# 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.

Scheme std syntax prints own syntax
KSi yes `(+inf.0 -inf.0 nan.0)` yes
NexJ yes `(Infinity -Infinity NaN)` no
VX yes `(inf. -inf. -nan.)` no
SCM * `(+inf.0 -inf.0 0/0)` yes
S7 no `(inf.0 -inf.0 -nan.0)` no
SXM no `(inf.0 -inf.0 -nan.0)` no
Inlab no `(inf.0 -inf.0 -nan.0)` no
UMB no `(inf.0 -inf.0 -nan)` no
Shoe no `(inf -inf -nan)` no
!TinyScheme no `(inf -inf -nan)` no
XLisp no `(inf -inf -nan)` no
Schemik no `(inf -inf -nan)` no
scsh no `(inf. -inf. -nan.)` no
Rep no `(inf. -inf. -nan.)` no
RScheme no `(inf. -inf. -nan.)` no
Elk no `(inf -inf -nan.0)` no
SISC no `(infinity.0 -infinity.0 nan.0)` no
BDC no `(Infinity -Infinity NaN)` no
MIT no `(#[+inf] #[-inf] #[NaN])` no

[*] 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:

Scheme Result
Bigloo `(#f #f #f #f)`
Biwa `(#t #f #f #f)`
Chibi `(#f #f #f #f)`
Chicken `(#f #f #f #f)`
Cyclone `(#f #f #f #f)`
Foment error (division by zero)
Gambit `(#f #f #f #f)`
Gauche `(#f #f #f #f)`
Guile `(#t #t #f #t)`
Kawa `(#t #t #f #f)`
LIPS `(#t #t #f #f)`
Loko `(#f #f #f #f)`
MIT `(#f #f #f #f)`
Racket `(#t #t #f #f)`
Sagittarius `(#t #t #f #t)`
STklos `(#t #t #t #f)`
Unsyntax `(#f #f #f #f)`
Ypsilon `(#t #t #f #t)`

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``````

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