The ‘roundtonearest’ mode works as in the IEEE 754 standard: in case the number to be rounded lies exactly in the middle of two representable numbers, it is rounded to the one with the least significant bit set to zero. For example, the number 2.5, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the drift phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2).

The MPFR_RNDF mode works as follows: the computed value is either that corresponding to MPFR_RNDD or that corresponding to MPFR_RNDU. In particular when those values are identical, i.e., when the result of the corresponding operation is exactly representable, that exact result is returned. Thus, the computed result can take at most two possible values, and in absence of underflow/overflow, the corresponding error is strictly less than one ulp (unit in the last place) of that result and of the exact result. For MPFR_RNDF, the ternary value (defined below) and the inexact flag (defined later, as with the other flags) are unspecified, the divide-by-zero flag is as with other roundings, and the underflow and overflow flags match what would be obtained in the case the computed value is the same as with MPFR_RNDD or MPFR_RNDU. The results may not be reproducible.

Most MPFR functions take as first argument the destination variable, as second and following arguments the input variables, as last argument a rounding mode, and have a return value of type int, called the ternary value. The value stored in the destination variable is correctly rounded, i.e., MPFR behaves as if it computed the result with an infinite precision, then rounded it to the precision of this variable. The input variables are regarded as exact (in particular, their precision does not affect the result).

As a consequence, in case of a non-zero real rounded result, the error on the result is less or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding).

Unless documented otherwise, functions returning an int return a ternary value. If the ternary value is zero, it means that the value stored in the destination variable is the exact result of the corresponding mathematical function. If the ternary value is positive (resp. negative), it means the value stored in the destination variable is greater (resp. lower) than the exact result. For example with the MPFR_RNDU rounding mode, the ternary value is usually positive, except when the result is exact, in which case it is zero. In the case of an infinite result, it is considered as inexact when it was obtained by overflow, and exact otherwise. A NaN result (Not-a-Number) always corresponds to an exact return value. The opposite of a returned ternary value is guaranteed to be representable in an int.

Unless documented otherwise, functions returning as result the value 1 (or any other value specified in this manual) for special cases (like acos(0)) yield an overflow or an underflow if that value is not representable in the current exponent range.

Reference