IEEE 754 specifies four formats for representing floatingpoint values: singleprecision (32bit), doubleprecision (64bit), singleextended precision (>= 43bit, not commonly used) and doubleextended precision (>= 79bit, usually implemented with 80 bits). Only 32bit values are required by the standard, the others are optional. Many languages specify that they implement IEEE arithmetic, although sometimes it is optional. The C programming language for example allows but does not require IEEE arithmetic. IEEE is commonly used in C where float implemented IEEE single precision and double implements IEEE double precision.
Also known as IEEE Standard for Binary FloatingPoint Arithmetic (ANSI/IEEE Std 7541985) and IEC 559: "Binary floatingpoint arithmetic for microprocessor systems".
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Following is a description of the standard's format for floating point numbers.
A binary floating point number is stored in a 32 bit word:
1 8 23 width in bits ++++ S Exp  Fraction  ++++ 31 30 23 22 0 bit index (0 on right) bias +127
The set of possible data values can be divided into the following classes:
(NaNs are used to represent exceptional cases, such as the square root of a negative number)Each class can be distinguished by the value of the Exp field (well, nearly):
Consider the Exp and Fraction fields as unsigned binary numbers (Exp will be in the range 0255):
Class Exp FractionZeroes 0 0 Denormalised numbers 0 non zero Normalised numbers 1254 any Infinities 255 0 NaN (Not a Number) 255 non zero
For normalised numbers, the most common, Exp is the biased exponent and Fraction is the fractional part of the mantissa. The number has value v:
v = s * 2^e * m
Where
s = 1 (positive numbers) when S is 0
s = 1 (negative numbers) when S is 1
e = Exp  127 (in other words the exponent is stored with 127 added to it, also called "biased with 127")
m = 1.Fraction in binary (the binary number 1 followed by the point followed by the binary bits of Fraction). Note 1 <= m < 2
Denormalised numbers are the same but e = 126 and m is 0.Fraction. Note that 126 is the smallest exponent for a normalised number.
There are two Zeroes, +0 (S is 0) and 0 (S is 1), and two Infinities +Inf (S is 0) and Inf (S is 1).
Notice that NaNs and Infinities have all 1s in the Exp field.
Double precision is basically the same but the fields are wider:
1 11 52 ++++ S Exp  Fraction  ++++ 63 62 52 51 0 bias +1023
NaNs and Infinities are represented with Exp being all 1s (2047).
See also: Let's Go To The (Floating) Point by Chris Hecker
Also note that the IEEE 754 standard is currently (2004) under revision. See: http://grouper.ieee.org/groups/754/ and current draft at: http://754r.ucbtest.org/