Round off Errors and Computer Arithmatic

Round-off Errors and Computer Arithmetic

Binary Machine Numbers
A 64-bit (binary digit) representation is used for a real number (according to IEEE standards).
(-1)^s 2^(c-1023) (1+f)
This representation is called floating point representation.
The first bit is a sign indicator, denoted s. This is followed by an 11-bit exponent, c, called the characteristic, and a 52-bit binary fraction, f, called the mantissa. The base for the exponent is 2.
Example 1: consider the following machine number:

0 10000000011 1011100100010000000000000000000000000000000000000000




Sign: (0: positive; 1:negative)

Characteristic:
c=1×2^0+1×2^1+0×2^2+0×2^3+0×2^4+0×2^5+0×2^6+0×2^7+0×2^8+0×2^9+1×2^10
=1+2+1024=1027.

The exponential part of the number is, therefore:
2^(1027-1023)=2^4

Mantissa The final 52 bits is:
f=1×(1/2)^1+1×(1/2)^3+1×(1/2)^4+1×(1/2)^5+1×(1/2)^8+1×(1/2)^12

(-1)^s 2^(c-1023) (1+f)= (-1)^0 2^(1027-1023) (1+(1×(1/2)^1+1×(1/2)^3+1×(1/2)^4+1×(1/2)^5+1×(1/2)^8+1×(1/2)^12 ))
=27.56640625
Decimal Machine Numbers

Machine numbers are represented in the normalized decimal floating-point form