Average Error: 0.4 → 0.4
Time: 51.9s
Precision: 64
Internal Precision: 320
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right) \cdot \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right) + 0.5\]

Error

Bits error versus u1

Bits error versus u2

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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 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Runtime

Time bar (total: 51.9s)Debug logProfile

herbie shell --seed 2018274 
(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))