Adding two numbers in base 2(floating point) vs Multiplying two numbers in base 2(floating point)

Is it true that adding two numbers in base 2 is more complex than multiplying them? If so can someone please explain why this is the case?

• To add floating point numbers you first need to put them on a “common denominator”, which isn’t necessary for multiplication. – Yuval Filmus Mar 28 at 5:51

Floating point numbers are stored as $$x \cdot 2^e$$, where typically $$1 \leq x < 2$$ (unless your number is denormalized). When multiplying $$x \cdot 2^e$$ and $$y \cdot 2^f$$, we simply compute $$xy \cdot 2^{e+f}$$ (we have to truncate $$xy$$). When adding $$x \cdot 2^e$$ and $$y \cdot 2^f$$, we first have to shift one of the numbers (the one with smaller exponent) so that we have a "common denominator". For example, to add $$1 \cdot 2^0$$ and $$1 \cdot 2^{-3}$$, you have to rewrite the latter as $$0.125 \cdot 2^0$$.
Whether this makes addition more complex than multiplication is a matter of opinion. Multiplying $$x$$ and $$y$$ is still harder than addition.