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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. Initial simplification0.4

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

3. Taylor expanded around inf 0.4

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

4. Simplified0.4

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

6. Applied unpow-prod-down0.3

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

8. Applied associate-*r*0.4

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

9. Final simplification0.4

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018234 
(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))