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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-cbrt-cube0.6

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

5. Taylor expanded around -inf 62.0

   \[\leadsto 0.5 + \sqrt[3]{\color{blue}{\left(\frac{1}{36} \cdot {\left({\left(\log -1 - \log \left(\frac{-1}{u1}\right)\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{1.0}\right)} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

6. Simplified0.5

   \[\leadsto 0.5 + \sqrt[3]{\color{blue}{\left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}\right)} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]
7. Using strategy rm

8. Applied add-cbrt-cube0.5

   \[\leadsto 0.5 + \sqrt[3]{\color{blue}{\sqrt[3]{\left(\left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}\right)\right) \cdot \left({\left({\left(\log u1\right)}^{1.0} \cdot {-2}^{1.0}\right)}^{1.0} \cdot \frac{1}{36}\right)}} \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]

9. Final simplification0.5

   \[\leadsto 0.5 + \sqrt[3]{\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \sqrt[3]{\left(\frac{1}{36} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0}\right) \cdot \left(\left(\frac{1}{36} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0}\right) \cdot \left(\frac{1}{36} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{1.0}\right)\right)}} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)\]

Runtime

Time bar (total: 33.8s)Debug logProfile

herbie shell --seed 2018248 
(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))