Average Error: 0.4 → 0.4
Time: 28.2s
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 r719712 = 1.0;
        double r719713 = 6.0;
        double r719714 = r719712 / r719713;
        double r719715 = -2.0;
        double r719716 = u1;
        double r719717 = log(r719716);
        double r719718 = r719715 * r719717;
        double r719719 = 0.5;
        double r719720 = pow(r719718, r719719);
        double r719721 = r719714 * r719720;
        double r719722 = 2.0;
        double r719723 = atan2(1.0, 0.0);
        double r719724 = r719722 * r719723;
        double r719725 = u2;
        double r719726 = r719724 * r719725;
        double r719727 = cos(r719726);
        double r719728 = r719721 * r719727;
        double r719729 = r719728 + r719719;
        return r719729;
}
↓
double f(double u1, double u2) {
        double r719730 = u1;
        double r719731 = log(r719730);
        double r719732 = -2.0;
        double r719733 = r719731 * r719732;
        double r719734 = 0.5;
        double r719735 = pow(r719733, r719734);
        double r719736 = 0.16666666666666666;
        double r719737 = sqrt(r719736);
        double r719738 = r719735 * r719737;
        double r719739 = r719738 * r719737;
        double r719740 = 2.0;
        double r719741 = atan2(1.0, 0.0);
        double r719742 = u2;
        double r719743 = r719741 * r719742;
        double r719744 = r719740 * r719743;
        double r719745 = cos(r719744);
        double r719746 = r719739 * r719745;
        double r719747 = r719746 + r719734;
        return r719747;
}

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 2019155 
(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))