A data processing apparatus and method are provided for multiplying first and second normalized floating point operands in order to generate a result, each normalized floating point operand comprising a significand and an exponent. Exponent determination circuitry is used to compute a result exponent for a normalized version of the result, and rounding value generation circuitry then generates a rounding value by shifting a rounding constant in a first direction by a shift amount that is dependent on the result exponent. Partial product generation circuitry multiplies the significands of the first and second normalized floating point operands to generate the first and second partial products, and the first and second partial products are then added together, along with the rounding value, in order to generate a normalized result significand. Thereafter, the normalized result significand is shifted in a second direction opposite to the first direction, by the shift amount, in order to generate a rounded result significand. This provides a particularly efficient mechanism for multiplying floating point numbers, while correctly rounding the result in situations where the result is subnormal.