-;;;; This file contains the definitions of float specific number
+;;;; 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
+;;;; 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
(in-package "SB!KERNEL")
\f
-;;;; utilities
-
-(eval-when (:compile-toplevel :load-toplevel :execute)
-
-;;; These functions let us create floats from bits with the
-;;; significand uniformly represented as an integer. This is less
-;;; efficient for double floats, but is more convenient when making
-;;; special values, etc.
-(defun single-from-bits (sign exp sig)
- (declare (type bit sign) (type (unsigned-byte 24) sig)
- (type (unsigned-byte 8) exp))
- (make-single-float
- (dpb exp sb!vm:single-float-exponent-byte
- (dpb sig sb!vm:single-float-significand-byte
- (if (zerop sign) 0 -1)))))
-(defun double-from-bits (sign exp sig)
- (declare (type bit sign) (type (unsigned-byte 53) sig)
- (type (unsigned-byte 11) exp))
- (make-double-float (dpb exp sb!vm:double-float-exponent-byte
- (dpb (ash sig -32) sb!vm:double-float-significand-byte
- (if (zerop sign) 0 -1)))
- (ldb (byte 32 0) sig)))
-#!+(and long-float x86)
-(defun long-from-bits (sign exp sig)
- (declare (type bit sign) (type (unsigned-byte 64) sig)
- (type (unsigned-byte 15) exp))
- (make-long-float (logior (ash sign 15) exp)
- (ldb (byte 32 32) sig)
- (ldb (byte 32 0) sig)))
-
-) ; EVAL-WHEN
-\f
-;;;; float parameters
-
-(defconstant least-positive-single-float (single-from-bits 0 0 1))
-(defconstant least-positive-short-float least-positive-single-float)
-(defconstant least-negative-single-float (single-from-bits 1 0 1))
-(defconstant least-negative-short-float least-negative-single-float)
-(defconstant least-positive-double-float (double-from-bits 0 0 1))
-#!-long-float
-(defconstant least-positive-long-float least-positive-double-float)
-#!+(and long-float x86)
-(defconstant least-positive-long-float (long-from-bits 0 0 1))
-(defconstant least-negative-double-float (double-from-bits 1 0 1))
-#!-long-float
-(defconstant least-negative-long-float least-negative-double-float)
-#!+(and long-float x86)
-(defconstant least-negative-long-float (long-from-bits 1 0 1))
-
-(defconstant least-positive-normalized-single-float
- (single-from-bits 0 sb!vm:single-float-normal-exponent-min 0))
-(defconstant least-positive-normalized-short-float
- least-positive-normalized-single-float)
-(defconstant least-negative-normalized-single-float
- (single-from-bits 1 sb!vm:single-float-normal-exponent-min 0))
-(defconstant least-negative-normalized-short-float
- least-negative-normalized-single-float)
-(defconstant least-positive-normalized-double-float
- (double-from-bits 0 sb!vm:double-float-normal-exponent-min 0))
-#!-long-float
-(defconstant least-positive-normalized-long-float
- least-positive-normalized-double-float)
-#!+(and long-float x86)
-(defconstant least-positive-normalized-long-float
- (long-from-bits 0 sb!vm:long-float-normal-exponent-min
- (ash sb!vm:long-float-hidden-bit 32)))
-(defconstant least-negative-normalized-double-float
- (double-from-bits 1 sb!vm:double-float-normal-exponent-min 0))
-#!-long-float
-(defconstant least-negative-normalized-long-float
- least-negative-normalized-double-float)
-#!+(and long-float x86)
-(defconstant least-negative-normalized-long-float
- (long-from-bits 1 sb!vm:long-float-normal-exponent-min
- (ash sb!vm:long-float-hidden-bit 32)))
-
-(defconstant most-positive-single-float
- (single-from-bits 0 sb!vm:single-float-normal-exponent-max
- (ldb sb!vm:single-float-significand-byte -1)))
-(defconstant most-positive-short-float most-positive-single-float)
-(defconstant most-negative-single-float
- (single-from-bits 1 sb!vm:single-float-normal-exponent-max
- (ldb sb!vm:single-float-significand-byte -1)))
-(defconstant most-negative-short-float most-negative-single-float)
-(defconstant most-positive-double-float
- (double-from-bits 0 sb!vm:double-float-normal-exponent-max
- (ldb (byte sb!vm:double-float-digits 0) -1)))
-#!-long-float
-(defconstant most-positive-long-float most-positive-double-float)
-#!+(and long-float x86)
-(defconstant most-positive-long-float
- (long-from-bits 0 sb!vm:long-float-normal-exponent-max
- (ldb (byte sb!vm:long-float-digits 0) -1)))
-(defconstant most-negative-double-float
- (double-from-bits 1 sb!vm:double-float-normal-exponent-max
- (ldb (byte sb!vm:double-float-digits 0) -1)))
-#!-long-float
-(defconstant most-negative-long-float most-negative-double-float)
-#!+(and long-float x86)
-(defconstant most-negative-long-float
- (long-from-bits 1 sb!vm:long-float-normal-exponent-max
- (ldb (byte sb!vm:long-float-digits 0) -1)))
-
-;;; We don't want to do these DEFCONSTANTs at cross-compilation time,
-;;; because the cross-compilation host might not support floating
-;;; point infinities.
-(eval-when (:load-toplevel :execute)
-(defconstant single-float-positive-infinity
- (single-from-bits 0 (1+ sb!vm:single-float-normal-exponent-max) 0))
-(defconstant short-float-positive-infinity single-float-positive-infinity)
-(defconstant single-float-negative-infinity
- (single-from-bits 1 (1+ sb!vm:single-float-normal-exponent-max) 0))
-(defconstant short-float-negative-infinity single-float-negative-infinity)
-(defconstant double-float-positive-infinity
- (double-from-bits 0 (1+ sb!vm:double-float-normal-exponent-max) 0))
-#!+(not long-float)
-(defconstant long-float-positive-infinity double-float-positive-infinity)
-#!+(and long-float x86)
-(defconstant long-float-positive-infinity
- (long-from-bits 0 (1+ sb!vm:long-float-normal-exponent-max)
- (ash sb!vm:long-float-hidden-bit 32)))
-(defconstant double-float-negative-infinity
- (double-from-bits 1 (1+ sb!vm:double-float-normal-exponent-max) 0))
-#!+(not long-float)
-(defconstant long-float-negative-infinity double-float-negative-infinity)
-#!+(and long-float x86)
-(defconstant long-float-negative-infinity
- (long-from-bits 1 (1+ sb!vm:long-float-normal-exponent-max)
- (ash sb!vm:long-float-hidden-bit 32)))
-) ; EVAL-WHEN
-
-(defconstant single-float-epsilon
- (single-from-bits 0 (- sb!vm:single-float-bias
- (1- sb!vm:single-float-digits)) 1))
-(defconstant short-float-epsilon single-float-epsilon)
-(defconstant single-float-negative-epsilon
- (single-from-bits 0 (- sb!vm:single-float-bias sb!vm:single-float-digits) 1))
-(defconstant short-float-negative-epsilon single-float-negative-epsilon)
-(defconstant double-float-epsilon
- (double-from-bits 0 (- sb!vm:double-float-bias
- (1- sb!vm:double-float-digits)) 1))
-#!-long-float
-(defconstant long-float-epsilon double-float-epsilon)
-#!+(and long-float x86)
-(defconstant long-float-epsilon
- (long-from-bits 0 (- sb!vm:long-float-bias (1- sb!vm:long-float-digits))
- (+ 1 (ash sb!vm:long-float-hidden-bit 32))))
-(defconstant double-float-negative-epsilon
- (double-from-bits 0 (- sb!vm:double-float-bias sb!vm:double-float-digits) 1))
-#!-long-float
-(defconstant long-float-negative-epsilon double-float-negative-epsilon)
-#!+(and long-float x86)
-(defconstant long-float-negative-epsilon
- (long-from-bits 0 (- sb!vm:long-float-bias sb!vm:long-float-digits)
- (+ 1 (ash sb!vm:long-float-hidden-bit 32))))
-\f
;;;; float predicates and environment query
#!-sb-fluid
(and (zerop (ldb sb!vm:long-float-exponent-byte (long-float-exp-bits x)))
(not (zerop x))))))
-(macrolet ((def-frob (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)))))))
-
- (def-frob 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)))
-
- (def-frob float-nan-p
- "Return true if the float X is a NaN (Not a Number)."
- (not (zerop (ldb sb!vm:single-float-significand-byte bits)))
- (or (not (zerop (ldb sb!vm:double-float-significand-byte hi)))
- (not (zerop lo)))
- #!+(and long-float x86)
- (or (not (zerop (ldb sb!vm:long-float-significand-byte hi)))
- (not (zerop lo))))
-
- (def-frob float-trapping-nan-p
- "Return true if the float X is a trapping NaN (Not a Number)."
- (zerop (logand (ldb sb!vm:single-float-significand-byte bits)
- sb!vm:single-float-trapping-nan-bit))
- (zerop (logand (ldb sb!vm:double-float-significand-byte hi)
- sb!vm:double-float-trapping-nan-bit))
- #!+(and long-float x86)
- (zerop (logand (ldb sb!vm:long-float-significand-byte hi)
- sb!vm:long-float-trapping-nan-bit))))
+(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)."
+ (not (zerop (ldb sb!vm:single-float-significand-byte bits)))
+ (or (not (zerop (ldb sb!vm:double-float-significand-byte hi)))
+ (not (zerop lo)))
+ #!+(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)."
+ (zerop (logand (ldb sb!vm:single-float-significand-byte bits)
+ sb!vm:single-float-trapping-nan-bit))
+ (zerop (logand (ldb sb!vm:double-float-significand-byte hi)
+ sb!vm:double-float-trapping-nan-bit))
+ #!+(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
#!-sb-fluid (declaim (maybe-inline float-precision))
(defun float-precision (f)
#!+sb-doc
- "Returns a non-negative number of significant digits in its float argument.
+ "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)
(defun float-sign (float1 &optional (float2 (float 1 float1)))
#!+sb-doc
- "Returns a floating-point number that has the same sign as
- float1 and, if float2 is given, has the same absolute value
- as float2."
+ "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)))
(defun float-radix (x)
#!+sb-doc
"Return (as an integer) the radix b of its floating-point argument."
- (declare (type float x))
- ;; ANSI says this function "should signal an error if [..] argument
- ;; is not a float". Since X is otherwise ignored, Python doesn't
- ;; check the type by default, so we have to do it ourself:
- (unless (floatp x)
- (error 'type-error :datum x :expected-type 'float))
+ (declare (ignore x))
2)
\f
;;;; INTEGER-DECODE-FLOAT and DECODE-FLOAT
(t
(values (logior sig sb!vm:single-float-hidden-bit) biased sign)))))
-;;; Like INTEGER-DECODE-SINGLE-DENORM, only doubly so.
+;;; 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)))
(truly-the fixnum (- biased extra-bias))
sign)))))
-;;; Like INTEGER-DECODE-SINGLE-FLOAT, only doubly so.
+;;; like INTEGER-DECODE-SINGLE-FLOAT, only doubly so
(defun integer-decode-double-float (x)
(declare (double-float x))
(let* ((abs (abs x))
;;; Dispatch to the correct type-specific i-d-f function.
(defun integer-decode-float (x)
#!+sb-doc
- "Returns three values:
+ "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
bits))
biased sign)))))
-;;; Like DECODE-SINGLE-DENORM, only doubly so.
+;;; 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)
(truly-the fixnum (+ exp sb!vm:double-float-digits))
(float sign x))))
-;;; Like DECODE-SINGLE-FLOAT, only doubly so.
+;;; like DECODE-SINGLE-FLOAT, only doubly so
(defun decode-double-float (x)
(declare (double-float x))
(let* ((abs (abs x))
;;; Dispatch to the appropriate type-specific function.
(defun decode-float (f)
#!+sb-doc
- "Returns three values:
+ "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.
:operands (list x exp)))
(* (float-sign x)
(etypecase x
- (single-float single-float-positive-infinity)
- (double-float double-float-positive-infinity))))))
+ (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.
;;; Dispatch to the correct type-specific scale-float function.
(defun scale-float (f ex)
#!+sb-doc
- "Returns the value (* f (expt (float 2 f) ex)), but with no unnecessary loss
+ "Return the value (* f (expt (float 2 f) ex)), but with no unnecessary loss
of precision or overflow."
(number-dispatch ((f float))
((single-float)
(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:word-bits)
+ (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:word-bits sb!vm:double-float-digits))
+ (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:word-bits))))
+ (- shift sb!vm:n-word-bits))))
res-hi)))
(declare (type (unsigned-byte 31) res-hi res))
(if (minusp hi-bits)
(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
+;;; <http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture17/lecture17/>
+;;; <http://modular.fas.harvard.edu/edu/Fall2001/124/lectures/lecture17/lecture18/>
+
(defun rationalize (x)
- #!+sb-doc
- "Converts any REAL to a RATIONAL. Floats are converted to a simple rational
+ "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:
+ 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))
- ;; Thanks to Kim Fateman, who stole this function rationalize-float from
- ;; macsyma's rational. Macsyma'a rationalize was written by the legendary
- ;; Gosper (rwg). Guy Steele said about Gosper, "He has been called the
- ;; only living 17th century mathematician and is also the best pdp-10
- ;; hacker I know." So, if you can understand or debug this code you win
- ;; big.
- (cond ((minusp x) (- (rationalize (- x))))
- ((zerop x) 0)
- (t
- (let ((eps (etypecase x
- (single-float single-float-epsilon)
- (double-float double-float-epsilon)
- #!+long-float
- (long-float long-float-epsilon)))
- (y ())
- (a ()))
- (do ((xx x (setq y (/ (float 1.0 x) (- xx (float a x)))))
- (num (setq a (truncate x))
- (+ (* (setq a (truncate y)) num) onum))
- (den 1 (+ (* a den) oden))
- (onum 1 num)
- (oden 0 den))
- ((and (not (zerop den))
- (not (> (abs (/ (- x (/ (float num x)
- (float den x)))
- x))
- eps)))
- (integer-/-integer num den))
- (declare ((dispatch-type x) xx)))))))
+ ;; 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)))
projects / sbcl.git / blobdiff