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Zero

arcfide
2012-10-05 01:12:13
17Update about Scheme 9history
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This page talks about how implementations handle various aspects of zero. Thanks to Alexey Radul for the ideas here; the research is mine.

Exact division by zero

Racket, MIT, Gambit, Chicken, Bigloo, Scheme48/scsh, Chibi, Guile, SISC, Chez, Ikarus/Vicare, Larceny, Ypsilon, Mosh, IronScheme, NexJ, STklos, Shoe, Scheme 7, BDC, XLisp, Rep, Schemik, Elk, UMB, SigScheme, SXM, Sizzle, Spark, Dfsch, Inlab, VSCM report an error.

Gauche, SCM, KSi, VX return an inexact infinity.

Kawa returns one of two exact infinities depending on the sign of the numerator. These are notated 1/0 and -1/0. They are the same in the sense of eqv?, anomalously so since they are not operationally equivalent. They are also distinct in the sense of eqv? from +inf.0 and -inf.0, but 1/0 and +inf.0 are the same in the sense of =, as are -1/0 and -inf.0.

Scheme 9 returns +nan.0.

Femtolisp return an incorrect result.

Inexact division by exact zero

Racket, Gambit, Chicken, scsh, Chibi, Guile, Shoe, Scheme 7, BDC, XLisp, Rep, Schemik, Elk, SXM, Sizzle, Spark, Dfsch, Inlab, VSCM report an error.

MIT (with floating traps ignored), Gauche, Bigloo, Scheme48, Kawa, SISC, SCM, Chez, Ikarus/Vicare, Larceny, Ypsilon, Mosh, IronScheme, NexJ, STklos, UMB, VX return an inexact infinity.

SigScheme, Dream, Oaklisp, Owl Lisp do not support inexact numbers.

Scheme 9 returns +nan.0.

KSi, Femtolisp return an incorrect result.

Division by inexact zero

Plain Chicken, scsh, Shoe, TinyScheme, XLisp, Rep, Schemik, Scheme 7, SXM, Sizzle, Dfsch, Inlab, VSCM report an error.

Racket, Gauche, MIT (with floating traps ignored), Gambit, Chicken with the numbers egg, Bigloo, Scheme48, Guile, Kawa, SISC, Chibi, SCM, Chez, SCM, Ikarus/Vicare, Larceny, Ypsilon, Mosh, IronScheme, NexJ, STklos, BDC, Elk, UMB, VX, Spark, Femtolisp return an inexact infinity.

SigScheme, Dream, Oaklisp, Owl Lisp do not support inexact numbers.

Scheme 9 returns +nan.0

KSi return an incorrect result.

Inexact multiplication by exact zero

Racket, MIT, Gambit, Chez, Ypsilon, TinyScheme, XLisp, Elk, SXM, Sizzle, Spark, Inlab return exact 0.

Gauche, Chicken, Bigloo, Scheme48/scsh, Guile, Kawa, SISC, Chibi, SCM, Ikarus/Vicare, Larceny, Mosh, IronScheme, NexJ, STklos, KSi, Shoe, Scheme 9, Scheme 7, BDC, Rep, Schemik, UMB, VX, Femtolisp, Dfsch, VSCM return inexact 0.0.

SigScheme, Dream, Oaklisp, Owl Lisp do not support inexact numbers.

Multiplying NaN by exact 0

Racket, MIT, Chez, Ypsilon, Elk, SXM, Sizzle, Spark, Inlab return exact 0.

Gauche, Gambit, Chicken, Bigloo, Scheme48/scsh, Guile, Kawa, SISC, Chibi, SCM, Vicare, Larceny, IronScheme, NexJ, STklos, KSi, Shoe, TinyScheme, RScheme, Scheme 7, BDC, XLisp, Rep, Schemik, UMB, VX, Llava, FemtoLisp, Dfsch return +nan.0.

Scheme 9 does not have any form of NaN.

SigScheme, Dream, Oaklisp, Owl Lisp do not support inexact numbers.

Dividing exact zero by an inexact number

Racket, Gambit, TinyScheme, Sizzle, Spark return exact 0.

Gauche, Chicken (with or without the numbers egg), Bigloo, Scheme48/scsh, Guile, Kawa, SISC, Chibi, Vicare, Larceny, Ypsilon, Mosh, IronScheme, STklos, KSi, Shoe, Scheme 7, BDC, XLisp, Rep, Schemik, UMB, VX, Llava, SXM, Dfsch, Inlab return inexact 0.0.

SigScheme, Dream, Oaklisp, Owl Lisp do not support inexact numbers.

Femtolisp returns the wrong answer.

Complex numbers with 0.0 imaginary part.

Gauche, MIT, Chicken with the numbers egg, Scheme48/scsh, Kawa, SISC, SCM, STklos, KSi, Scheme 7, UMB, Spark, Dfsch, VSCM consider 3.0+0.0i to be a real number.

Racket, Gambit, Guile, Chibi, Chez, Vicare, Larceny, Ypsilon, Mosh, IronScheme do not.

Plain Chicken, Bigloo, Ikarus, NexJ, SigScheme, Shoe, TinyScheme, Dream, Scheme 9, BDC, XLisp, Rep, Schemik, Elk, VX, Oaklisp, SXM, Sizzle, Femtolisp, Inlab, Owl Lisp do not implement non-real numbers.

Negative zero

I did a test of:

(let* ((minf (* 1.0e200 -1.0e200)) (mzero (/ 1.0 minf))) (list minf mzero (eqv? mzero 0.0)))

Gauche, Chicken, Bigloo, Scheme48/scsh, SISC, Chibi, SCM, Ikarus, Ypsilon, Mosh, IronScheme, NexJ, STklos, Shoe, TinyScheme, RScheme, Scheme 7, BDC, XLisp, Schemik, Elk, UMB, VX return some variant of (-inf.0 -0.0 #t).

Racket, Gambit, Guile, Kawa, Chez, Vicare, Larceny, Rep return some variant of (-inf.0 -0.0 #f).

I wasn't able to generate -0.0 on MIT, KSi, SigScheme, Scheme 9, Dream, Oaklisp, Owl Lisp.

These are the R6RS examples involving -0.0 (already accounted for verbally in the "Implementation extensions" section of R7RS):

(zero? -0.0) => #t ; -0.0 is a Scheme zero (+ 0.0 0.0) => 0.0 ; Sum is -0.0 only if all arguments are -0.0 (+ 0.0 -0.0) => 0.0 (+ -0.0 0.0) => 0.0 (+ -0.0 -0.0) => -0.0 (- 0.0) => -0.0 ; Negation flips the sign of zero (- -0.0) => 0.0 (- 0.0 -0.0) => 0.0 ; Negate all arguments but the last and then add (- -0.0 0.0) => -0.0 (- 0.0 0.0) => 0.0 (- -0.0 -0.0) => 0.0 (log -1.0+0.0i) => 0.0+3.141592653589793i ; approximately (log -1.0-0.0i) => 0.0-3.141592653589793i ; approximately (angle -1.0+0.0i) => 3.141592653589793 ; approximately (angle -1.0-0.0i) => -3.141592653589793 ; approximately

Angle of negative zero

Because of the branch cut of atan, the values of (angle 0.0) and (angle -0.0) should be 0 and pi respectively. Furthermore, by R5RS and R6RS, (atan y x) is equivalent to (angle (make-rectangular x y)) even on systems that don't support non-real numbers, so (atan 0.0 -0.0) should also be pi. On many implementations, these turn out not to be the case.

Gambit, Chicken with the numbers egg, Chez, Vicare, Larceny, IronScheme, STklos, Sizzle return pi to both angle and atan.

Gauche, plain Chicken, Guile, SISC return 0.0 to angle, pi to atan.

Scheme48 returns -0.0 to angle, pi to atan.

scsh returns an error to angle, pi to atan.

KSi returns NaN to angle, an error to atan.

Racket, Spark return pi to angle, 0.0 to atan.

SISC, Mosh, Scheme 7 return 0.0 to both expressions.

MIT does not support -0.0.

NexJ, RScheme, BDC, XLisp, Rep, Elk, Inlab do not support angle, but returns pi to atan.

Bigloo, Scheme 9, SXM do not support angle, but raise an error to atan.

VX does not support angle, but returns -0.0 to atan.

UMB does not support angle, but returns 0.0 to atan.

Shoe, TinyScheme, Schemik, Llava, FemtoLisp do not support either angle or atan.

SigScheme, Dream, Oaklisp, Owl Lisp do not support inexact numbers.

Inexact zero raised to a negative real power

This situation involves the log of 0, and R6RS and R7RS permit implementations to either raise an exception or return an arbitrary number.

Racket, Gauche, Gambit, Chicken, Bigloo, scsh, Kawa, Chibi, SCM, Chez, Ypsilon, Mosh, NexJ, STklos, XLisp, Rep, UMB, SXM, Spark, Dfsch, Inlab return inf.0.

Guile, Larceny, Scheme 9 return +nan.0.

SISC, IronScheme return 0.0.

Elk returns 0.

Sizzle returns its smallest representable integer (-232).

MIT (even ignoring floating point traps), Scheme48, KSi, Scheme 7 raise an exception.

SigScheme, Dream, Oaklisp, Owl Lisp do not support inexact numbers.

TinyScheme, Schemik, Femtolisp, BDC either don't provide expt or don't provide log.

Inexact zero raised to a complex power whose real part is negative

Racket, Gauche, Gambit, Chicken with the numbers egg, Kawa, Chibi, Guile, Ypsilon, Spark return +nan.0+nan.0i. (On some platforms, Guile returns the wrong answer: this is a glibc bug.)

SISC, Chez, Vicare, IronScheme return 0.0.

SCM, Dfsch return +inf.0.

Larceny returns +nan.0.

MIT (even when ignoring floating point traps), Scheme48/scsh, STklos, UMB raise an exception.

Mosh returns its implementation-specific "undefined" value.

Plain Chicken, Bigloo, Ikarus, NexJ, SigScheme, Shoe, TinyScheme, Scheme 9, Dream, RScheme, BDC, XLisp, Rep, Schemik, Elk, VX, Oaklisp, Femtolisp, Inlab, Owl Lisp do not support non-real numbers.