Prerequisite – IEEE Standard 754 Floating Point Numbers
Here, we have discussed an algorithm to multiply two floating point numbers, x and y.
- Convert these numbers in scientific notation, so that we can explicitly represent hidden 1.
- Let ‘a’ be the exponent of x and ‘b’ be the exponent of y.
- Assume resulting exponent c = a+b. It can be adjusted after the next step.
- Multiply mantissa of x to mantissa of y. Call this result m.
- If m does not have a single 1 left of radix point, then adjust radix point so it does, and adjust exponent c to compensate.
- Add sign bits, mod 2, to get sign of resulting multiplication.
- Convert back to one byte floating point representation, truncating bits if needed.
Negative values are simple to take care of in floating point multiplication. Treat sign bit as 1 bit unsigned binary, add mod 2. This is the same as XORing the sign bit.
Suppose you want to multiply following two numbers:
Now, these are steps according to above algorithm:
- Given, A = 1.11 x 2^0 and B = 1.01 x 2^2
- So, exponent c = a + b = 0 + 2 = 2 is the resulting exponent.
- Now, multiply 1.11 by 1.01, so result will be 10.0011
- We need to normalize 10.0011 to 1.00011 and adjust exponent 1 by 3 appropriately.
- Resulting sign bit 0 (XOR) 0 = 0, means positive.
- Now, truncate and normalite it 1.00011 x 2^3 to 1.000 x 2^3.
Therefore, resultant number is,
Similarly, we can multiply other floating point numbers.
Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready.