Average Error: 0.4 → 0.3
Time: 1.3m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)\]
\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 + \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(u2 \cdot \left(\pi \cdot 2\right)\right)
double f(double u1, double u2) {
        double r5152600 = 1.0;
        double r5152601 = 6.0;
        double r5152602 = r5152600 / r5152601;
        double r5152603 = -2.0;
        double r5152604 = u1;
        double r5152605 = log(r5152604);
        double r5152606 = r5152603 * r5152605;
        double r5152607 = 0.5;
        double r5152608 = pow(r5152606, r5152607);
        double r5152609 = r5152602 * r5152608;
        double r5152610 = 2.0;
        double r5152611 = atan2(1.0, 0.0);
        double r5152612 = r5152610 * r5152611;
        double r5152613 = u2;
        double r5152614 = r5152612 * r5152613;
        double r5152615 = cos(r5152614);
        double r5152616 = r5152609 * r5152615;
        double r5152617 = r5152616 + r5152607;
        return r5152617;
}
↓
double f(double u1, double u2) {
        double r5152618 = 0.5;
        double r5152619 = 0.16666666666666666;
        double r5152620 = sqrt(r5152619);
        double r5152621 = u1;
        double r5152622 = log(r5152621);
        double r5152623 = -2.0;
        double r5152624 = r5152622 * r5152623;
        double r5152625 = pow(r5152624, r5152618);
        double r5152626 = r5152620 * r5152625;
        double r5152627 = r5152626 * r5152620;
        double r5152628 = u2;
        double r5152629 = atan2(1.0, 0.0);
        double r5152630 = 2.0;
        double r5152631 = r5152629 * r5152630;
        double r5152632 = r5152628 * r5152631;
        double r5152633 = cos(r5152632);
        double r5152634 = r5152627 * r5152633;
        double r5152635 = r5152618 + r5152634;
        return r5152635;
}

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-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 
(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))