Average Error: 0.4 → 0.3
Time: 28.8s
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 r741687 = 1.0;
        double r741688 = 6.0;
        double r741689 = r741687 / r741688;
        double r741690 = -2.0;
        double r741691 = u1;
        double r741692 = log(r741691);
        double r741693 = r741690 * r741692;
        double r741694 = 0.5;
        double r741695 = pow(r741693, r741694);
        double r741696 = r741689 * r741695;
        double r741697 = 2.0;
        double r741698 = atan2(1.0, 0.0);
        double r741699 = r741697 * r741698;
        double r741700 = u2;
        double r741701 = r741699 * r741700;
        double r741702 = cos(r741701);
        double r741703 = r741696 * r741702;
        double r741704 = r741703 + r741694;
        return r741704;
}
↓
double f(double u1, double u2) {
        double r741705 = u1;
        double r741706 = log(r741705);
        double r741707 = -2.0;
        double r741708 = r741706 * r741707;
        double r741709 = 0.5;
        double r741710 = pow(r741708, r741709);
        double r741711 = 0.16666666666666666;
        double r741712 = sqrt(r741711);
        double r741713 = r741710 * r741712;
        double r741714 = r741713 * r741712;
        double r741715 = 2.0;
        double r741716 = atan2(1.0, 0.0);
        double r741717 = u2;
        double r741718 = r741716 * r741717;
        double r741719 = r741715 * r741718;
        double r741720 = cos(r741719);
        double r741721 = r741714 * r741720;
        double r741722 = r741721 + r741709;
        return r741722;
}

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*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 
(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))