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

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

3. Taylor expanded around inf 0.4

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

4. Simplified0.4

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

5. Taylor expanded around inf 0.4

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

6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019050 +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))