# Extreme floating point values

**Extreme floating point values**

You are encouraged to solve this task according to the task description, using any language you may know.

The IEEE floating point specification defines certain 'extreme' floating point values such as minus zero, -0.0, a value distinct from plus zero; not a number, NaN; and plus and minus infinity.

The task is to use expressions involving other 'normal' floating point values in your language to calculate these, (and maybe other), extreme floating point values in your language and assign them to variables.

Print the values of these variables if possible; and show some arithmetic with these values and variables.

If your language can directly enter these extreme floating point values then show it.

- See also

- Related tasks

## Contents

- 1 Ada
- 2 AWK
- 3 bc
- 4 C
- 5 Clojure
- 6 D
- 7 Delphi
- 8 Eiffel
- 9 Euphoria
- 10 Forth
- 11 Fortran
- 12 FreeBASIC
- 13 Go
- 14 Groovy
- 15 haskell
- 16 Icon and Unicon
- 17 J
- 18 Java
- 19 jq
- 20 Kotlin
- 21 Lua
- 22 Mathematica / Wolfram Language
- 23 Maxima
- 24 MUMPS
- 25 NetRexx
- 26 Nim
- 27 OCaml
- 28 Oforth
- 29 Oz
- 30 PARI/GP
- 31 Pascal
- 32 Perl
- 33 Perl 6
- 34 Phix
- 35 PicoLisp
- 36 PureBasic
- 37 Python
- 38 R
- 39 Racket
- 40 Ruby
- 41 S-lang
- 42 Scala
- 43 Scheme
- 44 Seed7
- 45 Sidef
- 46 Swift
- 47 Tcl

## Ada[edit]

The language specifies model floating-point numbers independent of the underlying hardware. Even if the machine numbers are IEEE 754, the user-defined floating-point numbers are guaranteed to have no IEEE 754 semantics. In particular, their values do not include any non-numeric ideals. Constraint_Error exception is propagated when the result of a numeric operation assigned to a floating-point variable is not in the range (the range is always **numeric**).

For performance reasons, the built-in floating-point types like Float and Long_Float are allowed to have IEEE 754 semantics if the machine numbers are IEEE 754. But the language provides means to exclude all non numbers from these types by defining a subtype with an explicit range:

subtype Consistent_Float is Float range Float'Range; -- No IEEE ideals

In general in properly written Ada programs variables may not become invalid when standard numeric operations are applied. The language also provides the attribute 'Valid to verify values obtained from unsafe sources e.g. from input, unchecked conversions etc.

As stated above on a machine where Float is implemented by an IEEE 754 machine number, IEEE 754 is permitted leak through. The following program illustrates how this leak can be exploited:

with Ada.Text_IO; use Ada.Text_IO;

procedure IEEE is -- Non portable, bad, never do this!

Zero : Float := 0.0;

PInf : Float := 1.0 / Zero;

NInf : Float := -PInf;

PZero : Float := 1.0 / PInf;

NZero : Float := 1.0 / NInf;

NaN : Float := 0.0 / Zero;

begin

Put_Line (" -oo = " & Float'Image (NInf));

Put_Line (" +oo = " & Float'Image (PInf));

Put_Line (" NaN = " & Float'Image (NaN));

Put_Line (" -0 = " & Float'Image (NZero));

Put_Line (" -oo < first " & Boolean'Image (NInf < Float'First));

Put_Line (" +oo > last " & Boolean'Image (PInf > Float'Last));

Put_Line (" NaN = NaN " & Boolean'Image (NaN = NaN));

Put_Line (" -0 = 0 " & Boolean'Image (NZero = 0.0));

Put_Line (" +0 = 0 " & Boolean'Image (PZero = 0.0));

Put_Line (" +0 < least positive " & Boolean'Image (PZero < Float'Succ (Zero)));

Put_Line (" -0 > biggest negative " & Boolean'Image (NZero > Float'Pred (Zero)));

-- Validness checks

Put_Line ("Valid -oo is " & Boolean'Image (NInf'Valid));

Put_Line ("Valid +oo is " & Boolean'Image (PInf'Valid));

Put_Line ("Valid NaN is " & Boolean'Image (NaN'Valid));

end IEEE;

The expression -1.0 / 0.0 were non-numeric and thus could not be used. To fool the compiler the variable Zero is used, which circumvents type checks giving desired broken result.

- Output:

-oo = -Inf******* +oo = +Inf******* NaN = NaN******** -0 = -0.00000E+00 -oo < first TRUE +oo > last TRUE NaN = NaN FALSE -0 = 0 TRUE +0 = 0 TRUE +0 < least positive TRUE -0 > biggest negative TRUE Valid -oo is FALSE Valid +oo is FALSE Valid NaN is FALSE

## AWK[edit]

The One True Awk (nawk) uses the native floating-point numbers. We can get the extreme values if these are IEEE numbers. (If you run Awk on a VAX, there are no signed zeros, infinities nor NaN on a VAX.)

Awk raises a fatal error if a program divides by zero. If a call to exp(x), log(x) and sqrt(x) goes out of range, Awk displays a warning and changes the result to 1. Therefore tricks like 1 / 0, or log(0), or sqrt(-1), will not provide the extreme values. There remains some loopholes. Awk never checks for overflow, so we can still get positive or negative infinity. When we have infinity, we can get NaN.

BEGIN {

# This requires 1e400 to overflow to infinity.

nzero = -0

nan = 0 * 1e400

pinf = 1e400

ninf = -1e400

print "nzero =", nzero

print "nan =", nan

print "pinf =", pinf

print "ninf =", ninf

# When y == 0, sign of x decides if atan2(y, x) is 0 or pi.

print "atan2(0, 0) =", atan2(0, 0)

print "atan2(0, pinf) =", atan2(0, pinf)

print "atan2(0, nzero) =", atan2(0, nzero)

print "atan2(0, ninf) =", atan2(0, ninf)

# From least to most: ninf, -1e200, 1e200, pinf.

print "ninf * -1 =", ninf * -1

print "pinf * -1 =", pinf * -1

print "-1e200 > ninf?", (-1e200 > ninf) ? "yes" : "no"

print "1e200 < pinf?", (1e200 < pinf) ? "yes" : "no"

# NaN spreads from input to output.

print "nan test:", (1 + 2 * 3 - 4) / (-5.6e7 + nan)

# NaN never equals anything. These tests should print "no".

print "nan == nan?", (nan == nan) ? "yes" : "no"

print "nan == 42?", (nan == 42) ? "yes" : "no"

}

- Output:

$ awk -f extreme.awk nzero = -0 nan = nan pinf = inf ninf = -inf atan2(0, 0) = 0 atan2(0, pinf) = 0 atan2(0, nzero) = 3.14159 atan2(0, ninf) = 3.14159 nan test: nan nan == nan? yes nan == 42? yes

The last two lines are wrong. IEEE says that NaN != NaN (and also NaN != 42).
The problem is that Awk assumes `a == b` unless `(a - b) < 0` or `(a - b) > 0`; but NaN - NaN (or NaN - 42) is NaN, and NaN < 0 is false, and NaN > 0 is false, so Awk supposes that NaN == NaN (or NaN == 42) is true.

- Output:

nzero = 0 nan = NaN pinf = Inf ninf = NaN atan2(0, 0) = 0 atan2(0, pinf) = 0 atan2(0, nzero) = 0 atan2(0, ninf) = 3.14159 ninf * -1 = Inf pinf * -1 = NaN -1e200 > ninf? yes 1e200 < pinf? yes nan test: NaN nan == nan? no nan == 42? no

The attempts to use negative zero have failed. GNU awk uses both integers and floating point; GNU awk converted negative zero to an integer and lost the negative sign.

NaN works. Negative infinity seems to work, except when printing. Whenever GNU awk tries to print negative infinity, it prints "NaN".

## bc[edit]

bc numbers are very different from IEEE floating-point numbers. bc numbers have a variable number of digits. They can always have more digits (until bc has no memory, runs too slow or crashes), so there is no overflow, and no way to reach infinity.

bc also has no negative zero, and no NaN.

$bc# trying for negative zero-00# trying to underflow to negative zero-1 / 20# trying for NaN (not a number)0 / 0dc: divide by zero 0sqrt(-1)dc: square root of negative number dc: stack empty dc: stack empty

## C[edit]

#include <stdio.h>

int main()

{

double inf = 1/0.0;

double minus_inf = -1/0.0;

double minus_zero = -1/ inf ;

double nan = 0.0/0.0;

printf("positive infinity: %f\n",inf);

printf("negative infinity: %f\n",minus_inf);

printf("negative zero: %f\n",minus_zero);

printf("not a number: %f\n",nan);

/* some arithmetic */

printf("+inf + 2.0 = %f\n",inf + 2.0);

printf("+inf - 10.1 = %f\n",inf - 10.1);

printf("+inf + -inf = %f\n",inf + minus_inf);

printf("0.0 * +inf = %f\n",0.0 * inf);

printf("1.0/-0.0 = %f\n",1.0/minus_zero);

printf("NaN + 1.0 = %f\n",nan + 1.0);

printf("NaN + NaN = %f\n",nan + nan);

/* some comparisons */

printf("NaN == NaN = %s\n",nan == nan ? "true" : "false");

printf("0.0 == -0.0 = %s\n",0.0 == minus_zero ? "true" : "false");

return 0;

}

- Output:

positive infinity: inf negative infinity: -inf negative zero: -0.000000 not a number: -nan +inf + 2.0 = inf +inf - 10.1 = inf +inf + -inf = -nan 0.0 * +inf = -nan 1.0/-0.0 = -inf NaN + 1.0 = -nan NaN + NaN = -nan NaN == NaN = false 0.0 == -0.0 = true

- Output:

positive infinity: 1.#INF00 negative infinity: -1.#INF00 negative zero: -0.000000 not a number: -1.#IND00 +inf + 2.0 = 1.#INF00 +inf - 10.1 = 1.#INF00 +inf + -inf = -1.#IND00 0.0 * +inf = -1.#IND00 1.0/-0.0 = -1.#INF00 NaN + 1.0 = -1.#IND00 NaN + NaN = -1.#IND00 NaN == NaN = false 0.0 == -0.0 = true

- Output:

positive infinity: inf negative infinity: -inf negative zero: -0.000000 not a number: -nan +inf + 2.0 = inf +inf - 10.1 = inf +inf + -inf = -nan 0.0 * +inf = 0.000000 1.0/-0.0 = -inf NaN + 1.0 = -nan NaN + NaN = -nan NaN == NaN = false 0.0 == -0.0 = true

Some values may be directly defined in various headers. Following code also shows some of those values' bit patterns (most significant bit first for each byte). It should be pretty portable.

#include <stdio.h>

#include <values.h>

#include <math.h>

char * bits(double v) {

static char s[sizeof(double) * (CHARBITS + 1)];

int n, i, j;

unsigned char *c = (void*)&v;

for (i = n = 0; i < sizeof(double); i++) {

for (j = 1 << (CHARBITS - 1); j; j >>= 1)

s[n++] = (c[i] & j) ? '1' : '.';

s[n++] = ' ';

}

s[n-1] = 0;

return s;

}

int main(void)

{

double x[] = {

1.0, -1.0, 1.0/256, 0.0, // "normal" values

-0.0, INFINITY, -INFINITY, NAN, -NAN, // special

DBL_MAX, DBL_MIN // not required by task

};

int i;

for (i = 0; i < sizeof(x) / sizeof(x[0]); i++)

printf("%s | %g\n", bits(x[i]), x[i]);

return 0;

}

## Clojure[edit]

(def neg-inf (/ -1.0 0.0)) ; Also Double/NEGATIVE_INFINITY

(def inf (/ 1.0 0.0)) ; Also Double/POSITIVE_INFINITY

(def nan (/ 0.0 0.0)) ; Also Double/NaN

(def neg-zero (/ -2.0 Double/POSITIVE_INFINITY)) ; Also -0.0

(println " Negative inf: " neg-inf)

(println " Positive inf: " inf)

(println " NaN: " nan)

(println " Negative 0: " neg-zero)

(println " inf + -inf: " (+ inf neg-inf))

(println " NaN == NaN: " (= Double/NaN Double/NaN))

(println "NaN equals NaN: " (.equals Double/NaN Double/NaN))

- Output:

Negative inf: -Infinity Positive inf: Infinity NaN: NaN Negative 0: -0.0 inf + -inf: NaN NaN == NaN: false NaN equals NaN: true

## D[edit]

D V.2 has a pretty comprehensive approach to floating point values, and unlike Ada embraces IEEE 754. This program shows only part of the floating point features supported by D and its Phobos standard library.

// Compile this module without -O

import std.stdio: writeln, writefln;

import std.string: format;

import std.math: NaN, getNaNPayload;

void show(T)() {

static string toHex(T x) {

string result;

auto ptr = cast(ubyte*)&x;

foreach_reverse (immutable i; 0 .. T.sizeof)

result ~= format("%02x", ptr[i]);

return result;

}

enum string name = T.stringof;

writeln("Computed extreme ", name, " values:");

T zero = 0.0;

T pos_inf = T(1.0) / zero;

writeln(" ", name, " +oo = ", pos_inf);

T neg_inf = -pos_inf;

writeln(" ", name, " -oo = ", neg_inf);

T pos_zero = T(1.0) / pos_inf;

writeln(" ", name, " +0 (pos_zero) = ", pos_zero);

T neg_zero = T(1.0) / neg_inf;

writeln(" ", name, " -0 = ", neg_zero);

T nan = zero / pos_zero;

writefln(" " ~ name ~ " zero / pos_zero = %f %s", nan, toHex(nan));

writeln();

writeln("Some ", T.stringof, " properties and literals:");

writeln(" ", name, " +oo = ", T.infinity);

writeln(" ", name, " -oo = ", -T.infinity);

writeln(" ", name, " +0 = ", T(0.0));

writeln(" ", name, " -0 = ", T(-0.0));

writefln(" " ~ name ~ " nan = %f %s", T.nan, toHex(T.nan));

writefln(" " ~ name ~ " init = %f %s", T.init, toHex(T.init));

writeln(" ", name, " epsilon = ", T.epsilon);

writeln(" ", name, " max = ", T.max);

writeln(" ", name, " -max = ", -T.max);

writeln(" ", name, " min_normal = ", -T.min_normal);

writeln("-----------------------------");

}

void main() {

show!float;

show!double;

show!real;

writeln("Largest possible payload for float, double and real NaNs:");

immutable float f1 = NaN(0x3F_FFFF);

writeln(getNaNPayload(f1));

immutable double f2 = NaN(0x3_FFFF_FFFF_FFFF);

writeln(getNaNPayload(f2));

immutable real f3 = NaN(0x3FFF_FFFF_FFFF_FFFF);

writeln(getNaNPayload(f3));

}

- Output:

Computed extreme float values: float +oo = inf float -oo = -inf float +0 = 0 float -0 = -0 float init = -nan ffc00000 Some float properties and literals: float +oo = inf float -oo = -inf float +0 = 0 float -0 = -0 float nan = nan 7fc00000 float init = nan 7fa00000 float epsilon = 1.19209e-07 float max = 3.40282e+38 float -max = -3.40282e+38 float min_normal = -1.17549e-38 ----------------------------- Computed extreme double values: double +oo = inf double -oo = -inf double +0 = 0 double -0 = -0 double init = -nan fff8000000000000 Some double properties and literals: double +oo = inf double -oo = -inf double +0 = 0 double -0 = -0 double nan = nan 7ff8000000000000 double init = nan 7ff4000000000000 double epsilon = 2.22045e-16 double max = 1.79769e+308 double -max = -1.79769e+308 double min_normal = -2.22507e-308 ----------------------------- Computed extreme real values: real +oo = inf real -oo = -inf real +0 = 0 real -0 = -0 real init = -nan ffffc000000000000000 Some real properties and literals: real +oo = inf real -oo = -inf real +0 = 0 real -0 = -0 real nan = nan 7fffc000000000000000 real init = nan 7fffa000000000000000 real epsilon = 1.0842e-19 real max = 1.18973e+4932 real -max = -1.18973e+4932 real min_normal = -3.3621e-4932 ----------------------------- Largest possible payload for float, double and real NaNs: 4194303 1125899906842623 4610560118520545279

If you compile it with -O you get results like:

Computed extreme float values: float +oo = 2.9411e-36 float -oo = -2.9411e-36 float +0 (pos_zero) = 3.40008e+35 float -0 = -3.40008e+35 float zero / pos_zero = 0.000000 00000000 Some float properties and literals: float +oo = inf float -oo = -inf float +0 = 0 float -0 = -0 float nan = nan 7fc00000 float init = nan 7fa00000 float epsilon = 1.19209e-07 float max = 3.40282e+38 float -max = -3.40282e+38 float min_normal = -1.17549e-38 ----------------------------- Computed extreme double values: double +oo = 2.04581e-275 double -oo = -2.04581e-275 double +0 (pos_zero) = 4.88804e+274 double -0 = -4.88804e+274 double zero / pos_zero = 0.000000 0000000000000000 Some double properties and literals: double +oo = inf double -oo = -inf double +0 = 0 double -0 = -0 double nan = nan 7ff8000000000000 double init = nan 7ff4000000000000 double epsilon = 2.22045e-16 double max = 1.79769e+308 double -max = -1.79769e+308 double min_normal = -2.22507e-308 ----------------------------- Computed extreme real values: real +oo = 1.81242e-4933 real -oo = -1.81242e-4933 real +0 (pos_zero) = inf real -0 = -inf real zero / pos_zero = 0.000000 00000000000000000000 Some real properties and literals: real +oo = inf real -oo = -inf real +0 = 0 real -0 = -0 real nan = nan 7fffc000000000000000 real init = nan 7fffa000000000000000 real epsilon = 1.0842e-19 real max = 1.18973e+4932 real -max = -1.18973e+4932 real min_normal = -3.3621e-4932 ----------------------------- Largest possible payload for float, double and real NaNs: 4194303 1125899906842623 4610560118520545279

Among other things, it is possible to trap FP hardware exceptions:

import std.math: FloatingPointControl;

void main() {

// Enable hardware exceptions for division by zero, overflow

// to infinity, invalid operations, and uninitialized

// floating-point variables.

FloatingPointControl fpc;

fpc.enableExceptions(FloatingPointControl.severeExceptions);

double f0 = 0.0;

double y1 = f0 / f0; // generates hardware exception

// unless it's compiled with -O)

}

- Output:

object.Error: Invalid Floating Point Operation

## Delphi[edit]

Tested on Delphi 2009:

program Floats;

{$APPTYPE CONSOLE}

uses

SysUtils;

var

PlusInf, MinusInf, NegZero, NotANum: Double;

begin

PlusInf:= 1.0/0.0;

MinusInf:= -1.0/0.0;

NegZero:= -1.0/PlusInf;

NotANum:= 0.0/0.0;

Writeln('Positive Infinity: ', PlusInf); // +Inf

Writeln('Negative Infinity: ', MinusInf); // -Inf

Writeln('Negative Zero: ', NegZero); // -0.0

Writeln('Not a Number: ', NotANum); // Nan

// allowed arithmetic

Writeln('+Inf + 2.0 = ', PlusInf + 2.0); // +Inf

Writeln('+Inf - 10.1 = ', PlusInf - 10.1); // +Inf

Writeln('NaN + 1.0 = ', NotANum + 1.0); // Nan

Writeln('NaN + NaN = ', NotANum + NotANum); // Nan

// throws exception

try

Writeln('+inf + -inf = ', PlusInf + MinusInf); // EInvalidOp

Writeln('0.0 * +inf = ', 0.0 * PlusInf); // EInlalidOp

Writeln('1.0/-0.0 = ', 1.0 / NegZero); // EZeroDivide

except

on E:Exception do

Writeln(E.Classname, ': ', E.Message);

end;

Readln;

end.

## Eiffel[edit]

class

APPLICATION

inherit

ARGUMENTS

create

make

feature {NONE} -- Initialization

make

-- Run application.

local

negInf, posInf, negZero, nan: REAL_64

do

negInf := -1. / 0. -- also {REAL_64}.negative_infinity

posInf := 1. / 0. -- also {REAL_64}.positive_infinity

negZero := -1. / posInf

nan := 0. / 0. -- also {REAL_64}.nan

print("Negative Infinity: ") print(negInf) print("%N")

print("Positive Infinity: ") print(posInf) print("%N")

print("Negative Zero: ") print(negZero) print("%N")

print("NaN: ") print(nan) print("%N%N")

print("1.0 + Infinity = ") print((1.0 + posInf)) print("%N")

print("1.0 - Infinity = ") print((1.0 - posInf)) print("%N")

print("-Infinity + Infinity = ") print((negInf + posInf)) print("%N")

print("-0.0 * Infinity = ") print((negZero * posInf)) print("%N")

print("NaN + NaN = ") print((nan + nan)) print("%N")

print("(NaN = NaN) = ") print((nan = nan)) print("%N")

print("(0.0 = -0.0) = ") print((0.0 = negZero)) print("%N")

end

end

- Output:

Negative Infinity: -Infinity Positive Infinity: Infinity Negative Zero: -0 NaN: NaN 1.0 + Infinity = Infinity 1.0 - Infinity = -Infinity -Infinity + Infinity = NaN -0.0 * Infinity = NaN NaN + NaN = NaN (NaN = NaN) = False (0.0 = -0.0) = True

## Euphoria[edit]

constant inf = 1E400

constant minus_inf = -inf

constant nan = 0*inf

printf(1,"positive infinity: %f\n", inf)

printf(1,"negative infinity: %f\n", minus_inf)

printf(1,"not a number: %f\n", nan)

-- some arithmetic

printf(1,"+inf + 2.0 = %f\n", inf + 2.0)

printf(1,"+inf - 10.1 = %f\n", inf - 10.1)

printf(1,"+inf + -inf = %f\n", inf + minus_inf)

printf(1,"0.0 * +inf = %f\n", 0.0 * inf)

printf(1,"NaN + 1.0 = %f\n", nan + 1.0)

printf(1,"NaN + NaN = %f\n", nan + nan)

- Output:

positive infinity: inf negative infinity: -inf not a number: -nan +inf + 2.0 = inf +inf - 10.1 = inf +inf + -inf = -nan 0.0 * +inf = -nan NaN + 1.0 = -nan NaN + NaN = -nan

## Forth[edit]

1e 0e f/ f. \ inf

-1e 0e f/ f. \ inf (output bug: should say "-inf")

-1e 0e f/ f0< . \ -1 (true, it is -inf)

0e 0e f/ f. \ nan

-1e 0e f/ 1/f f0< . \ 0 (false, can't represent IEEE negative zero)

## Fortran[edit]

### Honest numbers[edit]

The floating-point number services offered by computers over the decades have varied greatly in format and behaviour, being in base ten, two, four, eight, or sixteen, and of various storage sizes, with various choices for precision and exponent range - for a fixed size, more precision means a smaller dynamic range and *vice-versa*. The IBM1620 offered (discrete transistor) floating-point hardware that allowed two decimal digits for the exponent and from two to 100 decimal digits for the mantissa; extreme values are easily deduced. For base two (or four, etc.) computers, presenting the exact extreme value in decimal produces troublesomely long strings of digits, and there is no clear guarantee that such a value, when converted by the compiler, will in fact manifest the desired extreme value in binary - the compiler is itself using computer arithmetic of limited precision. One must know *exactly* what format is used for floating-point numbers, on the specific computer in question. And if a programme using those values is moved to a different computer or a different compiler, you may well have to start again.

F90 however contains facilities to help. Pseudo-function HUGE(x) returns the largest possible number of the type of its parameter - whether an integer or a floating-point variable, of single or double precision, etc. But, this does not solve the problem, as if you are dealing with a computer that represents integers in two's complement, the maximum sixteen-bit number is 32767 but for negative numbers it is -32768. Thus, if you intend to find the extrema of a set of numbers and it is not convenient to set MinX and MaxX to the first value, and you don't want to have special case code testing for N = 1, you might try MinX = HUGE(Minx) and MaxX = -HUGE(MaxX) and gain wrong results should there for instance be only one value and it -32768.

There is a TINY(x) pseudo-function, for floating-point types only, that gives the smallest possible floating-point number, but, it is not clear whether this is the smallest possible normalised floating-point number, or, does it allow "denormalised" floating-point numbers that are even smaller?

Still further pseudo-functions offer PRECISION(x), EPSILON(x) and RADIX(x) whereby one can determine whether the implicit leading-one of normalised base two floating-point numbers is in use or not. Thanks to the proliferation of Intel 8087 *et seq* floating-point processors, single and double precision numbers on many modern computers use an implicit leading-one bit, but, the 80-bit format does not, and it allows denormalised numbers.

### Peculiar "numbers"[edit]

Modern computers following the Intel 8087 also reserve some bit patterns to represent what really aren't floating-point numbers at all. To be finicky, for example, zero cannot be represented as a normalised floating-point number, but nearly every design finds a way to represent zero - possibly as the smallest possible number if not an actual zero. Useful additions include ±underflow, ±overflow for finite numbers resulting from arithmetic that require an exponent part that is too large or too small to be represented. Thus, underflow is not zero, and overflow is not infinity. As well, there are representations of ±infinity, though this doesn't solve the MinX, MaxX problem above as these states are not available for integer variables. Oddest of all is "Not a Number" - which, to be finicky can't be called an *extreme* floating-point value and it declares itself not to be a *number* anyway. But, it *is* a possible state of a variable of the *modern* floating-point type.

Certain calculations are said to benefit from the states "positive zero", and "negative zero" being available in recondite ways, and theoretical investigations of differentiation can be recast to use an "infinitesimal" adduced to the Real number system, but these notions are even further away from "normal" number crunching.

Fortran does not recognise names for these states, as in `X = +Inf`

, though later compilers for systems that do offer these states *do* produce +Infinity on output, or NaN, and also recognise these texts when being read in a numeric field. Further, the special logical function IsNaN(x) is the only safe way to detect a bit pattern representing NaN (there are many) because the comparison operators behave oddly by design. A test X = X returns *false* if x has the NaN state (if not optimised away by the compiler to always be *true* as per the millenia-old definition of equality), but x ¬= NaN may be compiled as ¬(x = Nan) to further confusion.

Similarly, the library functions may or may not recognise these special values and "pass them through" in ways that might be expected. For instance, ABS(NaN) returns NaN, but EXP(NaN) delivers a run-time error - at least for Compaq Visual Fortran 6.6 F90/95. Nor might they generate them as could be hoped for. For instance, ATAN(x,y) would be used in converting from Cartesian (x,y) coordinates to angular coordinates <r,a> = <sqrt(x^2 + y^2),atan(x,y)> where a is the angle. ATAN(0,0) could return NaN, since a zero-length vector points in no direction, but if so, converting back via (x,y) = (r*cos(a),r*sin(a)) will not return (0,0) unless 0*NaN gives 0, which it doesn't. And it may not be clear what special value should be generated anyway. For instance, TAND(90) - which represents ninety degrees *exactly* unlike TAN(pi/2) which cannot - should yield Infinity as its result, but, which sign?

Since there are multiple bit patterns that constitute a NaN state, there is an opportunity for an affronted function to set a specific bit according to the error. Thus there could be one bit to mark a sqrt(neg), another for log(not positive), and so on. Later examination of a variable containing a bad state could give some provenance to the problem. Organising this would require a lot of work in standardisation fora.

It is also possible to set various options for the processing of floating-point numbers that affects rounding and much else. Confusion will almost certainly be the result.

### Pragmatics[edit]

To prepare variables with these non-numerical states is troublesome, because attaining infinity by x = 1/0 or the like is not only bad behaviour, it invites complaint from the compiler or the generation of a run-time error and immediate cancellation of the run. One could mess about by using a READ statement on special texts, but that prevents the results being constants. Instead, one studies the definitions and devises code such as ...

REAL*8 BAD,NaN !Sometimes a number is not what is appropriate.

PARAMETER (NaN = Z'FFFFFFFFFFFFFFFF') !This value is recognised in floating-point arithmetic.

PARAMETER (BAD = Z'FFFFFFFFFFFFFFFF') !I pay special attention to BAD values.

CHARACTER*3 BADASTEXT !Speakable form.

DATA BADASTEXT/" ? "/ !Room for "NaN", short for "Not a Number", if desired.

REAL*8 PINF,NINF !Special values. No sign of an "overflow" state, damnit.

PARAMETER (PINF = Z'7FF0000000000000') !May well cause confusion

PARAMETER (NINF = Z'FFF0000000000000') !On a cpu not using this scheme.

After experimenting with code such as

Cause various arithmetic errors to see what sort of hissy fit is thrown.

REAL X2,X3,X4,Y4,XX,ZERO

INTEGER IX4,IY4

EQUIVALENCE (X4,IX4),(Y4,IY4) !To view bits without provoking special fp handling.

REAL*4 NaN4

PARAMETER (NaN4 = Z'FFC00000') !FFFFFFFF

c PARAMETER (NaN4 = Z'FFFFFFFF') !FFFFFFFF

REAL*8 NaN8,X8(5),Y8,INF8

PARAMETER (NaN8 = Z'FFF8000000000000') !FFFFFFFF

c PARAMETER (NaN8 = Z'FFFFFFFFFFFFFFFF')

LOGICAL LX(5)

INTEGER I

X4 = NaN4

WRITE (6,1) X4,IX4

1 FORMAT ("X4 =",F12.4,' Hex ',Z8)

WRITE (6,*) "Test X4 .EQ. Bad? ",X4.EQ.NaN4

WRITE (6,*) "Test X4 .NE. Bad? ",X4.NE.NaN4

WRITE (6,*) "Test IsNaN(X4) ",ISNAN(X4)

WRITE (6,*) "Test Abs(bad) ",ABS(X4)

c WRITE (6,*) "Test Exp(bad)",EXP(X4)

Y8 = HUGE(Y8)

WRITE(6,*) "Huge",Y8,LOG(Y8)

Y8 = LOG(Y8)

WRITE (6,*) "Hic",EXP(Y8)

X2 = 0

X3 = 0

ZERO = 0

XX = 666.66

X2 = XX + X4

WRITE (6,*) "Test x + BAD ",X2

WRITE (6,*) "Test 0/0 ",X3/ZERO

WRITE (6,*) "Test 1/0 ",1/ZERO

WRITE (6,*) "Test-1/0 ",-1/ZERO

X2 = MIN(XX,X4)

WRITE (6,*) "Test min(x,Bad) ",X2

WRITE (6,*) "Test min(x,NaN4)",MIN(XX,NaN4)

c WRITE (6,*) "Test mod(x,Bad) ",MOD(XX,X4)

c WRITE (6,*) "Test mod(Bad,x) ",MOD(X4,XX)

c WRITE (6,*) "Test mod(x,0) ",MOD(XX,Z)

c WRITE (6,*) "Sqrt(Bad)",SQRT(X4)

DO I = 1,0,-1 !for sqrt(-1), a snarl.

X4 = I

X4 = X4/FLOAT(I)

Y4 = SQRT(FLOAT(I))

WRITE (6,10) I,I,X4,IX4,I,Y4,IY4

10 FORMAT (I3,"/",I3," gives",F9.5," Hex ",Z8,

1 ", Sqrt(",I3,") gives",F9.5," Hex ",Z8)

END DO

Contemplate double precision.

WRITE (6,*)

WRITE (6,*) "Problems with IsNaN and arrays..."

DO I = 1,5

X8(I) = I

END DO

X8(3:4) = NaN8

WRITE (6,*) "X=",X8

WRITE (6,*) "X(2:4)=",X8(2:4)

WRITE (6,*) "isnan(x(2:4))",ISNAN(X8(2:4))

WRITE (6,*) "isnan(x(2))..(4))",ISNAN(X8(2)),ISNAN(X8(3)),

1 ISNAN(X8(4))

WRITE (6,*) "abs(x(2:4))",ABS(X8(2:4))

WRITE (6,*) "isnan(abs(x(2:4)))",ISNAN(ABS(X8(2:4)))

LX = ISNAN(X8)

WRITE (6,*) "LX = isnan(X)",LX

XX = HUGE(XX)

WRITE(6,*) "Huge(x)=",XX,-XX

XX = 1/ZERO

WRITE(6,11) XX,-XX

11 FORMAT("1/Zero=",Z8,", neg ",Z8)

INF8 = XX

WRITE (6,12) INF8,-INF8

12 FORMAT("1/Zero=",Z16,", neg ",Z16)

WRITE (6,*) "Burp!"

END

Which provides output such as

X4 = NaN Hex FFC00000 Test X4 .EQ. Bad? F Test X4 .NE. Bad? T Test IsNaN(X4) T Test Abs(bad) NaN Huge 1.797693134862316E+308 709.782712893384 Hic 1.797693134862273E+308 Test x + BAD NaN Test 0/0 NaN Test 1/0 Infinity Test-1/0 -Infinity Test min(x,Bad) 666.6600 Test min(x,NaN4) 666.6600 1/ 1 gives 1.00000 Hex 3F800000, Sqrt( 1) gives 1.00000 Hex 3F800000 0/ 0 gives NaN Hex FFC00000, Sqrt( 0) gives 0.00000 Hex 0 Problems with IsNaN and arrays... X= 1.00000000000000 2.00000000000000 NaN NaN 5.00000000000000 X(2:4)= 2.00000000000000 NaN NaN isnan(x(2:4)) T T F isnan(x(2))..(4)) F T T abs(x(2:4)) 2.00000000000000 NaN NaN isnan(abs(x(2:4))) F T T LX = isnan(X) F F T T F Huge(x)= 3.4028235E+38 -3.4028235E+38 1/Zero=7F800000, neg FF800000 1/Zero=7FF0000000000000, neg FFF0000000000000 Burp!

Some functions "pass through" bad values, and some raise an error and stop the run.

## FreeBASIC[edit]

' FB 1.05.0 Win64

#Include "crt/math.bi"

Dim inf As Double = INFINITY

Dim negInf As Double = -INFINITY

Dim notNum As Double = NAN_

Dim negZero As Double = 1.0 / negInf

Print inf, inf / inf

Print negInf, negInf * negInf

Print notNum, notNum + inf + negInf

Print negZero, negZero - 1

Sleep

- Output:

1.#INF -1.#IND -1.#INF 1.#INF -1.#IND -1.#IND -0 -1

## Go[edit]

package main

import (

"fmt"

"math"

)

func main() {

// compute "extreme values" from non-extreme values

var zero float64 // zero is handy.

var negZero, posInf, negInf, nan float64 // values to compute.

negZero = zero * -1

posInf = 1 / zero

negInf = -1 / zero

nan = zero / zero

// print extreme values stored in variables

fmt.Println(negZero, posInf, negInf, nan)

// directly obtain extreme values

fmt.Println(math.Float64frombits(1<<63),

math.Inf(1), math.Inf(-1), math.NaN())

// validate some arithmetic on extreme values

fmt.Println()

validateNaN(negInf+posInf, "-Inf + Inf")

validateNaN(0*posInf, "0 * Inf")

validateNaN(posInf/posInf, "Inf / Inf")

// mod is specifically named in "What every computer scientist..."

// Go math package doc lists many special cases for other package functions.

validateNaN(math.Mod(posInf, 1), "Inf % 1")

validateNaN(1+nan, "1 + NaN")

validateZero(1/posInf, "1 / Inf")

validateGT(posInf, math.MaxFloat64, "Inf > max value")

validateGT(-math.MaxFloat64, negInf, "-Inf < max neg value")

validateNE(nan, nan, "NaN != NaN")

validateEQ(negZero, 0, "-0 == 0")

}

func validateNaN(n float64, op string) {

if math.IsNaN(n) {

fmt.Println(op, "-> NaN")

} else {

fmt.Println("!!! Expected NaN from", op, " Found", n)

}

}

func validateZero(n float64, op string) {

if n == 0 {

fmt.Println(op, "-> 0")

} else {

fmt.Println("!!! Expected 0 from", op, " Found", n)

}

}

func validateGT(a, b float64, op string) {

if a > b {

fmt.Println(op)

} else {

fmt.Println("!!! Expected", op, " Found not true.")

}

}

func validateNE(a, b float64, op string) {

if a == b {

fmt.Println("!!! Expected", op, " Found not true.")

} else {

fmt.Println(op)

}

}

func validateEQ(a, b float64, op string) {

if a == b {

fmt.Println(op)

} else {

fmt.Println("!!! Expected", op, " Found not true.")

}

}

- Output:

-0 +Inf -Inf NaN -0 +Inf -Inf NaN -Inf + Inf -> NaN 0 * Inf -> NaN Inf / Inf -> NaN Inf % 1 -> NaN 1 + NaN -> NaN 1 / Inf -> 0 Inf > max value -Inf < max neg value NaN != NaN -0 == 0

## Groovy[edit]

Solution:

def negInf = -1.0d / 0.0d; //also Double.NEGATIVE_INFINITY

def inf = 1.0d / 0.0d; //also Double.POSITIVE_INFINITY

def nan = 0.0d / 0.0d; //also Double.NaN

def negZero = -2.0d / inf;

println(" Negative inf: " + negInf);

println(" Positive inf: " + inf);

println(" NaN: " + nan);

println(" Negative 0: " + negZero);

println(" inf + -inf: " + (inf + negInf));

println(" 0 * NaN: " + (0 * nan));

println(" NaN == NaN: " + (nan == nan));

println("NaN equals NaN: " + (nan.equals(nan)));

- Output:

Negative inf: -Infinity Positive inf: Infinity NaN: NaN Negative 0: -0.0 inf + -inf: NaN 0 * NaN: NaN NaN == NaN: true NaN equals NaN: true

Note that the Groovy implementation of 'equals' incorrectly allows that "NaN == NaN" is true.
In a correct IEEE implementation NaN is never equal to **anything**, including itself.

## haskell[edit]

main = do

let inf = 1/0

let minus_inf = -1/0

let minus_zero = -1/inf

let nan = 0/0

putStrLn ("Positive infinity = "++(show inf))

putStrLn ("Negative infinity = "++(show minus_inf))

putStrLn ("Negative zero = "++(show minus_zero))

putStrLn ("Not a number = "++(show nan))

--Some Arithmetic

putStrLn ("inf + 2.0 = "++(show (inf+2.0)))

putStrLn ("inf - 10 = "++(show (inf-10)))

putStrLn ("inf - inf = "++(show (inf-inf)))

putStrLn ("inf * 0 = "++(show (inf * 0)))

putStrLn ("nan + 1.0= "++(show (nan+1.0)))

putStrLn ("nan + nan = "++(show (nan + nan)))

--Some Comparisons

putStrLn ("nan == nan = "++(show (nan == nan)))

putStrLn ("0.0 == - 0.0 = "++(show (0.0 == minus_zero)))

putStrLn ("inf == inf = "++(show (inf == inf)))

- Output:

Positive infinity = Infinity Negative infinity = -Infinity Negative Zero = -0.0 Not a number = NaN inf + 2.0 = Infinity inf - 10 = Infinity inf - inf = NaN inf * 0 = NaN nan + 1.0 = NaN nan + nan = NaN nan == nan = False 0.0 == -0.0 = True inf == inf = True

## Icon and Unicon[edit]

Icon and Unicon don't define minimum or maximum values of reals, or a negative 0.0. Real numbers are implemented as C doubles and the behavior could vary somewhat from platform to platform. Both explicitly check for divide by zero and treat it as a runtime error (201), so it's not clear how you could produce one of these with the possible exception of the value being introduced through externally called code.

## J[edit]

**Extreme values**

Inf=: _

NegInf=: __

NB. Negative zero cannot be represented in J to be distinct from 0.

NaN=. _.

The numeric atom `_.`

(Indeterminate) is provided as a means for dealing with NaN in data from sources outside J.
J itself generates NaN errors rather than NaN values and recommends that `_.`

be removed from data as soon as possible because, by definition, NaN values will produce inconsistent results in contexts where value is important.

**Extreme values from expressions**

(1 % 0) , (_1 % 0)

_ __

(1e234 * 1e234) , (_1e234 * 1e234)

_ __

_ + __ NB. generates NaN error, rather than NaN

|NaN error

| _ +__

_ - _ NB. generates NaN error, rather than NaN

|NaN error

| _ -_

%_

0

%__ NB. Under the covers, the reciprocal of NegInf produces NegZero, but this fact isn't exposed to the user, who just sees zero

0

**Some arithmetic**

_ + _

_

__ + __

__

Inf + 0

_

NegInf * 0

0

## Java[edit]

public class Extreme {

public static void main(String[] args) {

double negInf = -1.0 / 0.0; //also Double.NEGATIVE_INFINITY

double inf = 1.0 / 0.0; //also Double.POSITIVE_INFINITY

double nan = 0.0 / 0.0; //also Double.NaN

double negZero = -2.0 / inf;

System.out.println("Negative inf: " + negInf);

System.out.println("Positive inf: " + inf);

System.out.println("NaN: " + nan);

System.out.println("Negative 0: " + negZero);

System.out.println("inf + -inf: " + (inf + negInf));

System.out.println("0 * NaN: " + (0 * nan));

System.out.println("NaN == NaN: " + (nan == nan));

}

}

- Output:

Negative inf: -Infinity Positive inf: Infinity NaN: NaN Negative 0: -0.0 inf + -inf: NaN 0 * NaN: NaN NaN == NaN: false

## jq[edit]

jq uses IEEE 754 64-bit numbers, and certain numeric expressions yield the exceptional floating point values in the usual way. However, since JSON does not support such values, jq currently prints the NaN value as null, and the infinite value as a very large float, so some care is required in interpreting the printed values.

For example, here are two expressions and the result of displaying their values:

0/0 #=> null

1e1000 #=> 1.7976931348623157e+308

If your jq does not already have `infinite` and `nan` defined as built-in functions, they can be defined as follows:

def infinite: 1e1000;

def nan: 0/0;

Here are some further expressions with their results:

-0 #=> -0

0 == -0 # => true

infinite == infinite #=> true

infinite == -(-infinite) #=> true

(infinite + infinite) == infinite #=> true

1/infinite #=> 0

nan == nan #=> false # N.B.

Since `==` cannot be used to check if a value is IEEE NaN, jq 1.5 provides the builtin function `isnan` for doing so:

nan | isnan #=> true

infinite | isnan #=> false

Exceptional values can be assigned to jq variables in the usual way:

infinite as $inf | 1 / $inf #=> 0

-0 as $z | $z #=> -0

## Kotlin[edit]

// version 1.0.5-2

@Suppress("DIVISION_BY_ZERO")

fun main(args: Array<String>) {

val inf = 1.0 / 0.0

val negInf = -1.0 / 0.0

val nan = 0.0 / 0.0

val negZero = -1.0e-325

println("*** Indirect ***\n")

println("Infinity : $inf")

println("Negative infinity : $negInf")

println("Not a number : $nan")

println("Negative zero : $negZero")

println("\n*** Direct ***\n")

println("Infinity : ${Double.POSITIVE_INFINITY}")

println("Negative infinity : ${Double.NEGATIVE_INFINITY}")

println("Not a number : ${Double.NaN}")

println("Negative zero : ${-0.0}")

println("\n*** Calculations ***\n")

println("inf * inf : ${inf * inf}")

println("inf + negInf : ${inf + negInf}")

println("nan / nan : ${nan / nan}")

println("negZero + 0.0 : ${negZero + 0.0}")

}

- Output:

*** Indirect *** Infinity : Infinity Negative infinity : -Infinity Not a number : NaN Negative zero : -0.0 *** Direct *** Infinity : Infinity Negative infinity : -Infinity Not a number : NaN Negative zero : -0.0 *** Calculations *** inf * inf : Infinity inf + negInf : NaN nan / nan : NaN negZero + 0.0 : 0.0

## Lua[edit]

Infinity and NaN are straight forward. for negative 0 you need to resort to a literal with a large, negative exponent, which is not the same thing.

local inf=math.huge

local minusInf=-math.huge

local NaN=0/0

local negativeZeroSorta=-1E-240

Lua seems to break x==1/(1/x) for infinity:

1/(1/-math.huge)==math.huge

true

## Mathematica / Wolfram Language[edit]

Column@{ReleaseHold[

Function[expression,

Row@[email protected]@expression, " = ", [email protected]},

HoldAll] /@

Hold[1./0., 0./0., Limit[-Log[x], x -> 0], Limit[Log[x], x -> 0],

Infinity + 1, Infinity + Infinity, 2 Infinity,

Infinity - Infinity, 0 Infinity, ComplexInfinity + 1,

ComplexInfinity + ComplexInfinity, 2 ComplexInfinity,

0 ComplexInfinity, Indeterminate + 1, 0 Indeterminate]]}

- Output:

1./0. = ComplexInfinity 0./0. = Indeterminate Limit[-Log[x], x -> 0] = ∞ Limit[Log[x], x -> 0] = -∞ Infinity + 1 = ∞ Infinity + Infinity = ∞ 2*Infinity = ∞ Infinity - Infinity = Indeterminate 0*Infinity = Indeterminate ComplexInfinity + 1 = ComplexInfinity ComplexInfinity + ComplexInfinity = Indeterminate 2*ComplexInfinity = ComplexInfinity 0*ComplexInfinity = Indeterminate Indeterminate + 1 = Indeterminate 0*Indeterminate = Indeterminate

## Maxima[edit]

With ordinary floating point numbers, `1.0 / 0.0`

, `0.0 / 0.0`

or `1e300^2`

all throw an exception.

However, Maxima has big floats and knows how to manage the `inf`

, `minf`

and `infinity`

symbols
(resp. positive, negative and complex infinity), with the function `limit`

. It also has `zeroa`

and `zerob`

for positive and negative infinitesimal (though their usage is quite obscure),
and `und`

for undefined value.

## MUMPS[edit]

### ANSI MUMPS[edit]

The 1995 Standard MUMPS (X11.1–1995) implementations do not deal with floating point numbers following IEEE 754. Attempting to use a number over the precision of the system results in a <MAXNUMBER> error:

USER>write 3e145 30000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 USER>write 3e146 <MAXNUMBER>

### Intersystems Caché[edit]

Caché has the function $DOUBLE which complies with the IEEE 754 standard. The negative zero is indistinguishable from positive zero by operations. The special values evaluate to 0 when converted to a number in a later operation.

EXTREMES

NEW INF,NINF,ZERO,NOTNUM,NEGZERO

SET INF=$DOUBLE(3.0E310),NINF=$DOUBLE(-3.0E310),ZERO=$DOUBLE(0),NOTNUM=$DOUBLE(INF-INF),NEGZERO=$DOUBLE(ZERO*-1)

WRITE "Infinity: ",INF,!

WRITE "Infinity ",$SELECT($ISVALIDNUM(INF):"is a number",1:"is not a number"),!

WRITE "Negative Infinity: ",NINF,!

WRITE "Negative Infinity ",$SELECT($ISVALIDNUM(NINF):"is a number",1:"is not a number"),!

WRITE "Zero: ",ZERO,!

WRITE "Zero ",$SELECT($ISVALIDNUM(ZERO):"is a number",1:"is not a number"),!

WRITE "Negative Zero: ",NEGZERO,!

WRITE "Negative Zero ",$SELECT($ISVALIDNUM(NEGZERO):"is a number",1:"is not a number"),!

WRITE "Not a Number: ",NOTNUM,!

WRITE "Not a Number ",$SELECT($ISVALIDNUM(NOTNUM):"is a number",1:"is not a number"),!

KILL INF,NINF,ZERO,NONNUM,NEGZERO

QUIT

- Output:

USER>d EXTREMES^ROSETTA Infinity: INF Infinity is not a number Negative Infinity: -INF Negative Infinity is not a number Zero: 0 Zero is a number Negative Zero: 0 Negative Zero is a number Not a Number: NAN Not a Number is not a number

## NetRexx[edit]

While NetRexx native support for numbers allows for very large decimal precision, the Java primitives (`int`, `long`, `float`, `double` etc.), can use the constants and methods provided for "extreme" values:

/* NetRexx */

options replace format comments java crossref symbols binary

negInf = double -1.0 / 0.0; knegInf = Double.NEGATIVE_INFINITY

inf = double 1.0 / 0.0; kinf = Double.POSITIVE_INFINITY

nan = double 0.0 / 0.0; knan = Double.NaN

negZero = double -2.0 / inf; knegZero = -2.0 / Double.POSITIVE_INFINITY

say "Negative inf: " Rexx(negInf).right(10) '|' knegInf

say "Positive inf: " Rexx(inf).right(10) '|' kinf

say "NaN: " Rexx(nan).right(10) '|' knan

say "Negative 0: " Rexx(negZero).right(10) '|' knegZero

say "inf + -inf: " Rexx(inf + negInf).right(10) '|' (kinf + knegInf)

say "0 * NaN: " Rexx(0 * nan).right(10) '|' (0 * knan)

say "NaN == NaN: " Rexx(nan == nan).right(10) '|' (knan == knan)

return

- Output:

Negative inf: Infinity | Infinity Positive inf: Infinity | Infinity NaN: NaN | NaN Negative 0: 0 | 0 inf + -inf: NaN | NaN 0 * NaN: NaN | NaN NaN == NaN: 0 | 0

## Nim[edit]

echo 1e234 * 1e234 # inf

echo 1e234 * -1e234 # -inf

echo 1 / inf # 0

echo inf + -inf # -nan

echo nan # nan

echo nan == nan # false

echo 0.0 == -0.0 # true

echo 0.0 * nan # nan

echo nan * 0.0 # nan

echo 0.0 * inf # -nan

echo inf * 0.0 # -nan

## OCaml[edit]

# infinity;;

- : float = infinity

# neg_infinity;;

- : float = neg_infinity

# nan;;

- : float = nan

# -0.;;

- : float = -0.

# -. 0.;;

- : float = -0.

# 1. /. 0.;;

- : float = infinity

# -1. /. 0.;;

- : float = neg_infinity

# -. infinity;;

- : float = neg_infinity

# infinity +. neg_infinity;;

- : float = nan

# 0. /. 0.;;

- : float = nan

# infinity /. infinity;;

- : float = nan

# nan = nan;;

- : bool = false

# nan == nan;;

- : bool = true

# 0. *. infinity;;

- : float = nan

# 0. = -0.;;

- : bool = true

# 0. == -0.;;

- : bool = false

## Oforth[edit]

In Oforth, the only 'extreme' floating point values are PInfinity (+oo) and NInfinity (-oo).

- Output:

>10.0 1000.0 powf PInf == println 1 ok >10.0 1000.0 powf neg NInf == println 1 ok

## Oz[edit]

declare

Inf = 1.0e234 * 1.0e234

MinusInf = 1.0e234 * ~1.0e234

Zero = 1.0 / Inf

MinusZero = 1.0 / MinusInf

NaN = 0.0 / 0.0

{System.showInfo "infinite: "#Inf}

{System.showInfo "-infinite: "#MinusInf}

{System.showInfo "0: "#Zero}

{System.showInfo "-0: "#MinusZero} %% seems to be identical to Zero

{System.showInfo "NaN: "#NaN}

{System.showInfo "inf + -inf: "#Inf+MinusInf}

{System.showInfo "NaN * 0: "#NaN*0.0}

{System.showInfo "0 * NaN: "#0.0*NaN}

{System.showInfo "inf * 0: "#Inf*0.0}

{System.showInfo "0 * inf: "#0.0*Inf}

{Show NaN == NaN} %% shows 'true' !

{Show Zero == MinusZero}

{Show 1.0/0.0 == Inf} %% true

{Show 1.0/~0.0 == MinusInf} %% true

- Output:

infinite: 1.#INF

-infinite: -1.#INF

0: 0.0

-0: 0.0

NaN: -1.#IND

inf + -inf: -1.#IND

NaN * 0: -1.#IND

0 * NaN: -1.#IND

inf * 0: -1.#IND

0 * inf: -1.#IND

true

true

true

true

## PARI/GP[edit]

PARI t_REALs are not IEEE floating-point numbers; in particular they cannot store NaN or infinite values. (The latter have their own type, t_INFINITY, with values `+oo`

and `-oo`

.)

PARI t_REAL numbers have a maximum value of

32-bit | 161,614,249 decimal digits | |
---|---|---|

64-bit | 694,127,911,065,419,642 decimal digits |

where is the machine epsilon at the selected precision. The minimum value is the opposite of the maximum value (reverse the sign bit).

## Pascal[edit]

See Delphi

## Perl[edit]

Perl numbers have three formats (integer, floating-point, string) and perlnumber explains the automatic conversions. Arithmetic tends to convert numbers to integers.
To get negative zero, one must negate a floating-point zero, not an integer zero.
So `-0` is "0", `-0.0` is "-0", but `-(1.0 - 1.0)` is again "0" because the result of `1.0 - 1.0` is an integer zero.
Stringification of minus zero may or may not keep the sign in the string, depending on the platform and Perl version.
If the sign is important, use `printf "%f"`

instead (`"%g"`

won't work: it gives "0").

Division by zero, `sqrt(-1)` and `log(0)` are fatal errors.
To get infinity and NaN, use corresponding string and force a numeric conversion by adding zero to it, or prepending a "+" or "-":

#!/usr/bin/perl

use strict;

use warnings;

my $nzero = -0.0;

my $nan = 0 + "nan";

my $pinf = +"inf";

my $ninf = -"inf";

printf "\$nzero = %.1f\n", $nzero;

print "\$nan = $nan\n";

print "\$pinf = $pinf\n";

print "\$ninf = $ninf\n\n";

printf "atan2(0, 0) = %g\n", atan2(0, 0);

printf "atan2(0, \$nzero) = %g\n", atan2(0, $nzero);

printf "sin(\$pinf) = %g\n", sin($pinf);

printf "\$pinf / -1 = %g\n", $pinf / -1;

printf "\$ninf + 1e100 = %g\n\n", $ninf + 1e100;

printf "nan test: %g\n", (1 + 2 * 3 - 4) / (-5.6e7 * $nan);

printf "nan == nan? %s\n", ($nan == $nan) ? "yes" : "no";

printf "nan == 42? %s\n", ($nan == 42) ? "yes" : "no";

- Output:

$nzero = -0.0 $nan = nan $pinf = inf $ninf = -inf atan2(0, 0) = 0 atan2(0, $nzero) = 3.14159 sin($pinf) = nan $pinf / -1 = -inf $ninf + 1e100 = -inf nan test: nan nan == nan? no nan == 42? no

Here is a rare example of NaN and infinity for an *integer* type.
Math::BigInt, a module that comes with Perl, provides integers of arbitrary sizes,
but also has NaN, positive infinity, and negative infinity.
There is no negative zero.

#!/usr/bin/perl

use strict;

use warnings;

use Math::BigInt;

my $nan = Math::BigInt->bnan();

my $pinf = Math::BigInt->binf();

my $ninf = Math::BigInt->binf('-');

print "\$nan = $nan\n";

print "\$pinf = $pinf\n";

print "\$ninf = $ninf\n\n";

my $huge = Math::BigInt->new("123456789");

$huge->bmul($huge)->bmul($huge)->bmul($huge);

print "\$huge = $huge\n";

printf "\$ninf + \$huge = %s\n", $ninf->copy()->badd($huge);

printf "\$pinf - \$huge = %s\n", $pinf->copy()->bsub($huge);

printf "\$nan * \$huge = %s\n", $nan->copy()->bmul($huge);

printf "\$nan == \$nan? %s\n", defined($nan->bcmp($nan)) ? "maybe" : "no";

printf "\$nan == \$huge? %s\n", defined($nan->bcmp($huge)) ? "maybe" : "no";

- Output:

$nan = NaN $pinf = inf $ninf = -inf $huge = 53965948844821664748141453212125737955899777414752273389058576481 $ninf + $huge = -inf $pinf - $huge = inf $nan * $huge = NaN $nan == $nan? no $nan == $huge? no

## Perl 6[edit]

Floating point limits are to a large extent implementation dependent. Right now, the Rakudo perl 6 interpreter running on moar VM on a 64 bit OS has in infinity threshold of 1e309.

print qq:to 'END'

positive infinity: {1e309}

negative infinity: {-1e309}

negative zero: {0e0 * -1}

not a number: {0 * 1e309}

+Inf + 2.0 = {Inf + 2}

+Inf - 10.1 = {Inf - 10.1}

+Inf + -Inf = {Inf + -Inf}

0 * +Inf = {0 * Inf}

NaN + 1.0 = {NaN + 1.0}

NaN + NaN = {NaN + NaN}

NaN == NaN = {NaN == NaN}

0.0 == -0.0 = {0e0 == -0e0}

END

`0e0`

is used to have floating point number.
Simply using `0.0`

makes rational number that doesn't recognize `-0`

.
`qq:to`

is heredoc syntax, where `qq`

means
that variables and closures (between braces) are interpolated.

- Output:

positive infinity: Inf negative infinity: -Inf negative zero: -0 not a number: NaN +Inf + 2.0 = Inf +Inf - 10.1 = Inf +Inf + -Inf = NaN 0 * +Inf = NaN NaN + 1.0 = NaN NaN + NaN = NaN NaN == NaN = False 0.0 == -0.0 = True

## Phix[edit]

constant inf = 1e300*1e300, -- (works on both 32 and 64 bit)

ninf = -inf,

nan = -(inf/inf),

nzero = -1/inf -- (not supported)

printf(1," inf: %f\n",{inf})

printf(1," ninf: %f\n",{ninf})

printf(1," nan: %f\n",{nan})

printf(1,"*nzero: %f\n",{nzero})

printf(1," inf+2: %f\n",{inf+2})

printf(1," inf+ninf: %f\n",{inf+ninf})

printf(1," 0*inf: %f\n",{0*inf})

printf(1," nan+1: %f\n",{nan+1})

printf(1," nan+nan: %f\n",{nan+nan})

printf(1," inf>1e300: %d\n",{inf>1e300})

printf(1," ninf<1e300: %d\n",{ninf<-1e300})

printf(1,"*nan=nan: %d\n",{nan=nan})

printf(1," nan=42: %d\n",{nan=42})

printf(1,"*nan<0: %d\n",{nan<0})

printf(1," nan>0: %d\n",{nan>0})

- Output:

inf: inf ninf: -inf nan: nan *nzero: 0.000000 inf+2: inf inf+ninf: -nan 0*inf: -nan nan+1: nan nan+nan: nan inf>1e300: 1 ninf<1e300: 1 *nan=nan: 1 nan=42: 0 *nan<0: 1 nan>0: 0

The * lines are wrong. negative zero is not supported, and might not be practical. nan=nan should be false (0), as should nan<0. division by 0 is a fatal error.

If you fancy having a go at getting nan to work properly (x86 assembly required), see builtins\VM\pJcc.e (search for nan, 4 places, marked with --DEV this may be the wrong thing to do entirely) and also (if you succeed) test\t41infan.exw will need a few corrections.

If you fancy having a go at negative zero support (ditto), your first stop should be :%pStoreFlt in builtins\VM\pHeap.e and use whatever the test is for -0.0 there. I would be happiest if apps that needed support of -0.0 had to explicitly call something in pHeap.e to set a flag to enable any new code.

## PicoLisp[edit]

PicoLisp has only very limited built-in floating point support, and handles the rest by calling native (typically C) libraries. Minus zero and negative infinity cannot be represented, while NaN is represented by NIL

(load "@lib/math.l")

: (exp 1000.0) # Too large for IEEE floats

-> T

: (+ 1 2 NIL 3) # NaN propagates

-> NIL

## PureBasic[edit]

Define.f

If OpenConsole()

inf = Infinity() ; or 1/None ;None represents a variable of value = 0

minus_inf = -Infinity() ; or -1/None

minus_zero = -1/inf

nan = NaN() ; or None/None

PrintN("positive infinity: "+StrF(inf))

PrintN("negative infinity: "+StrF(minus_inf))

PrintN("positive zero: "+StrF(None))

PrintN("negative zero: "+StrF(minus_zero)) ; handles as 0.0

PrintN("not a number: "+StrF(nan))

PrintN("Arithmetics")

PrintN("+inf + 2.0 = "+StrF(inf + 2.0))

PrintN("+inf - 10.1 = "+StrF(inf - 10.1))

PrintN("+inf + -inf = "+StrF(inf + minus_inf))

PrintN("0.0 * +inf = "+StrF(0.0 * inf))

PrintN("1.0/-0.0 = "+StrF(1.0/minus_zero))

PrintN("NaN + 1.0 = "+StrF(nan + 1.0))

PrintN("NaN + NaN = "+StrF(nan + nan))

PrintN("Logics")

If IsInfinity(inf): PrintN("Variable 'Infinity' is infinite"): EndIf

If IsNAN(nan): PrintN("Variable 'nan' is not a number"): EndIf

Print(#CRLF$+"Press ENTER to EXIT"): Input()

EndIf

positive infinity: +Infinity negative infinity: -Infinity positive zero: 0.0000000000 negative zero: 0.0000000000 not a number: -1.#IND000000 Arithmetics +inf + 2.0 = +Infinity +inf - 10.1 = +Infinity +inf + -inf = -1.#IND000000 0.0 * +inf = -1.#IND000000 1.0/-0.0 = -Infinity NaN + 1.0 = -1.#IND000000 NaN + NaN = -1.#IND000000 Logics Variabel 'Infinity' is infinite Variable 'nan' is not a number Press ENTER to EXIT

## Python[edit]

>>> # Extreme values from expressions

>>> inf = 1e234 * 1e234

>>> _inf = 1e234 * -1e234

>>> _zero = 1 / _inf

>>> nan = inf + _inf

>>> inf, _inf, _zero, nan

(inf, -inf, -0.0, nan)

>>> for value in (inf, _inf, _zero, nan): print (value)

inf

-inf

-0.0

nan

>>> # Extreme values from other means

>>> float('nan')

nan

>>> float('inf')

inf

>>> float('-inf')

-inf

>>> -0.

-0.0

>>> # Some arithmetic

>>> nan == nan

False

>>> nan is nan

True

>>> 0. == -0.

True

>>> 0. is -0.

False

>>> inf + _inf

nan

>>> 0.0 * nan

nan

>>> nan * 0.0

nan

>>> 0.0 * inf

nan

>>> inf * 0.0

nan

>>> # But note!

>>> 1 / -0.0

Traceback (most recent call last):

File "<pyshell#106>", line 1, in <module>

1 / -0.0

ZeroDivisionError: float division by zero

>>> # (Not minus infinity)

## R[edit]

# 0 and -0 are recognized but are both printed as simply 0.

1/c(0, -0, Inf, -Inf, NaN)

# Inf -Inf 0 0 NaN

## Racket[edit]

#lang racket

(define division-by-zero (/ 1.0 0.0)) ;+inf.0

(define negative-inf (- (/ 1.0 0.0))) ;-inf.0

(define zero 0.0) ;0.0

(define negative-zero (- 0.0)) ;-0.0

(define nan (/ 0.0 0.0)) ;+nan.0

(displayln division-by-zero)

(displayln negative-inf)

(displayln zero)

(displayln negative-zero)

(displayln nan)

(+ zero negative-zero) ;0.0

(- negative-inf division-by-zero) ; +nan.0

(+ zero nan) ; +nan.0

(= nan +nan.0) ;#f

This values can be assigned to a variable just as normal values

## Ruby[edit]

inf = 1.0 / 0.0 #=> Infinity

nan = 0.0 / 0.0 #=> NaN

expression = [

"1.0 / 0.0", "-1.0 / 0.0", "0.0 / 0.0", "- 0.0",

"inf + 1", "5 - inf", "inf * 5", "inf / 5", "inf * 0",

"1.0 / inf", "-1.0 / inf", "inf + inf", "inf - inf",

"inf * inf", "inf / inf", "inf * 0.0", " 0 < inf", "inf == inf",

"nan + 1", "nan * 5", "nan - nan", "nan * inf", "- nan",

"nan == nan", "nan > 0", "nan < 0", "nan == 0", "nan <=> 0.0", "0.0 == -0.0",

]

expression.each do |exp|

puts "%15s => %p" % [exp, eval(exp)]

end

- Output:

1.0 / 0.0 => Infinity -1.0 / 0.0 => -Infinity 0.0 / 0.0 => NaN - 0.0 => -0.0 inf + 1 => Infinity 5 - inf => -Infinity inf * 5 => Infinity inf / 5 => Infinity inf * 0 => NaN 1.0 / inf => 0.0 -1.0 / inf => -0.0 inf + inf => Infinity inf - inf => NaN inf * inf => Infinity inf / inf => NaN inf * 0.0 => NaN 0 < inf => true inf == inf => true nan + 1 => NaN nan * 5 => NaN nan - nan => NaN nan * inf => NaN - nan => NaN nan == nan => false nan > 0 => false nan < 0 => false nan == 0 => false nan <=> 0.0 => nil 0.0 == -0.0 => true

## S-lang[edit]

Each of these can be directly input; I'll calc the Infs for good measure:

foreach $1 ([{-0.0}, {_Inf, "1.0/0"}, {-_Inf, "-1.0/0"}, {_NaN}]) {

() = printf("%S", $1[0]);

if (length($1) > 1) () = printf("\t%S\n", eval($1[1]));

else () = printf("\n");

}

- Output:

-0.0 inf inf -inf -inf nan

% And make some comparisons:

() = printf("-0.0 and 0.0 are %sequal\n", -0.0 == 0.0 ? "" : "not ");

() = printf("-_Inf == _Inf are %sequal\n", -_Inf == _Inf ? "" : "not ");

() = printf("-0.0 and 0.0 are %sthe 'same'\n", __is_same(-0.0, 0.0) ? "" : "not ");

- Output:

-0.0 and 0.0 are equal -_Inf == _Inf are not equal -0.0 and 0.0 are not the 'same'

## Scala[edit]

object ExtremeFloatingPoint extends App {

val negInf = -1.0 / 0.0 //also Double.NegativeInfinity

val inf = 1.0 / 0.0 // //also Double.PositiveInfinity

val nan = 0.0 / 0.0 // //also Double.NaN

val negZero = -2.0 / inf

println("Value: Result: Infinity? Whole?")

println(f"Negative inf: ${negInf}%9s ${negInf.isInfinity}%9s ${negInf.isWhole}%9s")

println(f"Positive inf: ${inf}%9s ${inf.isInfinity}%9s ${inf.isWhole}%9s")

println(f"NaN: ${nan}%9s ${nan.isInfinity}%9s ${nan.isWhole}%9s")

println(f"Negative 0: ${negZero}%9s ${negZero.isInfinity}%9s ${negZero.isWhole}%9s")

println(f"inf + -inf: ${inf + negInf}%9s ${(inf + negInf).isInfinity}%9s ${(inf + negInf).isWhole}%9s")

println(f"0 * NaN: ${0 * nan}%9s ${(inf + negInf).isInfinity}%9s ${(inf + negInf).isWhole}%9s")

println(f"NaN == NaN: ${nan == nan}%9s")

}

- Output:

Value: Result: Infinity? Whole? Negative inf: -Infinity true false Positive inf: Infinity true false NaN: NaN false false Negative 0: -0.0 false true inf + -inf: NaN false false 0 * NaN: NaN false false NaN == NaN: false

## Scheme[edit]

(define infinity (/ 1.0 0.0))

(define minus-infinity (- infinity))

(define zero 0.0)

(define minus-zero (- zero))

(define not-a-number (/ 0.0 0.0))

(equal? (list infinity minus-infinity zero minus-zero not-a-number)

(list +inf.0 -inf.0 0.0 -0.0 +nan.0))

; #t

## Seed7[edit]

The type float works according to IEEE 754.
Constants like Infinity and
NaN are predefined in the library
float.s7i.
A zero is always written without sign (e.g.: *write(-0.0)* writes *0.0*,
and *write(-0.004 digits 2);* writes *0.00*).
To recognize negative zero the function isNegativeZero
can be used. NaN can be checked with isNaN.

$ include "seed7_05.s7i";

include "float.s7i";

const proc: main is func

begin

writeln("positive infinity: " <& Infinity);

writeln("negative infinity: " <& -Infinity);

writeln("negative zero: " <& -0.0);

writeln("not a number: " <& NaN);

# some arithmetic

writeln("+Infinity + 2.0 = " <& Infinity + 2.0);

writeln("+Infinity - 10.1 = " <& Infinity - 10.1);

writeln("+Infinity + -Infinity = " <& Infinity + -Infinity);

writeln("0.0 * +Infinity = " <& 0.0 * Infinity);

writeln("1.0/-0.0 = " <& 1.0 / -0.0);

writeln("NaN + 1.0 = " <& NaN + 1.0);

writeln("NaN + NaN = " <& NaN + NaN);

# some comparisons

writeln("NaN = NaN = " <& NaN = NaN);

writeln("isNaN(NaN) = " <& isNaN(NaN));

writeln("0.0 = -0.0 = " <& 0.0 = -0.0);

writeln("isNegativeZero(-0.0) = " <& isNegativeZero(-0.0));

writeln("isNegativeZero(0.0) = " <& isNegativeZero(0.0));

end func;

- Output:

positive infinity: Infinity negative infinity: -Infinity negative zero: 0.0 not a number: NaN +Infinity + 2.0 = Infinity +Infinity - 10.1 = Infinity +Infinity + -Infinity = NaN 0.0 * +Infinity = NaN 1.0/-0.0 = -Infinity NaN + 1.0 = NaN NaN + NaN = NaN NaN = NaN = FALSE isNaN(NaN) = TRUE 0.0 = -0.0 = TRUE isNegativeZero(-0.0) = TRUE isNegativeZero(0.0) = FALSE

## Sidef[edit]

*NaN* and *Inf* literals can be used to represent the *Not-a-Number* and *Infinity* values, which are returned in special cases, such as *0/0* and *1/0*. However, one thing to notice, is that in Sidef there is no distinction between *0.0* and *-0.0* and can't be differentiated from each other.

var inf = (1 / 0) #=> Inf

var nan = (0 / 0) #=> NaN

var exprs = [

"1.0 / 0.0", "-1.0 / 0.0", "0.0 / 0.0", "- 0.0",

"inf + 1", "5 - inf", "inf * 5", "inf / 5", "inf * 0",

"1.0 / inf", "-1.0 / inf", "inf + inf", "inf - inf",

"inf * inf", "inf / inf", "inf * 0.0", " 0 < inf", "inf == inf",

"nan + 1", "nan * 5", "nan - nan", "nan * inf", "- nan",

"nan == nan", "nan > 0", "nan < 0", "nan == 0", "0.0 == -0.0",

]

exprs.each { |expr|

"%15s => %s\n".printf(expr, eval(expr))

}

say "-"*40

say("NaN equality: ", NaN == nan) #=> true

say("Infinity equality: ", Inf == inf) #=> true

say("-Infinity equality: ", -Inf == -inf) #=> true

say "-"*40

say("sqrt(-1) = ", sqrt(-1)); #=> i

say("tanh(-Inf) = ", tanh(-inf)); #=> -1

say("(-Inf)**2 = ", (-inf)**2); #=> Inf

say("(-Inf)**3 = ", (-inf)**3); #=> -Inf

say("acos(Inf) = ", acos(inf)); #=> Inf*i

say("atan(Inf) = ", atan(inf)); #=> pi/2

say("log(-1) = ", log(-1)); #=> pi*i

say("atanh(Inf) = ", atanh(inf)); #=> -pi/2*i

- Output:

1.0 / 0.0 => Inf -1.0 / 0.0 => -Inf 0.0 / 0.0 => NaN - 0.0 => 0 inf + 1 => Inf 5 - inf => -Inf inf * 5 => Inf inf / 5 => Inf inf * 0 => NaN 1.0 / inf => 0 -1.0 / inf => 0 inf + inf => Inf inf - inf => NaN inf * inf => Inf inf / inf => NaN inf * 0.0 => NaN 0 < inf => true inf == inf => true nan + 1 => NaN nan * 5 => NaN nan - nan => NaN nan * inf => NaN - nan => NaN nan == nan => true nan > 0 => false nan < 0 => false nan == 0 => false 0.0 == -0.0 => true ---------------------------------------- NaN equality: true Infinity equality: true -Infinity equality: true ---------------------------------------- sqrt(-1) = i tanh(-Inf) = -1 (-Inf)**2 = Inf (-Inf)**3 = -Inf acos(Inf) = Infi atan(Inf) = 1.5707963267948966192313216916397514421 log(-1) = 3.1415926535897932384626433832795028842i atanh(Inf) = -1.5707963267948966192313216916397514421i

## Swift[edit]

let negInf = -1.0 / 0.0

let inf = 1.0 / 0.0 //also Double.infinity

let nan = 0.0 / 0.0 //also Double.NaN

let negZero = -2.0 / inf

println("Negative inf: \(negInf)")

println("Positive inf: \(inf)")

println("NaN: \(nan)")

println("Negative 0: \(negZero)")

println("inf + -inf: \(inf + negInf)")

println("0 * NaN: \(0 * nan)")

println("NaN == NaN: \(nan == nan)")

- Output:

Negative inf: -inf Positive inf: inf NaN: nan Negative 0: -0.0 inf + -inf: nan 0 * NaN: nan NaN == NaN: false

## Tcl[edit]

Tcl includes support in expressions for all IEEE “extreme” values except for NaN, which it throws a catchable exception on encountering numerically. Moreover, all can be just written directly as literals (they are parsed case-insensitively). For example, see this log of an interactive session:

% package require Tcl 8.5

8.5.2

% expr inf+1

Inf

% set inf_val [expr {1.0 / 0.0}]

Inf

% set neginf_val [expr {-1.0 / 0.0}]

-Inf

% set negzero_val [expr {1.0 / $neginf_val}]

-0.0

% expr {0.0 / 0.0}

domain error: argument not in valid range

% expr nan

domain error: argument not in valid range

% expr {1/-inf}

-0.0

It *is* possible to introduce a real NaN though numeric computation,
but only by using the mechanisms for dealing with external binary data
(it being judged better to just deal with it in that case
rather than throwing an exception):

% binary scan [binary format q nan] q nan

1

% puts $nan

NaN

% # Show that it is a real NaN in there

% expr {$nan+0}

can't use non-numeric floating-point value as operand of "+"

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