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;;;***************************************************************
;;;An ACL2 Library of Floating Point Arithmetic
;;;David M. Russinoff
;;;Advanced Micro Devices, Inc.
;;;February, 1998
;;;***************************************************************
(in-package "ACL2")
(include-book "logdefs")
(include-book "float")
(defun PLISTP (x)
(if (consp x)
(plistp (cdr x))
(equal x ())))
(defun FORMATP (phi)
(and (plistp phi)
(= (len phi) 2)
(integerp (car phi))
(integerp (cadr phi))
(> (car phi) 0)
(> (cadr phi) 0)))
(defun GET-SBITS (phi) (ifix (car phi)))
(defun GET-EBITS (phi) (ifix (cadr phi)))
(defun sgnf (z phi) (bitn z (+ (get-sbits phi) (get-ebits phi))))
(defun expf (z phi) (bits z (+ (get-sbits phi) (get-ebits phi) -1) (get-sbits phi)))
(defun sigf (z phi) (bits z (1- (get-sbits phi)) 0))
(defun NORMAL-ENCODING-P (z phi)
(and (integerp z)
(>= z 0)
(>= (sigf z phi) (expt 2 (1- (get-sbits phi))))))
(local (defthm BITN-0-1
(or (= (bitn x n) 0) (= (bitn x n) 1))
:rule-classes ()
:hints (("Goal" :in-theory (enable bitn)))))
(local (defthm BITS-NAT
(implies (and (integerp x) (>= x 0)
(integerp i) (>= i 0)
(integerp j) (>= j 0))
(and (integerp (bits x i j))
(>= (bits x i j) 0)))
:rule-classes ()
:hints (("Goal" :in-theory (enable bits)
:use ((:instance integerp-expt (n (1+ i)))
(:instance integerp-expt (n j))
(:instance expt-pos (x (1+ i)))
(:instance expt-pos (x j))
(:instance rem>=0 (m x) (n (expt 2 (1+ i))))
(:instance fl-def (x (/ (rem x (expt 2 (1+ i))) (expt 2 j)))))))))
(local (defthm BITS<
(implies (and (integerp x) (>= x 0)
(integerp i) (>= i 0)
(integerp j) (>= j 0))
(< (bits x i j) (expt 2 (- (1+ i) j))))
:rule-classes ()
:hints (("Goal" :in-theory (enable bits)
:use ((:instance rem<n (m x) (n (expt 2 (1+ i))))
(:instance expt-pos (x (- j)))
(:instance expt-pos (x (1+ i)))
(:instance integerp-expt (n (1+ i)))
(:instance *-strongly-monotonic
(x (expt 2 (- j)))
(y (rem x (expt 2 (1+ i))))
(y+ (expt 2 (1+ i))))
(:instance expo+ (m (1+ i)) (n (- j)))
(:instance fl-def (x (/ (rem x (expt 2 (1+ i))) (expt 2 j)))))))))
(defthm NORMAL-ENCODING-LEMMA
(implies (and (formatp phi)
(normal-encoding-p z phi))
(and (or (= (sgnf z phi) 0)
(= (sgnf z phi) 1))
(integerp (expf z phi))
(>= (expf z phi) 0)
(< (expf z phi) (expt 2 (get-ebits phi)))
(integerp (sigf z phi))
(>= (sigf z phi) 0)
(< (sigf z phi) (expt 2 (get-sbits phi)))))
:rule-classes ()
:hints (("Goal" :use ((:instance bitn-0-1 (x z) (n (+ (get-sbits phi) (get-ebits phi))))
(:instance bits<
(x z)
(i (+ (get-sbits phi) (get-ebits phi) -1))
(j (get-sbits phi)))
(:instance bits-nat
(x z)
(i (+ (get-sbits phi) (get-ebits phi) -1))
(j (get-sbits phi)))
(:instance bits<
(x z)
(i (1- (get-sbits phi)))
(j 0))
(:instance bits-nat
(x z)
(i (1- (get-sbits phi)))
(j 0))))))
(defun DECODE (z phi)
(* (if (= (sgnf z phi) 0) 1 -1)
(sigf z phi)
(expt 2 (+ (expf z phi) (- (expt 2 (1- (get-ebits phi)))) (- (get-sbits phi)) 2))))
(defun REPP (x phi)
(and (exactp x (get-sbits phi))
(<= (- 1 (expt 2 (1- (get-ebits phi))))
(expo x))
(<= (expo x)
(expt 2 (1- (get-ebits phi))))))
(defun ENCODE (x phi)
(cat (cat (if (= (sgn x) 1) 0 1)
(+ (expo x) (1- (expt 2 (1- (get-ebits phi)))))
(get-ebits phi))
(* (sig x) (expt 2 (1- (get-sbits phi))))
(get-sbits phi)))
(local
(defthm code-1
(implies (and (formatp phi)
(normal-encoding-p z phi))
(= (decode z phi)
(* (if (= (sgnf z phi) 0) 1 -1)
(* (sigf z phi) (expt 2 (- 1 (get-sbits phi))))
(expt 2 (- (expf z phi) (1- (expt 2 (1- (get-ebits phi)))))))))
:rule-classes ()))
(local
(defthm code-2
(implies (and (formatp phi)
(normal-encoding-p z phi))
(= (abs (decode z phi))
(* (* (sigf z phi) (expt 2 (- 1 (get-sbits phi))))
(expt 2 (- (expf z phi) (1- (expt 2 (1- (get-ebits phi)))))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable decode)
:use (code-1 normal-encoding-lemma)))))
(local
(defthm code-3
(implies (and (formatp phi)
(normal-encoding-p z phi))
(<= 1 (* (sigf z phi) (expt 2 (- 1 (get-sbits phi))))))
:rule-classes ()))
(local
(defthm code-4
(implies (and (formatp phi)
(normal-encoding-p z phi))
(> 2 (* (sigf z phi) (expt 2 (- 1 (get-sbits phi))))))
:rule-classes ()
:hints (("Goal" :use (normal-encoding-lemma)))))
(defthm NORMAL-NON-ZERO
(implies (and (formatp phi)
(normal-encoding-p z phi))
(not (equal (decode z phi) 0)))
:hints (("Goal" :use (normal-encoding-lemma))))
(local
(defthm code-5
(implies (and (formatp phi)
(normal-encoding-p z phi))
(and (= (sig (decode z phi)) (* (sigf z phi) (expt 2 (- 1 (get-sbits phi)))))
(= (expo (decode z phi)) (- (expf z phi) (1- (expt 2 (1- (get-ebits phi))))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable decode normal-encoding-p)
:use ((:instance fp-rep-unique
(x (decode z phi))
(m (* (sigf z phi) (expt 2 (- 1 (get-sbits phi)))))
(e (- (expf z phi) (1- (expt 2 (1- (get-ebits phi)))))))
(:instance code-4)
(:instance code-2)
(:instance code-3))))))
(local (IN-THEORY (disable expf get-ebits sigf get-sbits sgnf)))
(local
(defthm code-6
(implies (and (formatp phi)
(normal-encoding-p z phi))
(= (decode z phi)
(* (if (= (sgnf z phi) 0) 1 -1)
(sig (decode z phi))
(expt 2 (expo (decode z phi))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable expo a15 a2 formatp normal-encoding-p decode)
:use (normal-encoding-lemma
code-1
code-5
(:instance expo+
(m (+ 1 (- (GET-SBITS PHI))))
(n (+ (- (+ -1 (EXPT 2 (+ -1 (GET-EBITS PHI)))))
(EXPF Z PHI)))))))))
(defthm CODE-A
(implies (and (formatp phi)
(normal-encoding-p z phi))
(= (sgn (decode z phi))
(if (= (sgnf z phi) 0) 1 -1)))
:rule-classes ()
:hints (("Goal" :in-theory (disable decode formatp normal-non-zero normal-encoding-p)
:use ((:instance fp-rep (x (decode z phi)))
(:instance normal-non-zero)
(:instance code-6)))))
(defthm CODE-B
(implies (and (formatp phi)
(normal-encoding-p z phi))
(= (sig (decode z phi))
(* (sigf z phi) (expt 2 (- 1 (get-sbits phi))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable decode formatp normal-encoding-p)
:use ((:instance code-5)))))
(defthm CODE-C
(implies (and (formatp phi)
(normal-encoding-p z phi))
(= (expo (decode z phi))
(- (expf z phi) (1- (expt 2 (1- (get-ebits phi)))))))
:rule-classes ()
:hints (("Goal" :in-theory (disable decode formatp normal-encoding-p)
:use ((:instance code-5)))))
(defthm EXACTP-DECODE
(implies (and (formatp phi)
(normal-encoding-p z phi))
(exactp (decode z phi) (get-sbits phi)))
:rule-classes ()
:hints (("Goal" :in-theory (disable formatp normal-encoding-p decode)
:use ((:instance code-b)
(:instance exactp (x (decode z phi)) (n (get-sbits phi)))))))
(local (in-theory (disable expo)))
(local
(defthm expt+expt
(implies (integerp n)
(= (expt 2 n) (+ (expt 2 (1- n)) (expt 2 (1- n)))))
:rule-classes ()))
(local
(defthm bad-hack
(implies (and (integerp e)
(integerp h)
(integerp z)
(< e (* 2 h))
(= (+ h z) (+ 1 e)))
(<= z h))
:rule-classes ()))
(local (in-theory (enable get-ebits get-sbits)))
(defthm CODE-D
(implies (and (formatp phi)
(normal-encoding-p z phi))
(repp (decode z phi) phi))
:rule-classes ()
:hints (("Goal" :in-theory (disable normal-encoding-p decode)
:use ((:instance exactp-decode)
(:instance bad-hack
(h (expt 2 (1- (get-ebits phi))))
(e (expf z phi))
(z (expo (decode z phi))))
(:instance expt+expt (n (get-ebits phi)))
(:instance normal-encoding-lemma)
(:instance code-c)))))
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