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      • Floating-Point Exceptions and Specification of Elementary Arithmetic Instructions
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• Floating-Point Formats

Classification of Formats

Classification of Formats

Definitions and Theorems

Function: formatp

(defun formatp (f)
  (declare (xargs :guard t))
  (and (consp f)
       (consp (cdr f))
       (consp (cddr f))
       (natp (cadr f))
       (> (cadr f) 1)
       (natp (caddr f))
       (> (caddr f) 1)))

Function: explicitp

(defun explicitp (f)
  (declare (xargs :guard (formatp f)))
  (car f))

Function: prec

(defun prec (f)
  (declare (xargs :guard (formatp f)))
  (cadr f))

Function: expw

(defun expw (f)
  (declare (xargs :guard (formatp f)))
  (caddr f))

Function: sigw

(defun sigw (f)
  (declare (xargs :guard (formatp f)))
  (if (explicitp f)
      (prec f)
    (1- (prec f))))

Function: encodingp

(defun encodingp (x f)
  (declare (xargs :guard t))
  (and (formatp f)
       (bvecp x (+ 1 (expw f) (sigw f)))))

Function: hp

(defun hp nil
  (declare (xargs :guard t))
  '(nil 11 5))

Function: sp

(defun sp nil
  (declare (xargs :guard t))
  '(nil 24 8))

Function: dp

(defun dp nil
  (declare (xargs :guard t))
  '(nil 53 11))

Function: bf

(defun bf nil
  (declare (xargs :guard t))
  '(nil 8 8))

Function: ep

(defun ep nil
  (declare (xargs :guard t))
  '(t 64 15))

Theorem: formatp-sp

(defthm formatp-sp (formatp (sp)))

Theorem: formatp-dp

(defthm formatp-dp (formatp (dp)))

Theorem: formatp-hp

(defthm formatp-hp (formatp (hp)))

Theorem: formatp-ep

(defthm formatp-ep (formatp (ep)))

Function: sgnf

(defun sgnf (x f)
  (declare (xargs :guard (encodingp x f)))
  (bitn x (+ (expw f) (sigw f))))

Function: expf

(defun expf (x f)
  (declare (xargs :guard (encodingp x f)))
  (bits x (1- (+ (expw f) (sigw f)))
        (sigw f)))

Function: sigf

(defun sigf (x f)
  (declare (xargs :guard (encodingp x f)))
  (bits x (1- (sigw f)) 0))

Function: manf

(defun manf (x f)
  (declare (xargs :guard (encodingp x f)))
  (bits x (- (prec f) 2) 0))

Function: bias

(defun bias (f)
  (declare (xargs :guard (formatp f)))
  (- (expt 2 (- (expw f) 1)) 1))