Non-finite numbers

Syntax

Here's how various Schemes deal with syntax for non-finite inexact numbers. "Standard syntax" means what R6RS prescribes: +inf.0 for positive infinity, -inf.0 for negative infinity, and both +nan.0 and -nan.0 for NaN.

Racket, Gauche, Chicken (with or without the numbers egg), Scheme48, Guile, Kawa, Chez, Ikarus/Vicare, Larceny, Ypsilon, Mosh, IronScheme, STklos, Spark, Sagittarius accept and print the standard syntax.

Gambit, Bigloo, Chibi accept and print the standard syntax, except that they do not accept -nan.0.

SigScheme, Scheme 9, Dream, Oaklisp, Owl Lisp are excluded because they do not have inexact non-finite numbers.

The following table concisely describes the other Schemes in the test suite. "Std syntax" is "yes" if the Scheme can read the standard syntax, "print" shows what (let* ((i (* 1.0e200 1.0e200)) (n (- i i))) (list i (- i) n)) prints, and "own syntax" is "yes" if the Scheme can reread what it prints. The implementations are listed in roughly decreasing order of standardosity.

[*] Accepts +inf.0 and -inf.0 but not +nan.0 or -nan.0

NaN equivalence

The following implementations return #t for (eqv +nan.0 +nan.0): Chez, Gambit, Guile, Ikarus/Vicare, Kawa, Larceny, Racket, STklos, Sagittarius.

The following implementations return #f for (eqv +nan.0 +nan.0): Bigloo, Chibi, Chicken, Gauche, MIT Scheme, Scheme48.

Different NaNs: Is (/ 0.0 0.0) the same as +nan.0?

I ran the following code:

(list (equal? +nan.0 (/ 0.0 0.0)) (eqv? +nan.0 (/ 0.0 0.0)) (= +nan.0 (/ 0.0 0.0)) (eq? +nan.0 (/ 0.0 0.0)) )

And got this:

Emacs Lisp returns (nil nil nil) (without the eqv part, which elisp does not have)

Infinity examples

These are the R6RS examples involving +inf.0 and -inf.0 (already accounted for verbally in the "Implementation extensions" section of R7RS):

(complex? +inf.0)    => #t     ; infinities are real but not rational
(real? -inf.0)       => #t
(rational? -inf.0)   => #f
(integer? -inf.0)    => #f

(inexact? +inf.0)    => #t     ; infinities are inexact

(= +inf.0 +inf.0)    => #t     ; infinities are signed
(= -inf.0 +inf.0)    => #f
(= -inf.0 -inf.0)    => #t
(positive? +inf.0)   => #t
(negative? -inf.0)   => #t
(abs -inf.0)         => +inf.0

(finite? +inf.0)     => #f     ; infinities are infinite
(infinite? +inf.0)   => #t

                               ; infinities are maximal
(max +inf.0 x)       => +inf.0 where x is real
(min -inf.0 x)       => -inf.0 where x is real
(< -inf.0 x +inf.0)) => #t where x is real and finite
(> +inf.0 x -inf.0)) => #t where x is real and finite
(floor +inf.0)       => +inf.0
(ceiling -inf.0)     => -inf.0

                               ; infinities are sticky
(+ +inf.0 x)         => +inf.0 where x is real and finite
(+ -inf.0 x)         => -inf.0 where x is real and finite
(+ +inf.0 +inf.0)    => +inf.0

(+ +inf.0 -inf.0)    => +nan.0 ; sum of oppositely signed infinities is NaN
(- +inf.0 +inf.0)    => +nan.0

(* 5 +inf.0)         => +inf.0 ; infinities are sticky
(* -5 +inf.0)        => -inf.0
(* +inf.0 +inf.0)    => +inf.0
(* +inf.0 -inf.0)    => -inf.0

(/ 0.0)              => +inf.0 ; infinities are reciprocals of zero
(/ 1.0 0)            => +inf.0
(/ -1 0.0)           => -inf.0
(/ +inf.0)           => 0.0
(/ -inf.0)           => -0.0 if distinct from 0.0

(rationalize +inf.0 3)      => +inf.0
(rationalize +inf.0 +inf.0) => +nan.0
(rationalize 3 +inf.0)      => 0.0

(exp +inf.0)         => +inf.0
(exp -inf.0)         => 0.0
(log +inf.0)         => +inf.0
(log 0.0)            => -inf.0
(log -inf.0)         => +inf.0+3.141592653589793i ; approximately
(atan -inf.0)        => -1.5707963267948965 ; approximately
(atan +inf.0)        => 1.5707963267948965 ; approximately

(sqrt +inf.0)        => +inf.0
(sqrt -inf.0)        => +inf.0i

(angle +inf.0)       => 0.0
(angle -inf.0)       => 3.141592653589793
(magnitude (make-rectangular x y)) => +inf.0 where x or y or both are infinite

NaN examples

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

(number? +nan.0)   => #t ; NaN is real but not rational
(complex? +nan.0)  => #t
(real? +nan.0)     => #t
(rational? +nan.0) => #f

                         ; NaN compares #f to anything
(= +nan.0 z)       => #f where z numeric
(< +nan.0 x)       => #f where x real
(> +nan.0 x)       => #f where x real

(zero? +nan.0)     => #f ; NaN is unsigned
(positive? +nan.0) => #f
(negative? +nan.0) => #f

                         ; NaN is mostly sticky
(* 0 +inf.0)       => 0 or +nan.0
(* 0 +nan.0)       => 0 or +nan.0
(+ +nan.0 x)       => +nan.0 where x real
(* +nan.0 x)       => +nan.0 where x real and not exact 0

                         ; Sum of +inf.0 and -inf.0 is NaN
(+ +inf.0 -inf.0)  => +nan.0
(- +inf.0 +inf.0)  => +nan.0

                         ; 0/0 is NaN unless both 0s are exact
(/ 0 0.0) => +nan.0
(/ 0.0 0) => +nan.0
(/ 0.0 0.0) => +nan.0

(round +nan.0)     => +nan.0 ; Nan rounds (etc.) to NaN

(rationalize +inf.0 +inf.0) => +nan.0 ; Rationalizing infinity to nearest infinity is NaN

expt and infinities

The following table shows how Scheme implementations deal with exponentiation of infinities.

• The exactness of Biwa's and LIPS' answer is likely due to Javascript, where (+Infinity)**(-Infinity) returns exact zero. The +inf.0 in the other case is also consistent with the Javescript result.
• The First case (inf^(-inf)) is zero for most implementations probably because it is what the C pow function returns.
• Similarly, its likely that the (-inf^inf) case is most usually +inf.0 for the same reason
• The nan cases in (-inf^inf) could happen when computing x^y as exp(y*log(x)), since ( +inf * log(-inf) ) = ( +inf * (+inf + PI*i) ), which becomes NaN. (Using exp(y*log(x)) instead of pow makes sense because it works for complexes.)

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