;;;; This file contains the definitions of float-specific number
;;;; support (other than irrational stuff, which is in irrat.) There is
;;;; code in here that assumes there are only two float formats: IEEE
;;;; single and double. (LONG-FLOAT support has been added, but bugs
;;;; may still remain due to old code which assumes this dichotomy.)
;;;; 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!KERNEL")
;;;; float predicates and environment query
#!-sb-fluid
(declaim (maybe-inline float-denormalized-p float-infinity-p float-nan-p
float-trapping-nan-p))
(defun float-denormalized-p (x)
#!+sb-doc
"Return true if the float X is denormalized."
(number-dispatch ((x float))
((single-float)
(and (zerop (ldb sb!vm:single-float-exponent-byte (single-float-bits x)))
(not (zerop x))))
((double-float)
(and (zerop (ldb sb!vm:double-float-exponent-byte
(double-float-high-bits x)))
(not (zerop x))))
#!+(and long-float x86)
((long-float)
(and (zerop (ldb sb!vm:long-float-exponent-byte (long-float-exp-bits x)))
(not (zerop x))))))
(defmacro !define-float-dispatching-function
(name doc single double #!+(and long-float x86) long)
`(defun ,name (x)
,doc
(number-dispatch ((x float))
((single-float)
(let ((bits (single-float-bits x)))
(and (> (ldb sb!vm:single-float-exponent-byte bits)
sb!vm:single-float-normal-exponent-max)
,single)))
((double-float)
(let ((hi (double-float-high-bits x))
(lo (double-float-low-bits x)))
(declare (ignorable lo))
(and (> (ldb sb!vm:double-float-exponent-byte hi)
sb!vm:double-float-normal-exponent-max)
,double)))
#!+(and long-float x86)
((long-float)
(let ((exp (long-float-exp-bits x))
(hi (long-float-high-bits x))
(lo (long-float-low-bits x)))
(declare (ignorable lo))
(and (> (ldb sb!vm:long-float-exponent-byte exp)
sb!vm:long-float-normal-exponent-max)
,long))))))
(!define-float-dispatching-function float-infinity-p
"Return true if the float X is an infinity (+ or -)."
(zerop (ldb sb!vm:single-float-significand-byte bits))
(and (zerop (ldb sb!vm:double-float-significand-byte hi))
(zerop lo))
#!+(and long-float x86)
(and (zerop (ldb sb!vm:long-float-significand-byte hi))
(zerop lo)))
(!define-float-dispatching-function float-nan-p
"Return true if the float X is a NaN (Not a Number)."
#!-(or mips hppa)
(not (zerop (ldb sb!vm:single-float-significand-byte bits)))
#!+(or mips hppa)
(zerop (logand (ldb sb!vm:single-float-significand-byte bits)
sb!vm:single-float-trapping-nan-bit))
#!-(or mips hppa)
(or (not (zerop (ldb sb!vm:double-float-significand-byte hi)))
(not (zerop lo)))
#!+(or mips hppa)
(zerop (logand (ldb sb!vm:double-float-significand-byte hi)
sb!vm:double-float-trapping-nan-bit))
#!+(and long-float x86)
(or (not (zerop (ldb sb!vm:long-float-significand-byte hi)))
(not (zerop lo))))
(!define-float-dispatching-function float-trapping-nan-p
"Return true if the float X is a trapping NaN (Not a Number)."
#!-(or mips hppa)
(zerop (logand (ldb sb!vm:single-float-significand-byte bits)
sb!vm:single-float-trapping-nan-bit))
#!+(or mips hppa)
(not (zerop (ldb sb!vm:single-float-significand-byte bits)))
#!-(or mips hppa)
(zerop (logand (ldb sb!vm:double-float-significand-byte hi)
sb!vm:double-float-trapping-nan-bit))
#!+(or mips hppa)
(or (not (zerop (ldb sb!vm:double-float-significand-byte hi)))
(not (zerop lo)))
#!+(and long-float x86)
(zerop (logand (ldb sb!vm:long-float-significand-byte hi)
sb!vm:long-float-trapping-nan-bit)))
;;; If denormalized, use a subfunction from INTEGER-DECODE-FLOAT to find the
;;; actual exponent (and hence how denormalized it is), otherwise we just
;;; return the number of digits or 0.
#!-sb-fluid (declaim (maybe-inline float-precision))
(defun float-precision (f)
#!+sb-doc
"Return a non-negative number of significant digits in its float argument.
Will be less than FLOAT-DIGITS if denormalized or zero."
(macrolet ((frob (digits bias decode)
`(cond ((zerop f) 0)
((float-denormalized-p f)
(multiple-value-bind (ignore exp) (,decode f)
(declare (ignore ignore))
(truly-the fixnum
(+ ,digits (1- ,digits) ,bias exp))))
(t
,digits))))
(number-dispatch ((f float))
((single-float)
(frob sb!vm:single-float-digits sb!vm:single-float-bias
integer-decode-single-denorm))
((double-float)
(frob sb!vm:double-float-digits sb!vm:double-float-bias
integer-decode-double-denorm))
#!+long-float
((long-float)
(frob sb!vm:long-float-digits sb!vm:long-float-bias
integer-decode-long-denorm)))))
(defun float-sign (float1 &optional (float2 (float 1 float1)))
#!+sb-doc
"Return a floating-point number that has the same sign as
FLOAT1 and, if FLOAT2 is given, has the same absolute value
as FLOAT2."
(declare (float float1 float2))
(* (if (etypecase float1
(single-float (minusp (single-float-bits float1)))
(double-float (minusp (double-float-high-bits float1)))
#!+long-float
(long-float (minusp (long-float-exp-bits float1))))
(float -1 float1)
(float 1 float1))
(abs float2)))
(defun float-format-digits (format)
(ecase format
((short-float single-float) sb!vm:single-float-digits)
((double-float #!-long-float long-float) sb!vm:double-float-digits)
#!+long-float
(long-float sb!vm:long-float-digits)))
#!-sb-fluid (declaim (inline float-digits float-radix))
(defun float-digits (f)
(number-dispatch ((f float))
((single-float) sb!vm:single-float-digits)
((double-float) sb!vm:double-float-digits)
#!+long-float
((long-float) sb!vm:long-float-digits)))
(defun float-radix (x)
#!+sb-doc
"Return (as an integer) the radix b of its floating-point argument."
(declare (ignore x) (type float x))
2)
;;;; INTEGER-DECODE-FLOAT and DECODE-FLOAT
#!-sb-fluid
(declaim (maybe-inline integer-decode-single-float
integer-decode-double-float))
;;; Handle the denormalized case of INTEGER-DECODE-FLOAT for SINGLE-FLOAT.
(defun integer-decode-single-denorm (x)
(declare (type single-float x))
(let* ((bits (single-float-bits (abs x)))
(sig (ash (ldb sb!vm:single-float-significand-byte bits) 1))
(extra-bias 0))
(declare (type (unsigned-byte 24) sig)
(type (integer 0 23) extra-bias))
(loop
(unless (zerop (logand sig sb!vm:single-float-hidden-bit))
(return))
(setq sig (ash sig 1))
(incf extra-bias))
(values sig
(- (- sb!vm:single-float-bias)
sb!vm:single-float-digits
extra-bias)
(if (minusp (float-sign x)) -1 1))))
;;; Handle the single-float case of INTEGER-DECODE-FLOAT. If an infinity or
;;; NaN, error. If a denorm, call i-d-s-DENORM to handle it.
(defun integer-decode-single-float (x)
(declare (single-float x))
(let* ((bits (single-float-bits (abs x)))
(exp (ldb sb!vm:single-float-exponent-byte bits))
(sig (ldb sb!vm:single-float-significand-byte bits))
(sign (if (minusp (float-sign x)) -1 1))
(biased (- exp sb!vm:single-float-bias sb!vm:single-float-digits)))
(declare (fixnum biased))
(unless (<= exp sb!vm:single-float-normal-exponent-max)
(error "can't decode NaN or infinity: ~S" x))
(cond ((and (zerop exp) (zerop sig))
(values 0 biased sign))
((< exp sb!vm:single-float-normal-exponent-min)
(integer-decode-single-denorm x))
(t
(values (logior sig sb!vm:single-float-hidden-bit) biased sign)))))
;;; like INTEGER-DECODE-SINGLE-DENORM, only doubly so
(defun integer-decode-double-denorm (x)
(declare (type double-float x))
(let* ((high-bits (double-float-high-bits (abs x)))
(sig-high (ldb sb!vm:double-float-significand-byte high-bits))
(low-bits (double-float-low-bits x))
(sign (if (minusp (float-sign x)) -1 1))
(biased (- (- sb!vm:double-float-bias) sb!vm:double-float-digits)))
(if (zerop sig-high)
(let ((sig low-bits)
(extra-bias (- sb!vm:double-float-digits 33))
(bit (ash 1 31)))
(declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
(loop
(unless (zerop (logand sig bit)) (return))
(setq sig (ash sig 1))
(incf extra-bias))
(values (ash sig (- sb!vm:double-float-digits 32))
(truly-the fixnum (- biased extra-bias))
sign))
(let ((sig (ash sig-high 1))
(extra-bias 0))
(declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
(loop
(unless (zerop (logand sig sb!vm:double-float-hidden-bit))
(return))
(setq sig (ash sig 1))
(incf extra-bias))
(values (logior (ash sig 32) (ash low-bits (1- extra-bias)))
(truly-the fixnum (- biased extra-bias))
sign)))))
;;; like INTEGER-DECODE-SINGLE-FLOAT, only doubly so
(defun integer-decode-double-float (x)
(declare (double-float x))
(let* ((abs (abs x))
(hi (double-float-high-bits abs))
(lo (double-float-low-bits abs))
(exp (ldb sb!vm:double-float-exponent-byte hi))
(sig (ldb sb!vm:double-float-significand-byte hi))
(sign (if (minusp (float-sign x)) -1 1))
(biased (- exp sb!vm:double-float-bias sb!vm:double-float-digits)))
(declare (fixnum biased))
(unless (<= exp sb!vm:double-float-normal-exponent-max)
(error "Can't decode NaN or infinity: ~S." x))
(cond ((and (zerop exp) (zerop sig) (zerop lo))
(values 0 biased sign))
((< exp sb!vm:double-float-normal-exponent-min)
(integer-decode-double-denorm x))
(t
(values
(logior (ash (logior (ldb sb!vm:double-float-significand-byte hi)
sb!vm:double-float-hidden-bit)
32)
lo)
biased sign)))))
#!+(and long-float x86)
(defun integer-decode-long-denorm (x)
(declare (type long-float x))
(let* ((high-bits (long-float-high-bits (abs x)))
(sig-high (ldb sb!vm:long-float-significand-byte high-bits))
(low-bits (long-float-low-bits x))
(sign (if (minusp (float-sign x)) -1 1))
(biased (- (- sb!vm:long-float-bias) sb!vm:long-float-digits)))
(if (zerop sig-high)
(let ((sig low-bits)
(extra-bias (- sb!vm:long-float-digits 33))
(bit (ash 1 31)))
(declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
(loop
(unless (zerop (logand sig bit)) (return))
(setq sig (ash sig 1))
(incf extra-bias))
(values (ash sig (- sb!vm:long-float-digits 32))
(truly-the fixnum (- biased extra-bias))
sign))
(let ((sig (ash sig-high 1))
(extra-bias 0))
(declare (type (unsigned-byte 32) sig) (fixnum extra-bias))
(loop
(unless (zerop (logand sig sb!vm:long-float-hidden-bit))
(return))
(setq sig (ash sig 1))
(incf extra-bias))
(values (logior (ash sig 32) (ash low-bits (1- extra-bias)))
(truly-the fixnum (- biased extra-bias))
sign)))))
#!+(and long-float x86)
(defun integer-decode-long-float (x)
(declare (long-float x))
(let* ((hi (long-float-high-bits x))
(lo (long-float-low-bits x))
(exp-bits (long-float-exp-bits x))
(exp (ldb sb!vm:long-float-exponent-byte exp-bits))
(sign (if (minusp exp-bits) -1 1))
(biased (- exp sb!vm:long-float-bias sb!vm:long-float-digits)))
(declare (fixnum biased))
(unless (<= exp sb!vm:long-float-normal-exponent-max)
(error "can't decode NaN or infinity: ~S" x))
(cond ((and (zerop exp) (zerop hi) (zerop lo))
(values 0 biased sign))
((< exp sb!vm:long-float-normal-exponent-min)
(integer-decode-long-denorm x))
(t
(values (logior (ash hi 32) lo) biased sign)))))
;;; Dispatch to the correct type-specific i-d-f function.
(defun integer-decode-float (x)
#!+sb-doc
"Return three values:
1) an integer representation of the significand.
2) the exponent for the power of 2 that the significand must be multiplied
by to get the actual value. This differs from the DECODE-FLOAT exponent
by FLOAT-DIGITS, since the significand has been scaled to have all its
digits before the radix point.
3) -1 or 1 (i.e. the sign of the argument.)"
(number-dispatch ((x float))
((single-float)
(integer-decode-single-float x))
((double-float)
(integer-decode-double-float x))
#!+long-float
((long-float)
(integer-decode-long-float x))))
#!-sb-fluid (declaim (maybe-inline decode-single-float decode-double-float))
;;; Handle the denormalized case of DECODE-SINGLE-FLOAT. We call
;;; INTEGER-DECODE-SINGLE-DENORM and then make the result into a float.
(defun decode-single-denorm (x)
(declare (type single-float x))
(multiple-value-bind (sig exp sign) (integer-decode-single-denorm x)
(values (make-single-float
(dpb sig sb!vm:single-float-significand-byte
(dpb sb!vm:single-float-bias
sb!vm:single-float-exponent-byte
0)))
(truly-the fixnum (+ exp sb!vm:single-float-digits))
(float sign x))))
;;; Handle the single-float case of DECODE-FLOAT. If an infinity or NaN,
;;; error. If a denorm, call d-s-DENORM to handle it.
(defun decode-single-float (x)
(declare (single-float x))
(let* ((bits (single-float-bits (abs x)))
(exp (ldb sb!vm:single-float-exponent-byte bits))
(sign (float-sign x))
(biased (truly-the single-float-exponent
(- exp sb!vm:single-float-bias))))
(unless (<= exp sb!vm:single-float-normal-exponent-max)
(error "can't decode NaN or infinity: ~S" x))
(cond ((zerop x)
(values 0.0f0 biased sign))
((< exp sb!vm:single-float-normal-exponent-min)
(decode-single-denorm x))
(t
(values (make-single-float
(dpb sb!vm:single-float-bias
sb!vm:single-float-exponent-byte
bits))
biased sign)))))
;;; like DECODE-SINGLE-DENORM, only doubly so
(defun decode-double-denorm (x)
(declare (double-float x))
(multiple-value-bind (sig exp sign) (integer-decode-double-denorm x)
(values (make-double-float
(dpb (logand (ash sig -32) (lognot sb!vm:double-float-hidden-bit))
sb!vm:double-float-significand-byte
(dpb sb!vm:double-float-bias
sb!vm:double-float-exponent-byte 0))
(ldb (byte 32 0) sig))
(truly-the fixnum (+ exp sb!vm:double-float-digits))
(float sign x))))
;;; like DECODE-SINGLE-FLOAT, only doubly so
(defun decode-double-float (x)
(declare (double-float x))
(let* ((abs (abs x))
(hi (double-float-high-bits abs))
(lo (double-float-low-bits abs))
(exp (ldb sb!vm:double-float-exponent-byte hi))
(sign (float-sign x))
(biased (truly-the double-float-exponent
(- exp sb!vm:double-float-bias))))
(unless (<= exp sb!vm:double-float-normal-exponent-max)
(error "can't decode NaN or infinity: ~S" x))
(cond ((zerop x)
(values 0.0d0 biased sign))
((< exp sb!vm:double-float-normal-exponent-min)
(decode-double-denorm x))
(t
(values (make-double-float
(dpb sb!vm:double-float-bias
sb!vm:double-float-exponent-byte hi)
lo)
biased sign)))))
#!+(and long-float x86)
(defun decode-long-denorm (x)
(declare (long-float x))
(multiple-value-bind (sig exp sign) (integer-decode-long-denorm x)
(values (make-long-float sb!vm:long-float-bias (ash sig -32)
(ldb (byte 32 0) sig))
(truly-the fixnum (+ exp sb!vm:long-float-digits))
(float sign x))))
#!+(and long-float x86)
(defun decode-long-float (x)
(declare (long-float x))
(let* ((hi (long-float-high-bits x))
(lo (long-float-low-bits x))
(exp-bits (long-float-exp-bits x))
(exp (ldb sb!vm:long-float-exponent-byte exp-bits))
(sign (if (minusp exp-bits) -1l0 1l0))
(biased (truly-the long-float-exponent
(- exp sb!vm:long-float-bias))))
(unless (<= exp sb!vm:long-float-normal-exponent-max)
(error "can't decode NaN or infinity: ~S" x))
(cond ((zerop x)
(values 0.0l0 biased sign))
((< exp sb!vm:long-float-normal-exponent-min)
(decode-long-denorm x))
(t
(values (make-long-float
(dpb sb!vm:long-float-bias sb!vm:long-float-exponent-byte
exp-bits)
hi
lo)
biased sign)))))
;;; Dispatch to the appropriate type-specific function.
(defun decode-float (f)
#!+sb-doc
"Return three values:
1) a floating-point number representing the significand. This is always
between 0.5 (inclusive) and 1.0 (exclusive).
2) an integer representing the exponent.
3) -1.0 or 1.0 (i.e. the sign of the argument.)"
(number-dispatch ((f float))
((single-float)
(decode-single-float f))
((double-float)
(decode-double-float f))
#!+long-float
((long-float)
(decode-long-float f))))
;;;; SCALE-FLOAT
#!-sb-fluid (declaim (maybe-inline scale-single-float scale-double-float))
;;; Handle float scaling where the X is denormalized or the result is
;;; denormalized or underflows to 0.
(defun scale-float-maybe-underflow (x exp)
(multiple-value-bind (sig old-exp) (integer-decode-float x)
(let* ((digits (float-digits x))
(new-exp (+ exp old-exp digits
(etypecase x
(single-float sb!vm:single-float-bias)
(double-float sb!vm:double-float-bias))))
(sign (if (minusp (float-sign x)) 1 0)))
(cond
((< new-exp
(etypecase x
(single-float sb!vm:single-float-normal-exponent-min)
(double-float sb!vm:double-float-normal-exponent-min)))
(when (sb!vm:current-float-trap :inexact)
(error 'floating-point-inexact :operation 'scale-float
:operands (list x exp)))
(when (sb!vm:current-float-trap :underflow)
(error 'floating-point-underflow :operation 'scale-float
:operands (list x exp)))
(let ((shift (1- new-exp)))
(if (< shift (- (1- digits)))
(float-sign x 0.0)
(etypecase x
(single-float (single-from-bits sign 0 (ash sig shift)))
(double-float (double-from-bits sign 0 (ash sig shift)))))))
(t
(etypecase x
(single-float (single-from-bits sign new-exp sig))
(double-float (double-from-bits sign new-exp sig))))))))
;;; Called when scaling a float overflows, or the original float was a
;;; NaN or infinity. If overflow errors are trapped, then error,
;;; otherwise return the appropriate infinity. If a NaN, signal or not
;;; as appropriate.
(defun scale-float-maybe-overflow (x exp)
(cond
((float-infinity-p x)
;; Infinity is infinity, no matter how small...
x)
((float-nan-p x)
(when (and (float-trapping-nan-p x)
(sb!vm:current-float-trap :invalid))
(error 'floating-point-invalid-operation :operation 'scale-float
:operands (list x exp)))
x)
(t
(when (sb!vm:current-float-trap :overflow)
(error 'floating-point-overflow :operation 'scale-float
:operands (list x exp)))
(when (sb!vm:current-float-trap :inexact)
(error 'floating-point-inexact :operation 'scale-float
:operands (list x exp)))
(* (float-sign x)
(etypecase x
(single-float
;; SINGLE-FLOAT-POSITIVE-INFINITY
(single-from-bits 0 (1+ sb!vm:single-float-normal-exponent-max) 0))
(double-float
;; DOUBLE-FLOAT-POSITIVE-INFINITY
(double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0)))))))
;;; Scale a single or double float, calling the correct over/underflow
;;; functions.
(defun scale-single-float (x exp)
(declare (single-float x) (integer exp))
(etypecase exp
(fixnum
(let* ((bits (single-float-bits x))
(old-exp (ldb sb!vm:single-float-exponent-byte bits))
(new-exp (+ old-exp exp)))
(cond
((zerop x) x)
((or (< old-exp sb!vm:single-float-normal-exponent-min)
(< new-exp sb!vm:single-float-normal-exponent-min))
(scale-float-maybe-underflow x exp))
((or (> old-exp sb!vm:single-float-normal-exponent-max)
(> new-exp sb!vm:single-float-normal-exponent-max))
(scale-float-maybe-overflow x exp))
(t
(make-single-float (dpb new-exp
sb!vm:single-float-exponent-byte
bits))))))
(unsigned-byte (scale-float-maybe-overflow x exp))
((integer * 0) (scale-float-maybe-underflow x exp))))
(defun scale-double-float (x exp)
(declare (double-float x) (integer exp))
(etypecase exp
(fixnum
(let* ((hi (double-float-high-bits x))
(lo (double-float-low-bits x))
(old-exp (ldb sb!vm:double-float-exponent-byte hi))
(new-exp (+ old-exp exp)))
(cond
((zerop x) x)
((or (< old-exp sb!vm:double-float-normal-exponent-min)
(< new-exp sb!vm:double-float-normal-exponent-min))
(scale-float-maybe-underflow x exp))
((or (> old-exp sb!vm:double-float-normal-exponent-max)
(> new-exp sb!vm:double-float-normal-exponent-max))
(scale-float-maybe-overflow x exp))
(t
(make-double-float (dpb new-exp sb!vm:double-float-exponent-byte hi)
lo)))))
(unsigned-byte (scale-float-maybe-overflow x exp))
((integer * 0) (scale-float-maybe-underflow x exp))))
#!+(and x86 long-float)
(defun scale-long-float (x exp)
(declare (long-float x) (integer exp))
(scale-float x exp))
;;; Dispatch to the correct type-specific scale-float function.
(defun scale-float (f ex)
#!+sb-doc
"Return the value (* f (expt (float 2 f) ex)), but with no unnecessary loss
of precision or overflow."
(number-dispatch ((f float))
((single-float)
(scale-single-float f ex))
((double-float)
(scale-double-float f ex))
#!+long-float
((long-float)
(scale-long-float f ex))))
;;;; converting to/from floats
(defun float (number &optional (other () otherp))
#!+sb-doc
"Converts any REAL to a float. If OTHER is not provided, it returns a
SINGLE-FLOAT if NUMBER is not already a FLOAT. If OTHER is provided, the
result is the same float format as OTHER."
(if otherp
(number-dispatch ((number real) (other float))
(((foreach rational single-float double-float #!+long-float long-float)
(foreach single-float double-float #!+long-float long-float))
(coerce number '(dispatch-type other))))
(if (floatp number)
number
(coerce number 'single-float))))
(macrolet ((frob (name type)
`(defun ,name (x)
(number-dispatch ((x real))
(((foreach single-float double-float #!+long-float long-float
fixnum))
(coerce x ',type))
((bignum)
(bignum-to-float x ',type))
((ratio)
(float-ratio x ',type))))))
(frob %single-float single-float)
(frob %double-float double-float)
#!+long-float
(frob %long-float long-float))
;;; Convert a ratio to a float. We avoid any rounding error by doing an
;;; integer division. Accuracy is important to preserve print-read
;;; consistency, since this is ultimately how the reader reads a float. We
;;; scale the numerator by a power of two until the division results in the
;;; desired number of fraction bits, then do round-to-nearest.
(defun float-ratio (x format)
(let* ((signed-num (numerator x))
(plusp (plusp signed-num))
(num (if plusp signed-num (- signed-num)))
(den (denominator x))
(digits (float-format-digits format))
(scale 0))
(declare (fixnum digits scale))
;; Strip any trailing zeros from the denominator and move it into the scale
;; factor (to minimize the size of the operands.)
(let ((den-twos (1- (integer-length (logxor den (1- den))))))
(declare (fixnum den-twos))
(decf scale den-twos)
(setq den (ash den (- den-twos))))
;; Guess how much we need to scale by from the magnitudes of the numerator
;; and denominator. We want one extra bit for a guard bit.
(let* ((num-len (integer-length num))
(den-len (integer-length den))
(delta (- den-len num-len))
(shift (1+ (the fixnum (+ delta digits))))
(shifted-num (ash num shift)))
(declare (fixnum delta shift))
(decf scale delta)
(labels ((float-and-scale (bits)
(let* ((bits (ash bits -1))
(len (integer-length bits)))
(cond ((> len digits)
(aver (= len (the fixnum (1+ digits))))
(scale-float (floatit (ash bits -1)) (1+ scale)))
(t
(scale-float (floatit bits) scale)))))
(floatit (bits)
(let ((sign (if plusp 0 1)))
(case format
(single-float
(single-from-bits sign sb!vm:single-float-bias bits))
(double-float
(double-from-bits sign sb!vm:double-float-bias bits))
#!+long-float
(long-float
(long-from-bits sign sb!vm:long-float-bias bits))))))
(loop
(multiple-value-bind (fraction-and-guard rem)
(truncate shifted-num den)
(let ((extra (- (integer-length fraction-and-guard) digits)))
(declare (fixnum extra))
(cond ((/= extra 1)
(aver (> extra 1)))
((oddp fraction-and-guard)
(return
(if (zerop rem)
(float-and-scale
(if (zerop (logand fraction-and-guard 2))
fraction-and-guard
(1+ fraction-and-guard)))
(float-and-scale (1+ fraction-and-guard)))))
(t
(return (float-and-scale fraction-and-guard)))))
(setq shifted-num (ash shifted-num -1))
(incf scale)))))))
;;; These might be useful if we ever have a machine without float/integer
;;; conversion hardware. For now, we'll use special ops that
;;; uninterruptibly frob the rounding modes & do ieee round-to-integer.
#+nil
(progn
;; The compiler compiles a call to this when we are doing %UNARY-TRUNCATE
;; and the result is known to be a fixnum. We can avoid some generic
;; arithmetic in this case.
(defun %unary-truncate-single-float/fixnum (x)
(declare (single-float x) (values fixnum))
(locally (declare (optimize (speed 3) (safety 0)))
(let* ((bits (single-float-bits x))
(exp (ldb sb!vm:single-float-exponent-byte bits))
(frac (logior (ldb sb!vm:single-float-significand-byte bits)
sb!vm:single-float-hidden-bit))
(shift (- exp sb!vm:single-float-digits sb!vm:single-float-bias)))
(when (> exp sb!vm:single-float-normal-exponent-max)
(error 'floating-point-invalid-operation :operator 'truncate
:operands (list x)))
(if (<= shift (- sb!vm:single-float-digits))
0
(let ((res (ash frac shift)))
(declare (type (unsigned-byte 31) res))
(if (minusp bits)
(- res)
res))))))
;; Double-float version of this operation (see above single op).
(defun %unary-truncate-double-float/fixnum (x)
(declare (double-float x) (values fixnum))
(locally (declare (optimize (speed 3) (safety 0)))
(let* ((hi-bits (double-float-high-bits x))
(exp (ldb sb!vm:double-float-exponent-byte hi-bits))
(frac (logior (ldb sb!vm:double-float-significand-byte hi-bits)
sb!vm:double-float-hidden-bit))
(shift (- exp (- sb!vm:double-float-digits sb!vm:n-word-bits)
sb!vm:double-float-bias)))
(when (> exp sb!vm:double-float-normal-exponent-max)
(error 'floating-point-invalid-operation :operator 'truncate
:operands (list x)))
(if (<= shift (- sb!vm:n-word-bits sb!vm:double-float-digits))
0
(let* ((res-hi (ash frac shift))
(res (if (plusp shift)
(logior res-hi
(the fixnum
(ash (double-float-low-bits x)
(- shift sb!vm:n-word-bits))))
res-hi)))
(declare (type (unsigned-byte 31) res-hi res))
(if (minusp hi-bits)
(- res)
res)))))))
;;; This function is called when we are doing a truncate without any funky
;;; divisor, i.e. converting a float or ratio to an integer. Note that we do
;;; *not* return the second value of truncate, so it must be computed by the
;;; caller if needed.
;;;
;;; In the float case, we pick off small arguments so that compiler
;;; can use special-case operations. We use an exclusive test, since
;;; (due to round-off error), (float most-positive-fixnum) is likely
;;; to be equal to (1+ most-positive-fixnum). An exclusive test is
;;; good enough, because most-positive-fixnum will be one less than a
;;; power of two, and that power of two will be exactly representable
;;; as a float (at least until we get 128-bit fixnums).
(defun %unary-truncate (number)
(number-dispatch ((number real))
((integer) number)
((ratio) (values (truncate (numerator number) (denominator number))))
(((foreach single-float double-float #!+long-float long-float))
(if (< (float most-negative-fixnum number)
number
(float most-positive-fixnum number))
(truly-the fixnum (%unary-truncate number))
(multiple-value-bind (bits exp) (integer-decode-float number)
(let ((res (ash bits exp)))
(if (minusp number)
(- res)
res)))))))
;;; Specialized versions for floats.
(macrolet ((def (type name)
`(defun ,name (number)
(if (< ,(coerce sb!xc:most-negative-fixnum type)
number
,(coerce sb!xc:most-positive-fixnum type))
(truly-the fixnum (,name number))
;; General -- slow -- case.
(multiple-value-bind (bits exp) (integer-decode-float number)
(let ((res (ash bits exp)))
(if (minusp number)
(- res)
res)))))))
(def single-float %unary-truncate/single-float)
(def double-float %unary-truncate/double-float)
#!+long-float
(def double-float %unary-truncate/long-float))
;;; Similar to %UNARY-TRUNCATE, but rounds to the nearest integer. If we
;;; can't use the round primitive, then we do our own round-to-nearest on the
;;; result of i-d-f. [Note that this rounding will really only happen with
;;; double floats, since the whole single-float fraction will fit in a fixnum,
;;; so all single-floats larger than most-positive-fixnum can be precisely
;;; represented by an integer.]
(defun %unary-round (number)
(number-dispatch ((number real))
((integer) number)
((ratio) (values (round (numerator number) (denominator number))))
(((foreach single-float double-float #!+long-float long-float))
(if (< (float most-negative-fixnum number)
number
(float most-positive-fixnum number))
(truly-the fixnum (%unary-round number))
(multiple-value-bind (bits exp) (integer-decode-float number)
(let* ((shifted (ash bits exp))
(rounded (if (minusp exp)
(let ((fractional-bits (logand bits (lognot (ash -1 (- exp)))))
(0.5bits (ash 1 (- -1 exp))))
(cond
((> fractional-bits 0.5bits) (1+ shifted))
((< fractional-bits 0.5bits) shifted)
(t (if (oddp shifted) (1+ shifted) shifted))))
shifted)))
(if (minusp number)
(- rounded)
rounded)))))))
(defun %unary-ftruncate (number)
(number-dispatch ((number real))
((integer) (float number))
((ratio) (float (truncate (numerator number) (denominator number))))
(((foreach single-float double-float #!+long-float long-float))
(%unary-ftruncate number))))
(defun rational (x)
#!+sb-doc
"RATIONAL produces a rational number for any real numeric argument. This is
more efficient than RATIONALIZE, but it assumes that floating-point is
completely accurate, giving a result that isn't as pretty."
(number-dispatch ((x real))
(((foreach single-float double-float #!+long-float long-float))
(multiple-value-bind (bits exp) (integer-decode-float x)
(if (eql bits 0)
0
(let* ((int (if (minusp x) (- bits) bits))
(digits (float-digits x))
(ex (+ exp digits)))
(if (minusp ex)
(integer-/-integer int (ash 1 (+ digits (- ex))))
(integer-/-integer (ash int ex) (ash 1 digits)))))))
((rational) x)))
;;; This algorithm for RATIONALIZE, due to Bruno Haible, is included
;;; with permission.
;;;
;;; Algorithm (recursively presented):
;;; If x is a rational number, return x.
;;; If x = 0.0, return 0.
;;; If x < 0.0, return (- (rationalize (- x))).
;;; If x > 0.0:
;;; Call (integer-decode-float x). It returns a m,e,s=1 (mantissa,
;;; exponent, sign).
;;; If m = 0 or e >= 0: return x = m*2^e.
;;; Search a rational number between a = (m-1/2)*2^e and b = (m+1/2)*2^e
;;; with smallest possible numerator and denominator.
;;; Note 1: If m is a power of 2, we ought to take a = (m-1/4)*2^e.
;;; But in this case the result will be x itself anyway, regardless of
;;; the choice of a. Therefore we can simply ignore this case.
;;; Note 2: At first, we need to consider the closed interval [a,b].
;;; but since a and b have the denominator 2^(|e|+1) whereas x itself
;;; has a denominator <= 2^|e|, we can restrict the seach to the open
;;; interval (a,b).
;;; So, for given a and b (0 < a < b) we are searching a rational number
;;; y with a <= y <= b.
;;; Recursive algorithm fraction_between(a,b):
;;; c := (ceiling a)
;;; if c < b
;;; then return c ; because a <= c < b, c integer
;;; else
;;; ; a is not integer (otherwise we would have had c = a < b)
;;; k := c-1 ; k = floor(a), k < a < b <= k+1
;;; return y = k + 1/fraction_between(1/(b-k), 1/(a-k))
;;; ; note 1 <= 1/(b-k) < 1/(a-k)
;;;
;;; You can see that we are actually computing a continued fraction expansion.
;;;
;;; Algorithm (iterative):
;;; If x is rational, return x.
;;; Call (integer-decode-float x). It returns a m,e,s (mantissa,
;;; exponent, sign).
;;; If m = 0 or e >= 0, return m*2^e*s. (This includes the case x = 0.0.)
;;; Create rational numbers a := (2*m-1)*2^(e-1) and b := (2*m+1)*2^(e-1)
;;; (positive and already in lowest terms because the denominator is a
;;; power of two and the numerator is odd).
;;; Start a continued fraction expansion
;;; p[-1] := 0, p[0] := 1, q[-1] := 1, q[0] := 0, i := 0.
;;; Loop
;;; c := (ceiling a)
;;; if c >= b
;;; then k := c-1, partial_quotient(k), (a,b) := (1/(b-k),1/(a-k)),
;;; goto Loop
;;; finally partial_quotient(c).
;;; Here partial_quotient(c) denotes the iteration
;;; i := i+1, p[i] := c*p[i-1]+p[i-2], q[i] := c*q[i-1]+q[i-2].
;;; At the end, return s * (p[i]/q[i]).
;;; This rational number is already in lowest terms because
;;; p[i]*q[i-1]-p[i-1]*q[i] = (-1)^i.
;;;
;;; See also
;;; Hardy, Wright: An introduction to number theory
;;; and/or
;;;
;;;
(defun rationalize (x)
"Converts any REAL to a RATIONAL. Floats are converted to a simple rational
representation exploiting the assumption that floats are only accurate to
their precision. RATIONALIZE (and also RATIONAL) preserve the invariant:
(= x (float (rationalize x) x))"
(number-dispatch ((x real))
(((foreach single-float double-float #!+long-float long-float))
;; This is a fairly straigtforward implementation of the
;; iterative algorithm above.
(multiple-value-bind (frac expo sign)
(integer-decode-float x)
(cond ((or (zerop frac) (>= expo 0))
(if (minusp sign)
(- (ash frac expo))
(ash frac expo)))
(t
;; expo < 0 and (2*m-1) and (2*m+1) are coprime to 2^(1-e),
;; so build the fraction up immediately, without having to do
;; a gcd.
(let ((a (build-ratio (- (* 2 frac) 1) (ash 1 (- 1 expo))))
(b (build-ratio (+ (* 2 frac) 1) (ash 1 (- 1 expo))))
(p0 0)
(q0 1)
(p1 1)
(q1 0))
(do ((c (ceiling a) (ceiling a)))
((< c b)
(let ((top (+ (* c p1) p0))
(bot (+ (* c q1) q0)))
(build-ratio (if (minusp sign)
(- top)
top)
bot)))
(let* ((k (- c 1))
(p2 (+ (* k p1) p0))
(q2 (+ (* k q1) q0)))
(psetf a (/ (- b k))
b (/ (- a k)))
(setf p0 p1
q0 q1
p1 p2
q1 q2))))))))
((rational) x)))