Average Error: 0.4 → 0.5
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \sqrt[3]{\left(\left(\frac{1}{6} \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right)\right) \cdot \left(\frac{1}{6} \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right)\right)\right)\right) \cdot {\left(\log u1 \cdot -2\right)}^{\left(\left(0.5 + 0.5\right) + 0.5\right)}}\]

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

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

  4. Applied add-cbrt-cube0.4

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

  5. Applied add-cbrt-cube0.6

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

  6. Applied cbrt-unprod0.5

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

  8. Applied pow-prod-up0.5

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

  9. Applied pow-prod-up0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019089 
(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))