Average Error: 0.4 → 0.5
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{\left(-0.25\right) + \left(\cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\right) \cdot \left(\frac{-1}{18} \cdot \log u1\right)}{\left(\cos \left(\left(\pi \cdot u2\right) \cdot 2\right) \cdot \frac{1}{6}\right) \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5} - 0.5}\]

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Taylor expanded around -inf 62.0

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

  4. Simplified0.4

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

  6. Applied flip-+0.4

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

  7. Taylor expanded around -inf 62.0

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018278 
(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))