Average Error: 0.4 → 0.6
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[e^{\log \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \left(\cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) \cdot \frac{1}{6}\right) + 0.5\right)}\]

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-exp-log0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019051 
(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))