Average Error: 0.4 → 0.4
Time: 2.3m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[(\left(\cos \left(\left(\pi \cdot 2\right) \cdot u2\right)\right) \cdot \left(\frac{1}{{\left(\frac{1}{{-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}}\right)}^{0.5} \cdot 6}\right) + 0.5)_*\]

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. Simplified0.3

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

  4. Applied clear-num0.3

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

  5. Taylor expanded around 0 0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(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))