Floating Point Arithmetic — Changing Decimal to FP: Computer Architecture

Consider the following:
We want 13-bit floating point representation of 27.37
with a 5 bit Bias
Format is : Sign _ , Biased Exp _ _ _ _ _ , Significand _ _ _ _ _ _ _
Actual Exp is also referred to as True Exp
Bias = 2^(k-1) — 1
Biased Exp = True Exp + Bias
So, True Exp = Biased Exp — Bias
Calculate 27.37 in Decimal:
27.37 = 11011.010
Calculate Bias in Decimal:
Bias = ²⁴ — 1 = 15
Normalize The Significand and gain True Exp
it is always in the format 1.significand
so, we get 1.1011010 x 2⁴ (which is taking the 11011.010 above and moving the point to the first position after the first “1”
4 is the “power” or superscript because the point was moved four places to the left.
The 4, which is the superscript of any normalized binary number, would be the True Exp.
So, True Exp = 4
Calculate Biased Exp:
Biased Exp = True Exp + Bias
Biased Exp = 4 + 15
Biased Exp = 19
Finding the Biased Exp in Binary:
19 = 10011
Assemble components into FP Format
Sign- positive since it is 27.37
Biased Exp- 10011
Significand- 1011010
Finding the Floating Point representation:
0 10011 1011010
