Average Error: 0.4 → 0.3
Time: 18.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\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
double f(double u1, double u2) {
        double r75583 = 1.0;
        double r75584 = 6.0;
        double r75585 = r75583 / r75584;
        double r75586 = -2.0;
        double r75587 = u1;
        double r75588 = log(r75587);
        double r75589 = r75586 * r75588;
        double r75590 = 0.5;
        double r75591 = pow(r75589, r75590);
        double r75592 = r75585 * r75591;
        double r75593 = 2.0;
        double r75594 = atan2(1.0, 0.0);
        double r75595 = r75593 * r75594;
        double r75596 = u2;
        double r75597 = r75595 * r75596;
        double r75598 = cos(r75597);
        double r75599 = r75592 * r75598;
        double r75600 = r75599 + r75590;
        return r75600;
}
↓
double f(double u1, double u2) {
        double r75601 = 1.0;
        double r75602 = 6.0;
        double r75603 = r75601 / r75602;
        double r75604 = sqrt(r75603);
        double r75605 = -2.0;
        double r75606 = u1;
        double r75607 = log(r75606);
        double r75608 = r75605 * r75607;
        double r75609 = 0.5;
        double r75610 = pow(r75608, r75609);
        double r75611 = r75604 * r75610;
        double r75612 = r75604 * r75611;
        double r75613 = 2.0;
        double r75614 = atan2(1.0, 0.0);
        double r75615 = r75613 * r75614;
        double r75616 = u2;
        double r75617 = r75615 * r75616;
        double r75618 = cos(r75617);
        double r75619 = r75612 * r75618;
        double r75620 = r75619 + r75609;
        return r75620;
}

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

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

Reproduce

herbie shell --seed 2020042 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))