Editing Box–Muller transform (section)

==Tails truncation==
When a computer is used to produce a uniform random variable it will inevitably have some inaccuracies because there is a lower bound on how close numbers can be to 0. If the generator uses 32 bits per output value, the smallest non-zero number that can be generated is <math>2^{-32}</math>. When <math>U_1</math> and <math>U_2</math> are equal to this the Box–Muller transform produces a normal random deviate equal to <math display="inline">\delta = \sqrt{-2 \ln(2^{-32})} \cos(2 \pi 2^{-32})\approx 6.660</math>. This means that the algorithm will not produce random variables more than 6.660 standard deviations from the mean. This corresponds to a proportion of <math>2(1-\Phi(\delta)) \simeq 2.738 \times 10^{-11}</math> lost due to the truncation, where <math>\Phi(\delta)</math> is the standard cumulative normal distribution. With 64 bits the limit is pushed to <math>\delta = 9.419</math> standard deviations, for which <math>2(1-\Phi(\delta)) < 5 \times 10^{-21}</math>.