The modified ziggurat algorithm implemented in
RNG-151 has a variation for sampling from the overhang regions at the ziggurat edges, region A below:
Note: Region B is the ziggurat layer which is entirely within the distribution PDF.
These overhangs can be convex, concave or an inflection (change from convex to concave). Samples from the region A must be below the distribution density curve (PDF). When sampling from region A, the sampler will create a random point (x,y) uniformly within the rectangle. This is tested to determine if it is below the curve. Convex and concave curves can use a fast method that knows the largest possible triangle that fits above or below the curve. If a sampled point (x,y) is within these triangles then the actual curve position (pdf( x)) for the point x does not need to be computed. This can save time if the pdf is computationally expensive. In the case of exponential (exp(-x)) or Gaussian (exp(-0.5 * x * x)) this involves a call to Math.exp.
The current sampler implements the fast look-up method. The alternative is to always compute pdf( x) and determine if y is below the curve. This is known as 'simple overhangs'.
Investigate the use of simple overhangs on the performance of the sampler.
Note: The Marsaglia version of the ziggurat sampler uses the 'simple overhangs' method.