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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. Using strategy rm

4. Applied add-sqr-sqrt0.6

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

5. Applied associate-*l*0.6

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

7. Applied add-cbrt-cube0.7

   \[\leadsto 0.5 + \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \color{blue}{\sqrt[3]{\left(\left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right) \cdot \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right)\right) \cdot \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right)}}\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

8. Applied add-cbrt-cube0.8

   \[\leadsto 0.5 + \left(\color{blue}{\sqrt[3]{\left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}}\right) \cdot \sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}}}} \cdot \sqrt[3]{\left(\left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right) \cdot \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right)\right) \cdot \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right)}\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

9. Applied cbrt-unprod0.7

   \[\leadsto 0.5 + \color{blue}{\sqrt[3]{\left(\left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}}\right) \cdot \sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}}\right) \cdot \left(\left(\left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right) \cdot \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right)\right) \cdot \left(\sqrt{{\left(-2 \cdot \log u1\right)}^{0.5}} \cdot \frac{1}{6}\right)\right)}} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

10. Simplified0.5

    \[\leadsto 0.5 + \sqrt[3]{\color{blue}{\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \left(\frac{1}{216} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)}} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

11. Taylor expanded around 0 0.5

    \[\leadsto 0.5 + \sqrt[3]{\color{blue}{\frac{1}{216} \cdot {\left({-2}^{3.0} \cdot {\left(\log u1\right)}^{3.0}\right)}^{0.5}}} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

12. Final simplification0.5

    \[\leadsto \sqrt[3]{\frac{1}{216} \cdot {\left({\left(\log u1\right)}^{3.0} \cdot {-2}^{3.0}\right)}^{0.5}} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) + 0.5\]

Runtime

Time bar (total: 37.3s)Debug logProfile

herbie shell --seed 2018277 
(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))