Average Error: 0.4 → 0.4
Time: 2.0m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \frac{\frac{\cos \left(\left(u2 \cdot 2\right) \cdot \pi\right)}{6}}{{\left(\frac{\frac{1}{{\left(-\log u1\right)}^{1.0}}}{{2}^{1.0}}\right)}^{0.5}}\]

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. Applied simplify0.4

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

  3. Taylor expanded around inf 0.4

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

  4. Applied simplify0.4

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(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))