Average Error: 0.4 → 0.4
Time: 32.3s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}\]
\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(\left({\left(-2 \cdot \log u1\right)}^{0.5} \cdot \sqrt{\frac{\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}
double f(double u1, double u2) {
        double r1699715 = 1.0;
        double r1699716 = 6.0;
        double r1699717 = r1699715 / r1699716;
        double r1699718 = -2.0;
        double r1699719 = u1;
        double r1699720 = log(r1699719);
        double r1699721 = r1699718 * r1699720;
        double r1699722 = 0.5;
        double r1699723 = pow(r1699721, r1699722);
        double r1699724 = r1699717 * r1699723;
        double r1699725 = 2.0;
        double r1699726 = atan2(1.0, 0.0);
        double r1699727 = r1699725 * r1699726;
        double r1699728 = u2;
        double r1699729 = r1699727 * r1699728;
        double r1699730 = cos(r1699729);
        double r1699731 = r1699724 * r1699730;
        double r1699732 = r1699731 + r1699722;
        return r1699732;
}
↓
double f(double u1, double u2) {
        double r1699733 = 0.5;
        double r1699734 = -2.0;
        double r1699735 = u1;
        double r1699736 = log(r1699735);
        double r1699737 = r1699734 * r1699736;
        double r1699738 = pow(r1699737, r1699733);
        double r1699739 = u2;
        double r1699740 = 2.0;
        double r1699741 = atan2(1.0, 0.0);
        double r1699742 = r1699740 * r1699741;
        double r1699743 = r1699739 * r1699742;
        double r1699744 = cos(r1699743);
        double r1699745 = 6.0;
        double r1699746 = r1699744 / r1699745;
        double r1699747 = sqrt(r1699746);
        double r1699748 = r1699738 * r1699747;
        double r1699749 = 0.16666666666666666;
        double r1699750 = sqrt(r1699749);
        double r1699751 = r1699748 * r1699750;
        double r1699752 = sqrt(r1699744);
        double r1699753 = r1699751 * r1699752;
        double r1699754 = r1699733 + r1699753;
        return r1699754;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied div-inv0.4

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

  8. Applied sqrt-prod0.4

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

  9. Applied associate-*l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019146 
(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))