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rounding rules: properties to be ... "Round to nearest, ties to even" is the default for binary floating point and the recommended default for decimal. "Round to ...
This variant of the round-to-nearest method is also called convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, odd–even rounding, [6] or bankers' rounding. [ 7 ] This is the default rounding mode used in IEEE 754 operations for results in binary floating-point formats.
This rounding rule is biased because it always moves the result toward zero. Round-to-nearest: () is set to the nearest floating-point number to . When there is a tie, the floating-point number whose last stored digit is even (also, the last digit, in binary form, is equal to 0) is used.
The decimal number 0.15625 10 represented in binary is 0.00101 2 (that is, 1/8 + 1/32). (Subscripts indicate the number base.) Analogous to scientific notation, where numbers are written to have a single non-zero digit to the left of the decimal point, we rewrite this number so it has a single 1 bit to the left of the "binary point". We simply ...
round to nearest, where ties round to the nearest even digit in the required position (the default and by far the most common mode) round to nearest, where ties round away from zero (optional for binary floating-point and commonly used in decimal) round up (toward +∞; negative results thus round toward zero)
The binary interchange formats have the "half precision" (16-bit storage format) and "quad precision" (128-bit format) added, together with generalized formulae for some wider formats; the basic formats have 32-bit, 64-bit, and 128-bit encodings. Three new decimal formats are described, matching the lengths of the 32–128-bit binary formats.
Variable length arithmetic represents numbers as a string of digits of a variable's length limited only by the memory available. Variable-length arithmetic operations are considerably slower than fixed-length format floating-point instructions.
The advantage of "round to even" (if the radix is not a multiple of 4) is that it prevents the problem of halves from propagating. Consider 23.55: with a "round to odd" rule this would become 23.5 and then 23, though 24 would be nearer. Instead, 23.55 is rounded to 23.6 and then to 24.