Average Error: 0.4 → 0.4
Time: 35.7s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right) \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right)\]
\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 + \cos \left(\left(u2 \cdot \pi\right) \cdot 2\right) \cdot \left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \frac{1}{6}\right)
double f(double u1, double u2) {
        double r2137396 = 1.0;
        double r2137397 = 6.0;
        double r2137398 = r2137396 / r2137397;
        double r2137399 = -2.0;
        double r2137400 = u1;
        double r2137401 = log(r2137400);
        double r2137402 = r2137399 * r2137401;
        double r2137403 = 0.5;
        double r2137404 = pow(r2137402, r2137403);
        double r2137405 = r2137398 * r2137404;
        double r2137406 = 2.0;
        double r2137407 = atan2(1.0, 0.0);
        double r2137408 = r2137406 * r2137407;
        double r2137409 = u2;
        double r2137410 = r2137408 * r2137409;
        double r2137411 = cos(r2137410);
        double r2137412 = r2137405 * r2137411;
        double r2137413 = r2137412 + r2137403;
        return r2137413;
}
↓
double f(double u1, double u2) {
        double r2137414 = 0.5;
        double r2137415 = u2;
        double r2137416 = atan2(1.0, 0.0);
        double r2137417 = r2137415 * r2137416;
        double r2137418 = 2.0;
        double r2137419 = r2137417 * r2137418;
        double r2137420 = cos(r2137419);
        double r2137421 = u1;
        double r2137422 = log(r2137421);
        double r2137423 = -2.0;
        double r2137424 = r2137422 * r2137423;
        double r2137425 = pow(r2137424, r2137414);
        double r2137426 = 0.16666666666666666;
        double r2137427 = r2137425 * r2137426;
        double r2137428 = r2137420 * r2137427;
        double r2137429 = r2137414 + r2137428;
        return r2137429;
}

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 +-commutative0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019163 
(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))