))
    (specifier-type `(complex ,float-type))))

;;; Compute a specifier like '(OR FLOAT (COMPLEX FLOAT)), except float
;;; should be the right kind of float. Allow bounds for the float
;;; part too.
(defun float-or-complex-float-type (arg &optional lo hi)
  (declare (type numeric-type arg))
  (let* ((format (case (numeric-type-class arg)
                   ((integer rational) 'single-float)
                   (t (numeric-type-format arg))))
         (float-type (or format 'float))
         (lo (coerce-numeric-bound lo float-type))
         (hi (coerce-numeric-bound hi float-type)))
    (specifier-type `(or (,float-type ,(or lo '*) ,(or hi '*))
                         (complex ,float-type)))))

) ; PROGN

(eval-when (:compile-toplevel :execute)
  ;; So the problem with this hack is that it's actually broken.  If
  ;; the host does not have long floats, then setting *R-D-F-F* to
  ;; LONG-FLOAT doesn't actually buy us anything.  FIXME.
  (setf *read-default-float-format*
        #!+long-float 'long-float #!-long-float 'double-float))
;;; Test whether the numeric-type ARG is within the domain specified by
;;; DOMAIN-LOW and DOMAIN-HIGH, consider negative and positive zero to
;;; be distinct.
#-sb-xc-host  ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defun domain-subtypep (arg domain-low domain-high)
  (declare (type numeric-type arg)
           (type (or real null) domain-low domain-high))
  (let* ((arg-lo (numeric-type-low arg))
         (arg-lo-val (type-bound-number arg-lo))
         (arg-hi (numeric-type-high arg))
         (arg-hi-val (type-bound-number arg-hi)))
    ;; Check that the ARG bounds are correctly canonicalized.
    (when (and arg-lo (floatp arg-lo-val) (zerop arg-lo-val) (consp arg-lo)
               (minusp (float-sign arg-lo-val)))
      (compiler-notify "float zero bound ~S not correctly canonicalized?" arg-lo)
      (setq arg-lo 0e0 arg-lo-val arg-lo))
    (when (and arg-hi (zerop arg-hi-val) (floatp arg-hi-val) (consp arg-hi)
               (plusp (float-sign arg-hi-val)))
      (compiler-notify "float zero bound ~S not correctly canonicalized?" arg-hi)
      (setq arg-hi (ecase *read-default-float-format*
                     (double-float (load-time-value (make-unportable-float :double-float-negative-zero)))
                     #!+long-float
                     (long-float (load-time-value (make-unportable-float :long-float-negative-zero))))
            arg-hi-val arg-hi))
    (flet ((fp-neg-zero-p (f)           ; Is F -0.0?
             (and (floatp f) (zerop f) (minusp (float-sign f))))
           (fp-pos-zero-p (f)           ; Is F +0.0?
             (and (floatp f) (zerop f) (plusp (float-sign f)))))
      (and (or (null domain-low)
               (and arg-lo (>= arg-lo-val domain-low)
                    (not (and (fp-pos-zero-p domain-low)
                              (fp-neg-zero-p arg-lo)))))
           (or (null domain-high)
               (and arg-hi (<= arg-hi-val domain-high)
                    (not (and (fp-neg-zero-p domain-high)
                              (fp-pos-zero-p arg-hi)))))))))
(eval-when (:compile-toplevel :execute)
  (setf *read-default-float-format* 'single-float))

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

;;; Handle monotonic functions of a single variable whose domain is
;;; possibly part of the real line. ARG is the variable, FUN is the
;;; function, and DOMAIN is a specifier that gives the (real) domain
;;; of the function. If ARG is a subset of the DOMAIN, we compute the
;;; bounds directly. Otherwise, we compute the bounds for the
;;; intersection between ARG and DOMAIN, and then append a complex
;;; result, which occurs for the parts of ARG not in the DOMAIN.
;;;
;;; Negative and positive zero are considered distinct within
;;; DOMAIN-LOW and DOMAIN-HIGH.
;;;
;;; DEFAULT-LOW and DEFAULT-HIGH are the lower and upper bounds if we
;;; can't compute the bounds using FUN.
(defun elfun-derive-type-simple (arg fun domain-low domain-high
                                     default-low default-high
                                     &optional (increasingp t))
  (declare (type (or null real) domain-low domain-high))
  (etypecase arg
    (numeric-type
     (cond ((eq (numeric-type-complexp arg) :complex)
            (complex-float-type arg))
           ((numeric-type-real-p arg)
            ;; The argument is real, so let's find the intersection
            ;; between the argument and the domain of the function.
            ;; We compute the bounds on the intersection, and for
            ;; everything else, we return a complex number of the
            ;; appropriate type.
            (multiple-value-bind (intersection difference)
                (interval-intersection/difference (numeric-type->interval arg)
                                                  (make-interval
                                                   :low domain-low
                                                   :high domain-high))
              (cond
                (intersection
                 ;; Process the intersection.
                 (let* ((low (interval-low intersection))
                        (high (interval-high intersection))
                        (res-lo (or (bound-func fun (if increasingp low high) nil)
                                    default-low))
                        (res-hi (or (bound-func fun (if increasingp high low) nil)
                                    default-high))
                        (format (case (numeric-type-class arg)
                                  ((integer rational) 'single-float)
                                  (t (numeric-type-format arg))))
                        (bound-type (or format 'float))
                        (result-type
                         (make-numeric-type
                          :class 'float
                          :format format
                          :low (coerce-numeric-bound res-lo bound-type)
                          :high (coerce-numeric-bound res-hi bound-type))))
                   ;; If the ARG is a subset of the domain, we don't
                   ;; have to worry about the difference, because that
                   ;; can't occur.
                   (if (or (null difference)
                           ;; Check whether the arg is within the domain.
                           (domain-subtypep arg domain-low domain-high))
                       result-type
                       (list result-type
                             (specifier-type `(complex ,bound-type))))))
                (t
                 ;; No intersection so the result must be purely complex.
                 (complex-float-type arg)))))
           (t
            (float-or-complex-float-type arg default-low default-high))))))

(macrolet
    ((frob (name domain-low domain-high def-low-bnd def-high-bnd
                 &key (increasingp t))
       (let ((num (gensym)))
         `(defoptimizer (,name derive-type) ((,num))
           (one-arg-derive-type
            ,num
            (lambda (arg)
              (elfun-derive-type-simple arg #',name
                                        ,domain-low ,domain-high
                                        ,def-low-bnd ,def-high-bnd
                                        ,increasingp))
            #',name)))))
  ;; These functions are easy because they are defined for the whole
  ;; real line.
  (frob exp nil nil 0 nil)
  (frob sinh nil nil nil nil)
  (frob tanh nil nil -1 1)
  (frob asinh nil nil nil nil)

  ;; These functions are only defined for part of the real line. The
  ;; condition selects the desired part of the line.
  (frob asin -1d0 1d0 (- (/ pi 2)) (/ pi 2))
  ;; Acos is monotonic decreasing, so we need to swap the function
  ;; values at the lower and upper bounds of the input domain.
  (frob acos -1d0 1d0 0 pi :increasingp nil)
  (frob acosh 1d0 nil nil nil)
  (frob atanh -1d0 1d0 -1 1)
  ;; Kahan says that (sqrt -0.0) is -0.0, so use a specifier that
  ;; includes -0.0.
  (frob sqrt (load-time-value (make-unportable-float :double-float-negative-zero)) nil 0 nil))

;;; Compute bounds for (expt x y). This should be easy since (expt x
;;; y) = (exp (* y (log x))). However, computations done this way
;;; have too much roundoff. Thus we have to do it the hard way.
(defun safe-expt (x y)
  (handler-case
      (when (< (abs y) 10000)
        (expt x y))
    (error ()
      nil)))

;;; Handle the case when x >= 1.
(defun interval-expt-> (x y)
  (case (sb!c::interval-range-info y 0d0)
    (+
     ;; Y is positive and log X >= 0. The range of exp(y * log(x)) is
     ;; obviously non-negative. We just have to be careful for
     ;; infinite bounds (given by nil).
     (let ((lo (safe-expt (type-bound-number (sb!c::interval-low x))
                          (type-bound-number (sb!c::interval-low y))))
           (hi (safe-expt (type-bound-number (sb!c::interval-high x))
                          (type-bound-number (sb!c::interval-high y)))))
       (list (sb!c::make-interval :low (or lo 1) :high hi))))
    (-
     ;; Y is negative and log x >= 0. The range of exp(y * log(x)) is
     ;; obviously [0, 1]. However, underflow (nil) means 0 is the
     ;; result.
     (let ((lo (safe-expt (type-bound-number (sb!c::interval-high x))
                          (type-bound-number (sb!c::interval-low y))))
           (hi (safe-expt (type-bound-number (sb!c::interval-low x))
                          (type-bound-number (sb!c::interval-high y)))))
       (list (sb!c::make-interval :low (or lo 0) :high (or hi 1)))))
    (t
     ;; Split the interval in half.
     (destructuring-bind (y- y+)
         (sb!c::interval-split 0 y t)
       (list (interval-expt-> x y-)
             (interval-expt-> x y+))))))

;;; Handle the case when x <= 1
(defun interval-expt-< (x y)
  (case (sb!c::interval-range-info x 0d0)
    (+
     ;; The case of 0 <= x <= 1 is easy
     (case (sb!c::interval-range-info y)
       (+
        ;; Y is positive and log X <= 0. The range of exp(y * log(x)) is
        ;; obviously [0, 1]. We just have to be careful for infinite bounds
        ;; (given by nil).
        (let ((lo (safe-expt (type-bound-number (sb!c::interval-low x))
                             (type-bound-number (sb!c::interval-high y))))
              (hi (safe-expt (type-bound-number (sb!c::interval-high x))
                             (type-bound-number (sb!c::interval-low y)))))
          (list (sb!c::make-interval :low (or lo 0) :high (or hi 1)))))
       (-
        ;; Y is negative and log x <= 0. The range of exp(y * log(x)) is
        ;; obviously [1, inf].
        (let ((hi (safe-expt (type-bound-number (sb!c::interval-low x))
                             (type-bound-number (sb!c::interval-low y))))
              (lo (safe-expt (type-bound-number (sb!c::interval-high x))
                             (type-bound-number (sb!c::interval-high y)))))
          (list (sb!c::make-interval :low (or lo 1) :high hi))))
       (t
        ;; Split the interval in half
        (destructuring-bind (y- y+)
            (sb!c::interval-split 0 y t)
          (list (interval-expt-< x y-)
                (interval-expt-< x y+))))))
    (-
     ;; The case where x <= 0. Y MUST be an INTEGER for this to work!
     ;; The calling function must insure this! For now we'll just
     ;; return the appropriate unbounded float type.
     (list (sb!c::make-interval :low nil :high nil)))
    (t
     (destructuring-bind (neg pos)
         (interval-split 0 x t t)
       (list (interval-expt-< neg y)
             (interval-expt-< pos y))))))

;;; Compute bounds for (expt x y).
(defun interval-expt (x y)
  (case (interval-range-info x 1)
    (+
     ;; X >= 1
         (interval-expt-> x y))
    (-
     ;; X <= 1
     (interval-expt-< x y))
    (t
     (destructuring-bind (left right)
         (interval-split 1 x t t)
       (list (interval-expt left y)
             (interval-expt right y))))))

(defun fixup-interval-expt (bnd x-int y-int x-type y-type)
  (declare (ignore x-int))
  ;; Figure out what the return type should be, given the argument
  ;; types and bounds and the result type and bounds.
  (cond ((csubtypep x-type (specifier-type 'integer))
         ;; an integer to some power
         (case (numeric-type-class y-type)
           (integer
            ;; Positive integer to an integer power is either an
            ;; integer or a rational.
            (let ((lo (or (interval-low bnd) '*))
                  (hi (or (interval-high bnd) '*)))
              (if (and (interval-low y-int)
                       (>= (type-bound-number (interval-low y-int)) 0))
                  (specifier-type `(integer ,lo ,hi))
                  (specifier-type `(rational ,lo ,hi)))))
           (rational
            ;; Positive integer to rational power is either a rational
            ;; or a single-float.
            (let* ((lo (interval-low bnd))
                   (hi (interval-high bnd))
                   (int-lo (if lo
                               (floor (type-bound-number lo))
                               '*))
                   (int-hi (if hi
                               (ceiling (type-bound-number hi))
                               '*))
                   (f-lo (or (bound-func #'float lo nil)
                             '*))
                   (f-hi (or (bound-func #'float hi nil)
                             '*)))
              (specifier-type `(or (rational ,int-lo ,int-hi)
                                (single-float ,f-lo, f-hi)))))
           (float
            ;; A positive integer to a float power is a float.
            (let ((format (numeric-type-format y-type)))
              (aver format)
              (modified-numeric-type
               y-type
               :low (coerce-numeric-bound (interval-low bnd) format)
               :high (coerce-numeric-bound (interval-high bnd) format))))
           (t
            ;; A positive integer to a number is a number (for now).
            (specifier-type 'number))))
        ((csubtypep x-type (specifier-type 'rational))
         ;; a rational to some power
         (case (numeric-type-class y-type)
           (integer
            ;; A positive rational to an integer power is always a rational.
            (specifier-type `(rational ,(or (interval-low bnd) '*)
                                       ,(or (interval-high bnd) '*))))
           (rational
            ;; A positive rational to rational power is either a rational
            ;; or a single-float.
            (let* ((lo (interval-low bnd))
                   (hi (interval-high bnd))
                   (int-lo (if lo
                               (floor (type-bound-number lo))
                               '*))
                   (int-hi (if hi
                               (ceiling (type-bound-number hi))
                               '*))
                   (f-lo (or (bound-func #'float lo nil)
                             '*))
                   (f-hi (or (bound-func #'float hi nil)
                             '*)))
              (specifier-type `(or (rational ,int-lo ,int-hi)
                                (single-float ,f-lo, f-hi)))))
           (float
            ;; A positive rational to a float power is a float.
            (let ((format (numeric-type-format y-type)))
              (aver format)
              (modified-numeric-type
               y-type
               :low (coerce-numeric-bound (interval-low bnd) format)
               :high (coerce-numeric-bound (interval-high bnd) format))))
           (t
            ;; A positive rational to a number is a number (for now).
            (specifier-type 'number))))
        ((csubtypep x-type (specifier-type 'float))
         ;; a float to some power
         (case (numeric-type-class y-type)
           ((or integer rational)
            ;; A positive float to an integer or rational power is
            ;; always a float.
            (let ((format (numeric-type-format x-type)))
              (aver format)
              (make-numeric-type
               :class 'float
               :format format
               :low (coerce-numeric-bound (interval-low bnd) format)
               :high (coerce-numeric-bound (interval-high bnd) format))))
           (float
            ;; A positive float to a float power is a float of the
            ;; higher type.
            (let ((format (float-format-max (numeric-type-format x-type)
                                            (numeric-type-format y-type))))
              (aver format)
              (make-numeric-type
               :class 'float
               :format format
               :low (coerce-numeric-bound (interval-low bnd) format)
               :high (coerce-numeric-bound (interval-high bnd) format))))
           (t
            ;; A positive float to a number is a number (for now)
            (specifier-type 'number))))
        (t
         ;; A number to some power is a number.
         (specifier-type 'number))))

(defun merged-interval-expt (x y)
  (let* ((x-int (numeric-type->interval x))
         (y-int (numeric-type->interval y)))
    (mapcar (lambda (type)
              (fixup-interval-expt type x-int y-int x y))
            (flatten-list (interval-expt x-int y-int)))))

(defun expt-derive-type-aux (x y same-arg)
  (declare (ignore same-arg))
  (cond ((or (not (numeric-type-real-p x))
             (not (numeric-type-real-p y)))
         ;; Use numeric contagion if either is not real.
         (numeric-contagion x y))
        ((csubtypep y (specifier-type 'integer))
         ;; A real raised to an integer power is well-defined.
         (merged-interval-expt x y))
        ;; A real raised to a non-integral power can be a float or a
        ;; complex number.
        ((or (csubtypep x (specifier-type '(rational 0)))
             (csubtypep x (specifier-type '(float (0d0)))))
         ;; But a positive real to any power is well-defined.
         (merged-interval-expt x y))
        ((and (csubtypep x (specifier-type 'rational))
              (csubtypep y (specifier-type 'rational)))
         ;; A rational to the power of a rational could be a rational
         ;; or a possibly-complex single float
         (specifier-type '(or rational single-float (complex single-float))))
        (t
         ;; a real to some power. The result could be a real or a
         ;; complex.
         (float-or-complex-float-type (numeric-contagion x y)))))

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

;;; Note we must assume that a type including 0.0 may also include
;;; -0.0 and thus the result may be complex -infinity + i*pi.
(defun log-derive-type-aux-1 (x)
  (elfun-derive-type-simple x #'log 0d0 nil nil nil))

(defun log-derive-type-aux-2 (x y same-arg)
  (let ((log-x (log-derive-type-aux-1 x))
        (log-y (log-derive-type-aux-1 y))
        (accumulated-list nil))
    ;; LOG-X or LOG-Y might be union types. We need to run through
    ;; the union types ourselves because /-DERIVE-TYPE-AUX doesn't.
    (dolist (x-type (prepare-arg-for-derive-type log-x))
      (dolist (y-type (prepare-arg-for-derive-type log-y))
        (push (/-derive-type-aux x-type y-type same-arg) accumulated-list)))
    (apply #'type-union (flatten-list accumulated-list))))

(defoptimizer (log derive-type) ((x &optional y))
  (if y
      (two-arg-derive-type x y #'log-derive-type-aux-2 #'log)
      (one-arg-derive-type x #'log-derive-type-aux-1 #'log)))

(defun atan-derive-type-aux-1 (y)
  (elfun-derive-type-simple y #'atan nil nil (- (/ pi 2)) (/ pi 2)))

(defun atan-derive-type-aux-2 (y x same-arg)
  (declare (ignore same-arg))
  ;; The hard case with two args. We just return the max bounds.
  (let ((result-type (numeric-contagion y x)))
    (cond ((and (numeric-type-real-p x)
                (numeric-type-real-p y))
           (let* (;; FIXME: This expression for FORMAT seems to
                  ;; appear multiple times, and should be factored out.
                  (format (case (numeric-type-class result-type)
                            ((integer rational) 'single-float)
                            (t (numeric-type-format result-type))))
                  (bound-format (or format 'float)))
             (make-numeric-type :class 'float
                                :format format
                                :complexp :real
                                :low (coerce (- pi) bound-format)
                                :high (coerce pi bound-format))))
          (t
           ;; The result is a float or a complex number
           (float-or-complex-float-type result-type)))))

(defoptimizer (atan derive-type) ((y &optional x))
  (if x
      (two-arg-derive-type y x #'atan-derive-type-aux-2 #'atan)
      (one-arg-derive-type y #'atan-derive-type-aux-1 #'atan)))

(defun cosh-derive-type-aux (x)
  ;; We note that cosh x = cosh |x| for all real x.
  (elfun-derive-type-simple
   (if (numeric-type-real-p x)
       (abs-derive-type-aux x)
       x)
   #'cosh nil nil 0 nil))

(defoptimizer (cosh derive-type) ((num))
  (one-arg-derive-type num #'cosh-derive-type-aux #'cosh))

(defun phase-derive-type-aux (arg)
  (let* ((format (case (numeric-type-class arg)
                   ((integer rational) 'single-float)
                   (t (numeric-type-format arg))))
         (bound-type (or format 'float)))
    (cond ((numeric-type-real-p arg)
           (case (interval-range-info (numeric-type->interval arg) 0.0)
             (+
              ;; The number is positive, so the phase is 0.
              (make-numeric-type :class 'float
                                 :format format
                                 :complexp :real
                                 :low (coerce 0 bound-type)
                                 :high (coerce 0 bound-type)))
             (-
              ;; The number is always negative, so the phase is pi.
              (make-numeric-type :class 'float
                                 :format format
                                 :complexp :real
                                 :low (coerce pi bound-type)
                                 :high (coerce pi bound-type)))
             (t
              ;; We can't tell. The result is 0 or pi. Use a union
              ;; type for this.
              (list
               (make-numeric-type :class 'float
                                  :format format
                                  :complexp :real
                                  :low (coerce 0 bound-type)
                                  :high (coerce 0 bound-type))
               (make-numeric-type :class 'float
                                  :format format
                                  :complexp :real
                                  :low (coerce pi bound-type)
                                  :high (coerce pi bound-type))))))
          (t
           ;; We have a complex number. The answer is the range -pi
           ;; to pi. (-pi is included because we have -0.)
           (make-numeric-type :class 'float
                              :format format
                              :complexp :real
                              :low (coerce (- pi) bound-type)
                              :high (coerce pi bound-type))))))

(defoptimizer (phase derive-type) ((num))
  (one-arg-derive-type num #'phase-derive-type-aux #'phase))

) ; PROGN

(deftransform realpart ((x) ((complex rational)) *)
  '(sb!kernel:%realpart x))
(deftransform imagpart ((x) ((complex rational)) *)
  '(sb!kernel:%imagpart x))

;;; Make REALPART and IMAGPART return the appropriate types. This
;;; should help a lot in optimized code.
(defun realpart-derive-type-aux (type)
  (let ((class (numeric-type-class type))
        (format (numeric-type-format type)))
    (cond ((numeric-type-real-p type)
           ;; The realpart of a real has the same type and range as
           ;; the input.
           (make-numeric-type :class class
                              :format format
                              :complexp :real
                              :low (numeric-type-low type)
                              :high (numeric-type-high type)))
          (t
           ;; We have a complex number. The result has the same type
           ;; as the real part, except that it's real, not complex,
           ;; obviously.
           (make-numeric-type :class class
                              :format format
                              :complexp :real
                              :low (numeric-type-low type)
                              :high (numeric-type-high type))))))

(defoptimizer (realpart derive-type) ((num))
  (one-arg-derive-type num #'realpart-derive-type-aux #'realpart))

(defun imagpart-derive-type-aux (type)
  (let ((class (numeric-type-class type))
        (format (numeric-type-format type)))
    (cond ((numeric-type-real-p type)
           ;; The imagpart of a real has the same type as the input,
           ;; except that it's zero.
           (let ((bound-format (or format class 'real)))
             (make-numeric-type :class class
                                :format format
                                :complexp :real
                                :low (coerce 0 bound-format)
                                :high (coerce 0 bound-format))))
          (t
           ;; We have a complex number. The result has the same type as
           ;; the imaginary part, except that it's real, not complex,
           ;; obviously.
           (make-numeric-type :class class
                              :format format
                              :complexp :real
                              :low (numeric-type-low type)
                              :high (numeric-type-high type))))))

(defoptimizer (imagpart derive-type) ((num))
  (one-arg-derive-type num #'imagpart-derive-type-aux #'imagpart))

(defun complex-derive-type-aux-1 (re-type)
  (if (numeric-type-p re-type)
      (make-numeric-type :class (numeric-type-class re-type)
                         :format (numeric-type-format re-type)
                         :complexp (if (csubtypep re-type
                                                  (specifier-type 'rational))
                                       :real
                                       :complex)
                         :low (numeric-type-low re-type)
                         :high (numeric-type-high re-type))
      (specifier-type 'complex)))

(defun complex-derive-type-aux-2 (re-type im-type same-arg)
  (declare (ignore same-arg))
  (if (and (numeric-type-p re-type)
           (numeric-type-p im-type))
      ;; Need to check to make sure numeric-contagion returns the
      ;; right type for what we want here.

      ;; Also, what about rational canonicalization, like (complex 5 0)
      ;; is 5?  So, if the result must be complex, we make it so.
      ;; If the result might be complex, which happens only if the
      ;; arguments are rational, we make it a union type of (or
      ;; rational (complex rational)).
      (let* ((element-type (numeric-contagion re-type im-type))
             (rat-result-p (csubtypep element-type
                                      (specifier-type 'rational))))
        (if rat-result-p
            (type-union element-type
                        (specifier-type
                         `(complex ,(numeric-type-class element-type))))
            (make-numeric-type :class (numeric-type-class element-type)
                               :format (numeric-type-format element-type)
                               :complexp (if rat-result-p
                                             :real
                                             :complex))))
      (specifier-type 'complex)))

#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(defoptimizer (complex derive-type) ((re &optional im))
  (if im
      (two-arg-derive-type re im #'complex-derive-type-aux-2 #'complex)
      (one-arg-derive-type re #'complex-derive-type-aux-1 #'complex)))

;;; Define some transforms for complex operations. We do this in lieu
;;; of complex operation VOPs.
(macrolet ((frob (type)
             `(progn
                (deftransform complex ((r) (,type))
                  '(complex r ,(coerce 0 type)))
                (deftransform complex ((r i) (,type (and real (not ,type))))
                  '(complex r (truly-the ,type (coerce i ',type))))
                (deftransform complex ((r i) ((and real (not ,type)) ,type))
                  '(complex (truly-the ,type (coerce r ',type)) i))
               ;; negation
                #!-complex-float-vops
               (deftransform %negate ((z) ((complex ,type)) *)
                 '(complex (%negate (realpart z)) (%negate (imagpart z))))
               ;; complex addition and subtraction
               #!-complex-float-vops
               (deftransform + ((w z) ((complex ,type) (complex ,type)) *)
                 '(complex (+ (realpart w) (realpart z))
                           (+ (imagpart w) (imagpart z))))
               #!-complex-float-vops
               (deftransform - ((w z) ((complex ,type) (complex ,type)) *)
                 '(complex (- (realpart w) (realpart z))
                           (- (imagpart w) (imagpart z))))
               ;; Add and subtract a complex and a real.
               #!-complex-float-vops
               (deftransform + ((w z) ((complex ,type) real) *)
                 `(complex (+ (realpart w) z)
                           (+ (imagpart w) ,(coerce 0 ',type))))
               #!-complex-float-vops
               (deftransform + ((z w) (real (complex ,type)) *)
                 `(complex (+ (realpart w) z)
                           (+ (imagpart w) ,(coerce 0 ',type))))
               ;; Add and subtract a real and a complex number.
               #!-complex-float-vops
               (deftransform - ((w z) ((complex ,type) real) *)
                 `(complex (- (realpart w) z)
                           (- (imagpart w) ,(coerce 0 ',type))))
               #!-complex-float-vops
               (deftransform - ((z w) (real (complex ,type)) *)
                 `(complex (- z (realpart w))
                           (- ,(coerce 0 ',type) (imagpart w))))
               ;; Multiply and divide two complex numbers.
               #!-complex-float-vops
               (deftransform * ((x y) ((complex ,type) (complex ,type)) *)
                 '(let* ((rx (realpart x))
                         (ix (imagpart x))
                         (ry (realpart y))
                         (iy (imagpart y)))
                    (complex (- (* rx ry) (* ix iy))
                             (+ (* rx iy) (* ix ry)))))
               (deftransform / ((x y) ((complex ,type) (complex ,type)) *)
                 #!-complex-float-vops
                 '(let* ((rx (realpart x))
                         (ix (imagpart x))
                         (ry (realpart y))
                         (iy (imagpart y)))
                    (if (> (abs ry) (abs iy))
                        (let* ((r (/ iy ry))
                               (dn (+ ry (* r iy))))
                          (complex (/ (+ rx (* ix r)) dn)
                                   (/ (- ix (* rx r)) dn)))
                        (let* ((r (/ ry iy))
                               (dn (+ iy (* r ry))))
                          (complex (/ (+ (* rx r) ix) dn)
                                   (/ (- (* ix r) rx) dn)))))
                 #!+complex-float-vops
                 `(let* ((cs (conjugate (sb!vm::swap-complex x)))
                         (ry (realpart y))
                         (iy (imagpart y)))
                    (if (> (abs ry) (abs iy))
                        (let* ((r (/ iy ry))
                               (dn (+ ry (* r iy))))
                          (/ (+ x (* cs r)) dn))
                        (let* ((r (/ ry iy))
                               (dn (+ iy (* r ry))))
                          (/ (+ (* x r) cs) dn)))))
               ;; Multiply a complex by a real or vice versa.
               #!-complex-float-vops
               (deftransform * ((w z) ((complex ,type) real) *)
                 '(complex (* (realpart w) z) (* (imagpart w) z)))
               #!-complex-float-vops
               (deftransform * ((z w) (real (complex ,type)) *)
                 '(complex (* (realpart w) z) (* (imagpart w) z)))
               ;; Divide a complex by a real or vice versa.
               #!-complex-float-vops
               (deftransform / ((w z) ((complex ,type) real) *)
                 '(complex (/ (realpart w) z) (/ (imagpart w) z)))
               (deftransform / ((x y) (,type (complex ,type)) *)
                 #!-complex-float-vops
                 '(let* ((ry (realpart y))
                         (iy (imagpart y)))
                    (if (> (abs ry) (abs iy))
                        (let* ((r (/ iy ry))
                               (dn (+ ry (* r iy))))
                          (complex (/ x dn)
                                   (/ (- (* x r)) dn)))
                        (let* ((r (/ ry iy))
                               (dn (+ iy (* r ry))))
                          (complex (/ (* x r) dn)
                                   (/ (- x) dn)))))
                 #!+complex-float-vops
                 '(let* ((ry (realpart y))
                         (iy (imagpart y)))
                   (if (> (abs ry) (abs iy))
                       (let* ((r (/ iy ry))
                              (dn (+ ry (* r iy))))
                         (/ (complex x (- (* x r))) dn))
                       (let* ((r (/ ry iy))
                              (dn (+ iy (* r ry))))
                         (/ (complex (* x r) (- x)) dn)))))
               ;; conjugate of complex number
               #!-complex-float-vops
               (deftransform conjugate ((z) ((complex ,type)) *)
                 '(complex (realpart z) (- (imagpart z))))
               ;; CIS
               (deftransform cis ((z) ((,type)) *)
                 '(complex (cos z) (sin z)))
               ;; comparison
               #!-complex-float-vops
               (deftransform = ((w z) ((complex ,type) (complex ,type)) *)
                 '(and (= (realpart w) (realpart z))
                       (= (imagpart w) (imagpart z))))
               #!-complex-float-vops
               (deftransform = ((w z) ((complex ,type) real) *)
                 '(and (= (realpart w) z) (zerop (imagpart w))))
               #!-complex-float-vops
               (deftransform = ((w z) (real (complex ,type)) *)
                 '(and (= (realpart z) w) (zerop (imagpart z)))))))

  (frob single-float)
  (frob double-float))

;;; Here are simple optimizers for SIN, COS, and TAN. They do not
;;; produce a minimal range for the result; the result is the widest
;;; possible answer. This gets around the problem of doing range
;;; reduction correctly but still provides useful results when the
;;; inputs are union types.
#-sb-xc-host ; (See CROSS-FLOAT-INFINITY-KLUDGE.)
(progn
(defun trig-derive-type-aux (arg domain fun
                                 &optional def-lo def-hi (increasingp t))
  (etypecase arg
    (numeric-type
     (cond ((eq (numeric-type-complexp arg) :complex)
            (make-numeric-type :class (numeric-type-class arg)
                               :format (numeric-type-format arg)
                               :complexp :complex))
           ((numeric-type-real-p arg)
            (let* ((format (case (numeric-type-class arg)
                             ((integer rational) 'single-float)
                             (t (numeric-type-format arg))))
                   (bound-type (or format 'float)))
              ;; If the argument is a subset of the "principal" domain
              ;; of the function, we can compute the bounds because
              ;; the function is monotonic. We can't do this in
              ;; general for these periodic functions because we can't
              ;; (and don't want to) do the argument reduction in
              ;; exactly the same way as the functions themselves do
              ;; it.
              (if (csubtypep arg domain)
                  (let ((res-lo (bound-func fun (numeric-type-low arg) nil))
                        (res-hi (bound-func fun (numeric-type-high arg) nil)))
                    (unless increasingp
                      (rotatef res-lo res-hi))
                    (make-numeric-type
                     :class 'float
                     :format format
                     :low (coerce-numeric-bound res-lo bound-type)
                     :high (coerce-numeric-bound res-hi bound-type)))
                  (make-numeric-type
                   :class 'float
                   :format format
                   :low (and def-lo (coerce def-lo bound-type))
                   :high (and def-hi (coerce def-hi bound-type))))))
           (t
            (float-or-complex-float-type arg def-lo def-hi))))))

(defoptimizer (sin derive-type) ((num))
  (one-arg-derive-type
   num
   (lambda (arg)
     ;; Derive the bounds if the arg is in [-pi/2, pi/2].
     (trig-derive-type-aux
      arg
      (specifier-type `(float ,(- (/ pi 2)) ,(/ pi 2)))
      #'sin
      -1 1))
   #'sin))

(defoptimizer (cos derive-type) ((num))
  (one-arg-derive-type
   num
   (lambda (arg)
     ;; Derive the bounds if the arg is in [0, pi].
     (trig-derive-type-aux arg
                           (specifier-type `(float 0d0 ,pi))
                           #'cos
                           -1 1
                           nil))
   #'cos))

(defoptimizer (tan derive-type) ((num))
  (one-arg-derive-type
   num
   (lambda (arg)
     ;; Derive the bounds if the arg is in [-pi/2, pi/2].
     (trig-derive-type-aux arg
                           (specifier-type `(float ,(- (/ pi 2)) ,(/ pi 2)))
                           #'tan
                           nil nil))
   #'tan))

(defoptimizer (conjugate derive-type) ((num))
  (one-arg-derive-type num
    (lambda (arg)
      (flet ((most-negative-bound (l h)
               (and l h
                    (if (< (type-bound-number l) (- (type-bound-number h)))
                        l
                        (set-bound (- (type-bound-number h)) (consp h)))))
             (most-positive-bound (l h)
               (and l h
                    (if (> (type-bound-number h) (- (type-bound-number l)))
                        h
                        (set-bound (- (type-bound-number l)) (consp l))))))
        (if (numeric-type-real-p arg)
            (lvar-type num)
            (let ((low (numeric-type-low arg))
                  (high (numeric-type-high arg)))
              (let ((new-low (most-negative-bound low high))
                    (new-high (most-positive-bound low high)))
              (modified-numeric-type arg :low new-low :high new-high))))))
    #'conjugate))

(defoptimizer (cis derive-type) ((num))
  (one-arg-derive-type num
    (lambda (arg)
      (sb!c::specifier-type
       `(complex ,(or (numeric-type-format arg) 'float))))
    #'cis))

) ; PROGN

;;;; TRUNCATE, FLOOR, CEILING, and ROUND

(macrolet ((define-frobs (fun ufun)
             `(progn
                (defknown ,ufun (real) integer (movable foldable flushable))
                (deftransform ,fun ((x &optional by)
                                    (* &optional
                                       (constant-arg (member 1))))
                  '(let ((res (,ufun x)))
                     (values res (- x res)))))))
  (define-frobs truncate %unary-truncate)
  (define-frobs round %unary-round))

(deftransform %unary-truncate ((x) (single-float))
  `(%unary-truncate/single-float x))
(deftransform %unary-truncate ((x) (double-float))
  `(%unary-truncate/double-float x))

;;; Convert (TRUNCATE x y) to the obvious implementation.
;;;
;;; ...plus hair: Insert explicit coercions to appropriate float types: Python
;;; is reluctant it generate explicit integer->float coercions due to
;;; precision issues (see SAFE-SINGLE-COERCION-P &co), but this is not an
;;; issue here as there is no DERIVE-TYPE optimizer on specialized versions of
;;; %UNARY-TRUNCATE, so the derived type of TRUNCATE remains the best we can
;;; do here -- which is fine. Also take care not to add unnecassary division
;;; or multiplication by 1, since we are not able to always eliminate them,
;;; depending on FLOAT-ACCURACY. Finally, leave out the secondary value when
;;; we know it is unused: COERCE is not flushable.
(macrolet ((def (type other-float-arg-types)
             (let ((unary (symbolicate "%UNARY-TRUNCATE/" type))
                   (coerce (symbolicate "%" type)))
               `(deftransform truncate ((x &optional y)
                                        (,type
                                         &optional (or ,type ,@other-float-arg-types integer))
                                        * :result result)
                  (let* ((result-type (and result
                                           (lvar-derived-type result)))
                         (compute-all (and (values-type-p result-type)
                                           (not (type-single-value-p result-type)))))
                    (if (or (not y)
                            (and (constant-lvar-p y) (= 1 (lvar-value y))))
                        (if compute-all
                            `(let ((res (,',unary x)))
                               (values res (- x (,',coerce res))))
                            `(let ((res (,',unary x)))
                               ;; Dummy secondary value!
                               (values res x)))
                        (if compute-all
                            `(let* ((f (,',coerce y))
                                    (res (,',unary (/ x f))))
                               (values res (- x (* f (,',coerce res)))))
                            `(let* ((f (,',coerce y))
                                    (res (,',unary (/ x f))))
                               ;; Dummy secondary value!
                               (values res x)))))))))
  (def single-float ())
  (def double-float (single-float)))

(deftransform floor ((number &optional divisor)
                     (float &optional (or integer float)))
  (let ((defaulted-divisor (if divisor 'divisor 1)))
    `(multiple-value-bind (tru rem) (truncate number ,defaulted-divisor)
       (if (and (not (zerop rem))
                (if (minusp ,defaulted-divisor)
                    (plusp number)
                    (minusp number)))
           (values (1- tru) (+ rem ,defaulted-divisor))
           (values tru rem)))))

(deftransform ceiling ((number &optional divisor)
                       (float &optional (or integer float)))
  (let ((defaulted-divisor (if divisor 'divisor 1)))
    `(multiple-value-bind (tru rem) (truncate number ,defaulted-divisor)
       (if (and (not (zerop rem))
                (if (minusp ,defaulted-divisor)
                    (minusp number)
                    (plusp number)))
           (values (1+ tru) (- rem ,defaulted-divisor))
           (values tru rem)))))

(defknown %unary-ftruncate (real) float (movable foldable flushable))
(defknown %unary-ftruncate/single (single-float) single-float
  (movable foldable flushable))
(defknown %unary-ftruncate/double (double-float) double-float
  (movable foldable flushable))

(defun %unary-ftruncate/single (x)
  (declare (type single-float x))
  (declare (optimize speed (safety 0)))
  (let* ((bits (single-float-bits x))
         (exp (ldb sb!vm:single-float-exponent-byte bits))
         (biased (the single-float-exponent
                   (- exp sb!vm:single-float-bias))))
    (declare (type (signed-byte 32) bits))
    (cond
      ((= exp sb!vm:single-float-normal-exponent-max) x)
      ((<= biased 0) (* x 0f0))
      ((>= biased (float-digits x)) x)
      (t
       (let ((frac-bits (- (float-digits x) biased)))
         (setf bits (logandc2 bits (- (ash 1 frac-bits) 1)))
         (make-single-float bits))))))

(defun %unary-ftruncate/double (x)
  (declare (type double-float x))
  (declare (optimize speed (safety 0)))
  (let* ((high (double-float-high-bits x))
         (low (double-float-low-bits x))
         (exp (ldb sb!vm:double-float-exponent-byte high))
         (biased (the double-float-exponent
                   (- exp sb!vm:double-float-bias))))
    (declare (type (signed-byte 32) high)
             (type (unsigned-byte 32) low))
    (cond
      ((= exp sb!vm:double-float-normal-exponent-max) x)
      ((<= biased 0) (* x 0d0))
      ((>= biased (float-digits x)) x)
      (t
       (let ((frac-bits (- (float-digits x) biased)))
         (cond ((< frac-bits 32)
                (setf low (logandc2 low (- (ash 1 frac-bits) 1))))
               (t
                (setf low 0)
                (setf high (logandc2 high (- (ash 1 (- frac-bits 32)) 1)))))
         (make-double-float high low))))))

(macrolet
    ((def (float-type fun)
         `(deftransform %unary-ftruncate ((x) (,float-type))
            '(,fun x))))
  (def single-float %unary-ftruncate/single)
  (def double-float %unary-ftruncate/double))