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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.
Mathematically, an apportionment method is just a method of rounding real numbers to natural numbers. Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes, as proven by the Balinski-Young theorem. The mathematical theory of apportionment identifies what properties can be expected from an apportionment ...
If the n + 1 digit is 5 not followed by other digits or followed by only zeros, then rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant figures: Round half away from zero rounds up to 1.3. This is the default rounding method implied in many disciplines [citation needed] if the required rounding method is not ...
The two names for these methods—highest averages and divisors—reflect two different ways of thinking about them, and their two independent inventions. However, both procedures are equivalent and give the same answer. [1] Divisor methods are based on rounding rules, defined using a signpost sequence post(k), where k ≤ post(k) ≤ k+1.
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)
Round-by-chop: The base-expansion of is truncated after the ()-th digit. 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 ...
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This alternative definition is significantly more widespread: machine epsilon is the difference between 1 and the next larger floating point number.This definition is used in language constants in Ada, C, C++, Fortran, MATLAB, Mathematica, Octave, Pascal, Python and Rust etc., and defined in textbooks like «Numerical Recipes» by Press et al.