C why is the DBL_MANT_DIG macro defined as 53

Question

I want to know why DBL_MANT_DIG is set to 53, even though the ieee 754 standard uses 52 mantissa bits. Is it counting the sign bit as well?

Answer 1

In fact the binary IEE 754 representation includes a hidden bit: the leading bit of the mantissa is required to be 1 for normal numbers and because of that it has not to be stored. So the mantissa is in fact that leading 1 bit plus the 52 other bits which gives... 53

References from Wikipedia (emphasize mine):

For numbers with an exponent in the normal range (the exponent field being neither all ones nor all zeros), the leading bit of the significand will always be 1. Consequently, a leading 1 can be implied rather than explicitly present in the memory encoding, and under the standard the explicitly represented part of the significand will lie between 0 and 1. This rule is called leading bit convention, implicit bit convention, or hidden bit convention. This rule allows the binary format to have an extra bit of precision.