Error

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

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

3. Taylor expanded around -inf 62.0

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

4. Simplified0.4

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

6. Applied add-cbrt-cube0.6

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

7. Taylor expanded around inf 0.5

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

8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019026 +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))