Average Error: 0.4 → 0.4
Time: 33.4s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\]
\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(\frac{1}{6} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)
double f(double u1, double u2) {
        double r1145634 = 1.0;
        double r1145635 = 6.0;
        double r1145636 = r1145634 / r1145635;
        double r1145637 = -2.0;
        double r1145638 = u1;
        double r1145639 = log(r1145638);
        double r1145640 = r1145637 * r1145639;
        double r1145641 = 0.5;
        double r1145642 = pow(r1145640, r1145641);
        double r1145643 = r1145636 * r1145642;
        double r1145644 = 2.0;
        double r1145645 = atan2(1.0, 0.0);
        double r1145646 = r1145644 * r1145645;
        double r1145647 = u2;
        double r1145648 = r1145646 * r1145647;
        double r1145649 = cos(r1145648);
        double r1145650 = r1145643 * r1145649;
        double r1145651 = r1145650 + r1145641;
        return r1145651;
}
↓
double f(double u1, double u2) {
        double r1145652 = 0.5;
        double r1145653 = 0.16666666666666666;
        double r1145654 = u1;
        double r1145655 = log(r1145654);
        double r1145656 = -2.0;
        double r1145657 = r1145655 * r1145656;
        double r1145658 = pow(r1145657, r1145652);
        double r1145659 = r1145653 * r1145658;
        double r1145660 = u2;
        double r1145661 = 2.0;
        double r1145662 = atan2(1.0, 0.0);
        double r1145663 = r1145661 * r1145662;
        double r1145664 = r1145660 * r1145663;
        double r1145665 = cos(r1145664);
        double r1145666 = r1145659 * r1145665;
        double r1145667 = r1145652 + r1145666;
        return r1145667;
}

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*0.4

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

  6. Applied *-un-lft-identity0.4

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

  7. Applied associate-*l*0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019149 
(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))