Average Error: 0.4 → 0.4
Time: 31.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(0.1666666666666666574148081281236954964697 \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5}\right) \cdot \cos \left(\sqrt[3]{\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}\right) + 0.5\]
\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(0.1666666666666666574148081281236954964697 \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5}\right) \cdot \cos \left(\sqrt[3]{\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}\right) + 0.5
double f(double u1, double u2) {
        double r1865417 = 1.0;
        double r1865418 = 6.0;
        double r1865419 = r1865417 / r1865418;
        double r1865420 = -2.0;
        double r1865421 = u1;
        double r1865422 = log(r1865421);
        double r1865423 = r1865420 * r1865422;
        double r1865424 = 0.5;
        double r1865425 = pow(r1865423, r1865424);
        double r1865426 = r1865419 * r1865425;
        double r1865427 = 2.0;
        double r1865428 = atan2(1.0, 0.0);
        double r1865429 = r1865427 * r1865428;
        double r1865430 = u2;
        double r1865431 = r1865429 * r1865430;
        double r1865432 = cos(r1865431);
        double r1865433 = r1865426 * r1865432;
        double r1865434 = r1865433 + r1865424;
        return r1865434;
}
↓
double f(double u1, double u2) {
        double r1865435 = 0.16666666666666666;
        double r1865436 = -2.0;
        double r1865437 = 1.0;
        double r1865438 = pow(r1865436, r1865437);
        double r1865439 = u1;
        double r1865440 = log(r1865439);
        double r1865441 = pow(r1865440, r1865437);
        double r1865442 = r1865438 * r1865441;
        double r1865443 = 0.5;
        double r1865444 = pow(r1865442, r1865443);
        double r1865445 = r1865435 * r1865444;
        double r1865446 = u2;
        double r1865447 = 2.0;
        double r1865448 = atan2(1.0, 0.0);
        double r1865449 = r1865447 * r1865448;
        double r1865450 = r1865446 * r1865449;
        double r1865451 = r1865450 * r1865450;
        double r1865452 = r1865450 * r1865451;
        double r1865453 = cbrt(r1865452);
        double r1865454 = cos(r1865453);
        double r1865455 = r1865445 * r1865454;
        double r1865456 = r1865455 + r1865443;
        return r1865456;
}

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. Using strategy rm

  3. Applied add-cbrt-cube0.4

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

  5. Applied add-sqr-sqrt0.6

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

  6. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\left({\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5} \cdot {\left(\sqrt{0.1666666666666666574148081281236954964697}\right)}^{2}\right)} \cdot \cos \left(\sqrt[3]{\left(\left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right) + 0.5\]

  7. Simplified0.4

    \[\leadsto \color{blue}{\left({\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5} \cdot 0.1666666666666666574148081281236954964697\right)} \cdot \cos \left(\sqrt[3]{\left(\left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \left(\left(2 \cdot \pi\right) \cdot u2\right)}\right) + 0.5\]

  8. Final simplification0.4

    \[\leadsto \left(0.1666666666666666574148081281236954964697 \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5}\right) \cdot \cos \left(\sqrt[3]{\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(\left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}\right) + 0.5\]

Reproduce

herbie shell --seed 2019170 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))