28.7 Rounding

When floating-point arithmetic produces a result that can’t fit exactly in the significand of the type that’s in use, it has to round the value. The basic arithmetic operations—addition, subtraction, multiplication, division, and square root—always produce a result that is equivalent to the exact, possibly infinite-precision result rounded to storage precision according to the current rounding rule.

Rounding sets the FE_INEXACT exception flag (see Exception Flags). This enables programs to determine that rounding has occurred.

Rounding consists of adjusting the exponent to bring the significand back to the required base-point alignment, then applying the current rounding rule to squeeze the significand into the fixed available size.

The current rule is selected at run time from four options. Here they are:

• * round-to-nearest, with ties rounded to an even integer;
• * round-up, towards +Infinity;
• * round-down, towards -Infinity;
• * round-towards-zero.

Under those four rounding rules, a decimal value -1.2345 that is to be rounded to a four-digit result would become -1.234, -1.234, -1.235, and -1.234, respectively.

The default rounding rule is round-to-nearest, because that has the least bias, and produces the lowest average error. When the true result lies exactly halfway between two representable machine numbers, the result is rounded to the one that ends with an even digit.

The round-towards-zero rule was common on many early computer designs, because it is the easiest to implement: it just requires silent truncation of all extra bits.

The two other rules, round-up and round-down, are essential for implementing interval arithmetic, whereby each arithmetic operation produces lower and upper bounds that are guaranteed to enclose the exact result.

See Rounding Control, for details on getting and setting the current rounding mode.