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. Initial simplification0.4

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

3. Taylor expanded around -inf 62.0

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

4. Simplified0.4

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

6. Applied add-cbrt-cube0.6

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

8. Applied add-sqr-sqrt0.6

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

9. Applied associate-*l*0.6

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

10. Final simplification0.6

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

Reproduce

herbie shell --seed 2018365 +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))

Details

Time bar (total: 23.0s)Debug log