How to perform a multiplication to a floating point number represented in IEEE754

Question

Given a floating point number M represented in IEEE754 as
0 10000000 01010101010101010101010

How to compute the result of Round(Mx20) ?

I know how to translate M into normalized form,
M = 1.01010101010101010101010(base 2) x 2^1

But I don't know how to do the multiplication M x 20 by hand effectively, because there are so many numbers in mantissa.
any hint is appreciated!

Answer 1

The way forward is the standard Long Multiplication method, but in base 2:

1.01010101010101010101010₂•2¹ × 20
= 1.01010101010101010101010₂•2¹ × 5•4
= 1.01010101010101010101010₂•2¹ × 5•2²
= 1.01010101010101010101010₂•2¹ × 101₂•2²

    1.010101010101010101010102 • 21
                          1012 • 22
× _________________________________
    1.010101010101010101010102 • 23
   00.000000000000000000000002 • 23
  101.010101010101010101010002 • 23
+ _________________________________
  110.101010101010101010100102 • 23

Then we want to round the significand to 24 bits, shown in bold: 110.10101010101010101010010₂ • 2³.

So the result is 110.1010101010101010101₂ • 2³, or 1.1010101010101010101010010₂ • 2⁵.