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One may also round half away from zero (or round half toward infinity), a tie-breaking rule that is commonly taught and used, namely: If the fractional part of x is exactly 0.5, then y = x + 0.5 if x is positive, and y = x − 0.5 if x is negative.
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. For IEEE standard where the base β {\displaystyle \beta } is 2 {\displaystyle 2} , this means when there is a tie it is rounded so that the last digit is equal to 0 {\displaystyle 0} .
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.
For example, the decimal numbers 0.1 and 0.01 cannot be represented exactly as binary floating-point numbers. In the IEEE 754 binary32 format with its 24-bit significand, the result of attempting to square the approximation to 0.1 is neither 0.01 nor the representable number closest to it.
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".
For tie-breaking, Python 3 uses round to even: round(1.5) and round(2.5) both produce 2. [123] Versions before 3 used round-away-from-zero: round(0.5) is 1.0, round(-0.5) is −1.0. [124] Python allows Boolean expressions with multiple equality relations in a manner that is consistent with general use in mathematics.
function plot (x, y, c) is plot the pixel at (x, y) with brightness c (where 0 ≤ c ≤ 1) // integer part of x function ipart (x) is return floor (x) function round (x) is return ipart (x + 0.5) // fractional part of x function fpart (x) is return x-ipart (x) function rfpart (x) is return 1-fpart (x) function drawLine (x0, y0, x1, y1) is boolean steep:= abs (y1-y0) > abs (x1-x0) if steep ...
Sums with finite operands can give an infinite result (i.e. 14.0 + 3.0 = +Inf as a result is the cyan area, −Inf is the magenta area). The range of the finite operands is filled with the curves x + y = c , where c is always one of the representable float values (blue and red for positive and negative results respectively).