GMP floating point number.
The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on most systems. In the current implementation the exponent is a count of limbs, so for example on a 32-bit system this means a range of roughly 2^-68719476768 to 2^68719476736, or on a 64-bit system this will be greater.
Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high.
All calculations are performed to the precision of the destination variable. Each function is defined to calculate with "infinite precision" followed by a truncation to the destination precision, but of course the work done is only what's needed to determine a result under that definition.
The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn't be concerned by such details.
The mantissa in stored in binary, as might be imagined from the fact precisions are expressed in bits. One consequence of this is that decimal fractions like 0.1 cannot be represented exactly. The same is true of plain IEEE double floats. This makes both highly unsuitable for calculations involving money or other values that should be exact decimal fractions. (Suitably scaled integers, or perhaps rationals, are better choices.)
mpf functions and variables have no special notion of infinity or not-a-number, and applications must take care not to overflow the exponent or results will be unpredictable. This might change in a future release.
Note that the mpf functions are not intended as a smooth extension to IEEE P754 arithmetic. In particular results obtained on one computer often differ from the results on a computer with a different word size.