ARITHMETICS FOR COMPUTER: FLOATING-POINT REPRESENTATION

-Representation for non-integral numbers

-Including very small and very large numbers

-123,000,000,000,000 can be represented as 1.23 x 10¹⁴and 0.0000000000000123 can be represented as 1.23 x 10^-14

-What we have done, in effect, is dynamically to slide the decimal point to a convenient location and use the exponent of 10 to keep track of that decimal point. This allows a range of very large and very small numbers to be represented with only a few digits.

-In binary

To store a number in floating point representation, a computer word is divided into 3 fields, representing the sign, the exponent E, and the significand m respectively.

-Example:

       0.110 x 2⁵

0.0110 x 2⁶

110 x 2²

-The IEEE standard has three very important requirements:

l  consistent representation of floating point numbers across all machines adopting the standard

l  correctly rounded arithmetic (to be explained in the next section)

l  consistent and sensible treatment of exceptional situations such as division by zero (to be discussed in the following section).

IEEE Floating-Point Format

}  S: sign bit (0 Þ non-negative, 1 Þ negative)

}  Normalize significand: 1.0 ≤ |significand| < 2.0

◦      Always has a leading pre-binary-point 1 bit, so no need to represent it explicitly (hidden bit)

◦      Significand is Fraction with the “1.” restored

}  Exponent: excess representation: actual exponent + Bias

◦      Ensures exponent is unsigned

◦      Single: Bias = 127; Double: Bias = 1203

Example:

1) Represent -14.375

• Convert to binary number: 1110.011₂
• 1110.011₂ = 1.110011x 2³
• Sign = 1 (negative number)
• Exponent = 3
• bias = 2⁸ -  1 = 127 (8-bit exponent encoding for binary 32)

          adding this to the exponent gives:

          3+127=130= 10000010₂ 

• Mantissa = 110011 (fractional part to the right of the decimal point)

  SIGN	EXPONENT	MANTISSA
  1	10000010	11001100000000000000000

2) Represent 11000000101100110...00₂