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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. Using strategy rm

3. Applied associate-*l/0.3

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

4. Simplified0.3

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

6. Applied *-un-lft-identity0.3

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

7. Applied associate-/l*0.3

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

8. Taylor expanded around -inf 62.0

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

9. Simplified0.4

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

10. Final simplification0.4

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

Runtime

Time bar (total: 29.6s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(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))