Error

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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

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

4. Applied flip-+0.4

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

5. Taylor expanded around inf 0.5

   \[\leadsto \color{blue}{\frac{0.25 - \frac{1}{18} \cdot \left({\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}^{2} \cdot \log \left(\frac{1}{u1}\right)\right)}{0.5 - \frac{1}{6} \cdot \left({\left({\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0} \cdot \left({-2}^{1.0} \cdot {-1}^{1.0}\right)\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}}\]

6. Final simplification0.5

   \[\leadsto \frac{0.25 - \frac{1}{18} \cdot \left(\log \left(\frac{1}{u1}\right) \cdot {\left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right)}^{2}\right)}{0.5 - \left({\left(\left({-1}^{1.0} \cdot {-2}^{1.0}\right) \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1.0}\right)}^{0.5} \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right)\right) \cdot \frac{1}{6}}\]

Runtime

Time bar (total: 56.1s)Debug logProfile

herbie shell --seed 2018354 
(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))