20.6 Rounding Modes

Floating-point calculations are carried out internally with extra precision, and then rounded to fit into the destination type. This ensures that results are as precise as the input data. IEEE 754 defines four possible rounding modes:

Round to nearest.

This is the default mode. It should be used unless there is a specific need for one of the others. In this mode results are rounded to the nearest representable value. If the result is midway between two representable values, the even representable is chosen. Even here means the lowest-order bit is zero. This rounding mode prevents statistical bias and guarantees numeric stability: round-off errors in a lengthy calculation will remain smaller than half of FLT_EPSILON.

Round toward plus Infinity.

All results are rounded to the smallest representable value which is greater than the result.

Round toward minus Infinity.

All results are rounded to the largest representable value which is less than the result.

Round toward zero.

All results are rounded to the largest representable value whose magnitude is less than that of the result. In other words, if the result is negative it is rounded up; if it is positive, it is rounded down.

fenv.h defines constants which you can use to refer to the various rounding modes. Each one will be defined if and only if the FPU supports the corresponding rounding mode.

FE_TONEAREST

Round to nearest.

FE_UPWARD

Round toward +∞.

FE_DOWNWARD

Round toward -∞.

FE_TOWARDZERO

Round toward zero.

Underflow is an unusual case. Normally, IEEE 754 floating point numbers are always normalized (see Floating Point Concepts). Numbers smaller than 2^r (where r is the minimum exponent, FLT_MIN_RADIX-1 for float) cannot be represented as normalized numbers. Rounding all such numbers to zero or 2^r would cause some algorithms to fail at 0. Therefore, they are left in denormalized form. That produces loss of precision, since some bits of the mantissa are stolen to indicate the decimal point.

If a result is too small to be represented as a denormalized number, it is rounded to zero. However, the sign of the result is preserved; if the calculation was negative, the result is negative zero. Negative zero can also result from some operations on infinity, such as 4/-∞.

At any time one of the above four rounding modes is selected. You can find out which one with this function:

To change the rounding mode, use this function:

You should avoid changing the rounding mode if possible. It can be an expensive operation; also, some hardware requires you to compile your program differently for it to work. The resulting code may run slower. See your compiler documentation for details.