Average Error: 0.4 → 0.4
Time: 33.6s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]
\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({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5
double f(double u1, double u2) {
        double r819357 = 1.0;
        double r819358 = 6.0;
        double r819359 = r819357 / r819358;
        double r819360 = -2.0;
        double r819361 = u1;
        double r819362 = log(r819361);
        double r819363 = r819360 * r819362;
        double r819364 = 0.5;
        double r819365 = pow(r819363, r819364);
        double r819366 = r819359 * r819365;
        double r819367 = 2.0;
        double r819368 = atan2(1.0, 0.0);
        double r819369 = r819367 * r819368;
        double r819370 = u2;
        double r819371 = r819369 * r819370;
        double r819372 = cos(r819371);
        double r819373 = r819366 * r819372;
        double r819374 = r819373 + r819364;
        return r819374;
}
↓
double f(double u1, double u2) {
        double r819375 = u1;
        double r819376 = log(r819375);
        double r819377 = -2.0;
        double r819378 = r819376 * r819377;
        double r819379 = 0.5;
        double r819380 = pow(r819378, r819379);
        double r819381 = 0.16666666666666666;
        double r819382 = sqrt(r819381);
        double r819383 = r819380 * r819382;
        double r819384 = r819383 * r819382;
        double r819385 = 2.0;
        double r819386 = atan2(1.0, 0.0);
        double r819387 = u2;
        double r819388 = r819386 * r819387;
        double r819389 = r819385 * r819388;
        double r819390 = cos(r819389);
        double r819391 = r819384 * r819390;
        double r819392 = r819391 + r819379;
        return r819392;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019151 
(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))