A floatingpoint number system is characterized by four integ

A floating-point number system is characterized by four integers: the base beta, the precision p, and the lower and upper limits L and U of the exponent range. (a) If beta = 10, what are the smallest values of p and U, and largest value of L, such that both 2365.27 and 0.0000512 can be represented exactly in a normalized floating-point system? (b) How would your answer change if the system is not normalized, i.e., if gradual underflow is allowed?

Solution

Floating-point number system characterized by four integers:

Beeta notation = base,

P = Precision,

[L,U] = Exponent Range

Number x represented as

x = ± (d₀ + d₁/ + d₂/ ² + · · · + d_p1/ ^p1 ) ^E ,

Where

0 <= d_i <= -1, i=0,......, p-1, and L<= E <= U

d₀d₁.....d_p-1 called mantissa

E called exponent

d₁d₂.....d_p-1 called fraction.

Normalization

Floating-point system normalized if leading digit d₀ always nonzero unless number represented is zero.

1 <= m <

Reasons for normalization:

1) Representation of each number unique.

2) no digits wasted on leading zeros

3) Leading bit need not be stored(In Binary System)

a) Assume = 10, p = 6

Let X = 2.36527 * 10^{3 ,} Y = 5.12 * 10^-5

Floating-point addition gives..

X+Y = 2.365270051 * 10³

Assuming rounding to nearest

Last two digits of Y do not affect to result, and with even smaller exponent, Y could have had

no effect onfinal result.

Floating-point multiplication gives

X*Y = 1.21101824 * 10^-1.........................

Real result may also fail to be representable because its exponent is beyond available range.

Overflow usually more serious than underflow because there is no good approximation to arbitrarily

large magnitudes in floating-point-system, whereas zero is often reasonable approximation for arbitrarily small

magnitudes

On many computer systems overflow is fatal but an underflow may be silently set to zero

b) when the answer changed if the numer system is not normalized

gradual underflow is allowed?

Computers satisfying IEEE floating-point standard achieve this ideal as long as x op y is with in range

of floating-point-system. But some familiar laws of real arithmetic not necessarily valid in floating-point

system.

Smallest positive normalized number :

Underflow Level = UFL = ^L

According to given scenario..

L means lower value = 10

Then 10 *^{( 5.12*10-5)} = UFL

Not all real numbers exactly representable; those that are called machine numbers.