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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. Simplified0.4

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

4. Applied flip-+0.4

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

6. Applied *-un-lft-identity0.4

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

7. Applied *-un-lft-identity0.4

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

8. Applied times-frac0.4

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

9. Simplified0.4

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

10. Simplified0.4

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

11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019002 
(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))

Details

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