Mathematical Fundamentals

137

mantissa

exponent

biased representation

quantization step

numbers. Often, the rational numbers are considered to be normalized to one,

i.e., to be limited to the range [-1, 1). In such a case, the decimal point is placed

before the leftmost binary digit.

For the floating point representation we can follow different conventions. In

particular, the IEEE 754 floating-point single-precision numbers obey to the

following rules

· the number is represented as

1.xx . . . x

2

× 2

yy...y

2

,

(65)

where x are the binary digits of the mantissa and y are the binary digits

of the exponent

· The number is represented on 32 bits according to the following block

decomposition

bit 31: sign bit

bits 2330: exponent yy . . . y in biased representation
most negative 00 . . . 0 to the most positive 11 . . . 1

bits 022: mantissa in unsigned binary representation

The IEEE 754 standard of double-precision floating-point numbers uses 11 bits

for the exponent and 52 bits for the mantissa.

It should be clear that both the fixed- and the floating-point representations

take a subset of rational numbers. Fixed-point numbers are equally spaced be-

tween the minimum and the maximum representable value with a quantization

step equal to 2

-d

, where d is the number of digits on the right of the deci-

mal point. Floating-point numbers are unevenly distributed, being more sparse

for large values of the exponent and more dense for little exponents. Floating-

point numbers have the possibility to represent a large range, from 2 × 10

-38

to

2 × 10

38

in single precision, and from 2 × 10

-308

to 2 × 10

308

in double precision.

Therefore, it is possible to do many computations without worrying of errors

due to overflow. Moreover, the high density of small numbers reduces the prob-

lems due to the quantization step. This is paid in terms of a more complicated

arithmetics.

20

The bias is 127. Therefore, the exponent 1 is coded as 1 + 127 = 128 = 10000000

2

. The

biased representation simplifies the bit-oriented sorting operations.