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Box-Muller transform

A Box-Muller transform is a method of generating pairs of independent normally distributed random numbers, given a source of uniformly distributed random numbers. There are two kinds:

  1. Given r and φ independently uniformly distributed in (0,1], compute:
    <math>
    z_0 = \cos(2 \pi \varphi) \cdot \sqrt{-2 \ln r} </math>
    and
    <math>
    z_1 = \sin(2 \pi \varphi) \cdot \sqrt{-2 \ln r} </math>
  2. Given x and y independently uniformly distributed in [-1,1], set r = x2+y2. If r = 0 or r > 1, throw them away and try another pair (x, y). Thus, all points left after this filtering process will be uniformly distributed within a unit circle. Then, for these filtered points, compute:
    <math>
    z_0 = x \cdot \sqrt{\frac{-2 \ln r}{r}} </math>
    and
    <math>
    z_1 = y \cdot \sqrt{\frac{-2 \ln r}{r}} </math>

The second method is faster because it uses only one transcendental function[?] instead of three, even though it throws away 21% of the numbers.

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