Average Error: 0.4 → 0.5
Time: 50.3s
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \sqrt{\frac{\cos \left(\left(\pi \cdot u2\right) \cdot 2\right)}{6}} \cdot \left(\sqrt{\frac{\cos \left(\left(\pi \cdot u2\right) \cdot 2\right)}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\]

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. Applied simplify0.4

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-*l*0.5

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

Runtime

Time bar (total: 50.3s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(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))