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

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

4. Applied associate-*r*0.4

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

5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019053 
(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))