The unit of information is a word of 32 bits, (double-precision numbers require two words each). A word may contain:
- positive integer: true binary - negative integer: two's complement of the absolute value - floating point: IEEE standard as shown below - Hollerith: 4 character 8-bit ASCII, stored left to right - bit pattern: 32 bits of zero or one (no conversion)
The following reduction may occur when transforming to the exchange representation on machines with a word-size larger than 32 bits:
- integer: 'bad conversion' if more than 31 significant bits
- floating: reduced precision, but correctly rounded
- Hollerith: only the first 4 characters are taken
(trailing characters should be blank anyway)
- bit pattern: only the right-hand 32 bits are taken
The following expansions may occur for the inverse transformation:
- Hollerith: blank fill - bit pattern: zero-fill of the leading bits
The IEEE representation of single and double precision floating-point numbers is:
Single precision: 1 bit 32 the sign-bit 's'
8 bits 31 - 24 the characteristic 'c'
23 bits 23 - 1 the mantissa 'm'
Double precision: 1 bit 64 the sign-bit 's'
11 bits 63 - 53 the characteristic 'c'
52 bits 52 - 1 the mantissa 'm'
A normal floating point number is represented by 3 fields on one or two 32-bit words:
'm' is the fractional part of the mantissa M;
the binary point is assumed to be immediately to the left
of the most significant bit of m,
preceded by an implied 1 (hidden bit), for example:
m M mantissa decimal value
0000...0 1.0000...0 1.
1000...0 1.1000...0 1.5
1100...0 1.1100...0 1.75
'c' is the characteristic, being the binary exponent with a bias of
127 or 1023 added, for example:
c exponent c exp
011...101 -2 100...000 1
011...110 -1 100...001 2
011...111 0 100...011 3
A normal floating-point number does not have a c of all ones or all zeros, such characteristics are used to represent Exceptions.
's' is the sign-bit, the number is negative if this bit is on.
The representation of floating-point numbers is of the
sign/magnitude form, ie. only the sign-bit is complemented
to change the sign of the number.
The Exceptions are:
c=0, m=0 floating-point zero
c=0, m= not 0 'de-normalized number',
value: +/- 0.m * 2**(-126 or -1022)
c=all 1, m=0 signed infinity (c=255 or 2047)
c=all 1, m= not 0 'not a number (NaN)'