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Derivation

1. Initial program 0.4

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

2. Initial simplification0.4

   \[\leadsto 0.5 + \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

3. Taylor expanded around -inf 62.0

   \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({\left({\left(\log -1 - \log \left(\frac{-1}{u1}\right)\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) + 0.5}\]

4. Simplified0.4

   \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \cos \left(\pi \cdot \left(u2 \cdot 2\right)\right)\right) \cdot {\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} + 0.5}\]
5. Using strategy rm

6. Applied add-exp-log0.6

   \[\leadsto \color{blue}{e^{\log \left(\left(\frac{1}{6} \cdot \cos \left(\pi \cdot \left(u2 \cdot 2\right)\right)\right) \cdot {\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} + 0.5\right)}}\]

7. Final simplification0.6

   \[\leadsto e^{\log \left(\left(\frac{1}{6} \cdot \cos \left(\pi \cdot \left(u2 \cdot 2\right)\right)\right) \cdot {\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{0.5} + 0.5\right)}\]

Runtime

Time bar (total: 33.2s)Debug logProfile

herbie shell --seed 2018235 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))