Rounding - Floating-point Rounding

Floating-point Rounding

In floating-point arithmetic, rounding aims to turn a given value x into a value z with a specified number of significant digits. In other words, z should be a multiple of a number m that depends on the magnitude of x. The number m is a power of the base (usually 2 or 10) of the floating-point representation.

Apart from this detail, all the variants of rounding discussed above apply to the rounding of floating-point numbers as well. The algorithm for such rounding is presented in the Scaled rounding section above, but with a constant scaling factor s=1, and an integer base b>1.

For results where the rounded result would overflow the result for a directed rounding is either the appropriate signed infinity, or the highest representable positive finite number (or the lowest representable negative finite number if x is negative), depending on the direction of rounding. The result of an overflow for the usual case of round to even is always the appropriate infinity.

In addition, if the rounded result would underflow, i.e. if the exponent would exceed the lowest representable integer value, the effective result may be either zero (possibly signed if the representation can maintain a distinction of signs for zeroes), or the smallest representable positive finite number (or the highest representable negative finite number if x is negative), possibly a denormal positive or negative number (if the mantissa is storing all its significant digits, in which case the most significant digit may still be stored in a lower position by setting the highest stored digits to zero, and this stored mantissa does not drop the most significant digit, something that is possible when base b=2 because the most significant digit is always 1 in that base), depending on the direction of rounding. The result of an underflow for the usual case of round to even is always the appropriate zero.