Error

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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 pow10.4

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

7. Applied pow10.4

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

8. Applied pow10.4

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

9. Applied pow-prod-down0.4

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

10. Applied pow-prod-down0.4

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

11. Simplified0.5

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

12. Taylor expanded around inf 0.5

    \[\leadsto \color{blue}{\frac{0.25 - \frac{1}{18} \cdot \left({\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}^{2} \cdot \log \left(\frac{1}{u1}\right)\right)}{0.5 - \frac{1}{6} \cdot \left({\left({\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0} \cdot \left({-2}^{1.0} \cdot {-1}^{1.0}\right)\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}}\]

13. Final simplification0.5

    \[\leadsto \frac{0.25 - \frac{1}{18} \cdot \left(\log \left(\frac{1}{u1}\right) \cdot {\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}^{2}\right)}{0.5 - \left({\left(\left({-1}^{1.0} \cdot {-2}^{1.0}\right) \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0}\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \frac{1}{6}}\]

Reproduce

herbie shell --seed 2019026 
(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))