A quick intuition: magic number 7e ffffff is negating by two's complement both the mantissa and exponent. 1. 7e: the first sig bit has to be zero to preserve

A quick intuition: magic number 7e ffffff is negating by two's complement both the mantissa and exponent. 1. 7e: the first sig bit has to be zero to preserve the sign of the overall number. 2. ffffff: due to Taylor series 1/(1 + X) = 1 - X ..., negating gives the multiplicative inverse. Although an IEEE float has a sign bit (thus 2's complement does not work on the float itself) but mantissa and the exponent individually work with 2's complement for different reasons. The exponent has a biased range due to being chopped by half; where as the mantissa has a biased range due to its definition and constant offset. The exponent's 1 offset (7e instead of 7f) is a bit more difficult to see at first -- the devil is in the details, but 2's complement is the basic intuition.