Average Error: 0.4 → 0.5
Time: 34.8s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \log \left(\sqrt{\left(\sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}} \cdot \sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}}\right) \cdot \sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}}} \cdot \sqrt{e^{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right)}}\right)\]
\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 + \log \left(\sqrt{\left(\sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}} \cdot \sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}}\right) \cdot \sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}}} \cdot \sqrt{e^{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right)}}\right)
double f(double u1, double u2) {
        double r2104546 = 1.0;
        double r2104547 = 6.0;
        double r2104548 = r2104546 / r2104547;
        double r2104549 = -2.0;
        double r2104550 = u1;
        double r2104551 = log(r2104550);
        double r2104552 = r2104549 * r2104551;
        double r2104553 = 0.5;
        double r2104554 = pow(r2104552, r2104553);
        double r2104555 = r2104548 * r2104554;
        double r2104556 = 2.0;
        double r2104557 = atan2(1.0, 0.0);
        double r2104558 = r2104556 * r2104557;
        double r2104559 = u2;
        double r2104560 = r2104558 * r2104559;
        double r2104561 = cos(r2104560);
        double r2104562 = r2104555 * r2104561;
        double r2104563 = r2104562 + r2104553;
        return r2104563;
}
↓
double f(double u1, double u2) {
        double r2104564 = 0.5;
        double r2104565 = u2;
        double r2104566 = atan2(1.0, 0.0);
        double r2104567 = 2.0;
        double r2104568 = r2104566 * r2104567;
        double r2104569 = r2104565 * r2104568;
        double r2104570 = cos(r2104569);
        double r2104571 = 0.16666666666666666;
        double r2104572 = r2104570 * r2104571;
        double r2104573 = log(r2104572);
        double r2104574 = u1;
        double r2104575 = log(r2104574);
        double r2104576 = -2.0;
        double r2104577 = r2104575 * r2104576;
        double r2104578 = log(r2104577);
        double r2104579 = r2104564 * r2104578;
        double r2104580 = r2104573 + r2104579;
        double r2104581 = exp(r2104580);
        double r2104582 = exp(r2104581);
        double r2104583 = cbrt(r2104582);
        double r2104584 = r2104583 * r2104583;
        double r2104585 = r2104584 * r2104583;
        double r2104586 = sqrt(r2104585);
        double r2104587 = pow(r2104577, r2104564);
        double r2104588 = r2104587 * r2104572;
        double r2104589 = exp(r2104588);
        double r2104590 = sqrt(r2104589);
        double r2104591 = r2104586 * r2104590;
        double r2104592 = log(r2104591);
        double r2104593 = r2104564 + r2104592;
        return r2104593;
}

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

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

  4. Applied add-log-exp0.4

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

  6. Applied add-sqr-sqrt0.4

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

  8. Applied add-exp-log0.6

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

  9. Applied pow-to-exp0.5

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

  10. Applied prod-exp0.5

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

  12. Applied add-cube-cbrt0.5

    \[\leadsto 0.5 + \log \left(\sqrt{e^{{\left(-2 \cdot \log u1\right)}^{0.5} \cdot \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}} \cdot \sqrt{\color{blue}{\left(\sqrt[3]{e^{e^{\log \left(-2 \cdot \log u1\right) \cdot 0.5 + \log \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}}} \cdot \sqrt[3]{e^{e^{\log \left(-2 \cdot \log u1\right) \cdot 0.5 + \log \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}}}\right) \cdot \sqrt[3]{e^{e^{\log \left(-2 \cdot \log u1\right) \cdot 0.5 + \log \left(\frac{1}{6} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\right)}}}}}\right)\]

  13. Final simplification0.5

    \[\leadsto 0.5 + \log \left(\sqrt{\left(\sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}} \cdot \sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}}\right) \cdot \sqrt[3]{e^{e^{\log \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right) + 0.5 \cdot \log \left(\log u1 \cdot -2\right)}}}} \cdot \sqrt{e^{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \left(\cos \left(u2 \cdot \left(\pi \cdot 2\right)\right) \cdot \frac{1}{6}\right)}}\right)\]

Reproduce

herbie shell --seed 2019132 
(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))