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

↓

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

Derivation

Initial program 0.4
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
Initial simplification0.4
\[\leadsto 0.5 + \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]
Using strategy rm
Applied add-cbrt-cube0.6
\[\leadsto 0.5 + \color{blue}{\sqrt[3]{\left(\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)\right) \cdot \left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \frac{1}{6}\right)}} \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]
Final simplification0.6
\[\leadsto \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt[3]{\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right)\right)} + 0.5\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.6	0.6	0.0	0.5	0%

herbie shell --seed 2018286 
(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))

Error

Try it out

Derivation

Runtime

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