Bugs when multiplying numbers with bitwise operators

Question

I am trying to multiply two floating-point numbers using bitwise operators in IEEE-754 format. The 32-bit number is composed in the form sign - exponent - mantissa . After multiplying each number, the resultant answer is correct some of the time but not all of the time.

I think it has something to do with the resulting answer not being in normalized form (eg 1.1010101 * 2 ⁵ ), but I don't know how to fix it.

#include <csdtdio>

struct Real
{    
   int sign;
   long exponent;
   unsigned long fraction;
};

Real Multiply(Real Val1, Real Val2){
   Real answer;
   answer.fraction = left.fraction + right.fraction;
   answer.exponent = left.exponent  + right.exponent;
   answer.sign = left.sign ^ right.sign;
   return  answer;
}

Answer 1

While multiplying the mantissa parts must be multiplied together, not add

(-1) ^sign1 × 2 ^exp1 × mantissa1 * (-1) ^sign2 × 2 ^exp2 × mantissa2
= (-1) ^{sign1 + sign2} × 2 ^{exp1 + exp2} × mantissa1 × mantissa2

And you don't need a separate variable for returning

Real Multiply(Real Val1, Real Val2){
   Val1.fraction *= Val2.fraction;
   Val1.exponent += Val2.exponent;
   Val1.sign ^= Val2.sign;
   return Val1;
}

After those basic things you'll still have to do normalization, for which you need to get the full result instead of just the low bits like the normal non-widening multiplication. Therefore you must cast your "fraction" (if you're using IEEE-754 then the correct term for it is significand ) to a wider type. Depending on which platform you're on, you may or may not have a type twice as big as an unsigned long . It's better to used fixed-width types like int32_t , uint64_t in this case. That's all the hints needed to do this