Average Error: 0.4 → 0.5
Time: 2.1m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\sqrt{\frac{\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6}} \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6}}\right) + 0.5\]

Error

Bits error versus u1

Bits error versus u2

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

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-*l*0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019091 
(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))