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One method, more obscure than most, is to alternate direction when rounding a number with 0.5 fractional part. All others are rounded to the closest integer. Whenever the fractional part is 0.5, alternate rounding up or down: for the first occurrence of a 0.5 fractional part, round up, for the second occurrence, round down, and so on.
IEEE 754 requires correct rounding: that is, the rounded result is as if infinitely precise arithmetic was used to compute the value and then rounded (although in implementation only three extra bits are needed to ensure this). There are several different rounding schemes (or rounding modes). Historically, truncation was the typical approach.
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x , denoted ⌈ x ⌉ or ceil( x ) .
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.
Given numbers and , the naive attempt to compute the mathematical function by the floating-point arithmetic ( ()) is subject to catastrophic cancellation when and are close in magnitude, because the subtraction can expose the rounding errors in the squaring.
Here we start with 0 in single precision (binary32) and repeatedly add 1 until the operation does not change the value. Since the significand for a single-precision number contains 24 bits, the first integer that is not exactly representable is 2 24 +1, and this value rounds to 2 24 in round to nearest, ties to even.
This rounding rule is more accurate but more computationally expensive. Rounding so that the last stored digit is even when there is a tie ensures that it is not rounded up or down systematically. This is to try to avoid the possibility of an unwanted slow drift in long calculations due simply to a biased rounding.
decimals can be negative to round down to a multiple of a power of ten. NAN may be returned if very large values are used. Examples. Template Result Expected