Average Error: 0.4 → 0.4
Time: 11.9s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot {\left(\sqrt{0.1666666666666666574148081281236954964697}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot {\left(\sqrt{0.1666666666666666574148081281236954964697}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r73651 = 1.0;
        double r73652 = 6.0;
        double r73653 = r73651 / r73652;
        double r73654 = -2.0;
        double r73655 = u1;
        double r73656 = log(r73655);
        double r73657 = r73654 * r73656;
        double r73658 = 0.5;
        double r73659 = pow(r73657, r73658);
        double r73660 = r73653 * r73659;
        double r73661 = 2.0;
        double r73662 = atan2(1.0, 0.0);
        double r73663 = r73661 * r73662;
        double r73664 = u2;
        double r73665 = r73663 * r73664;
        double r73666 = cos(r73665);
        double r73667 = r73660 * r73666;
        double r73668 = r73667 + r73658;
        return r73668;
}
↓
double f(double u1, double u2) {
        double r73669 = -1.0;
        double r73670 = 1.0;
        double r73671 = pow(r73669, r73670);
        double r73672 = -2.0;
        double r73673 = pow(r73672, r73670);
        double r73674 = 1.0;
        double r73675 = u1;
        double r73676 = r73674 / r73675;
        double r73677 = log(r73676);
        double r73678 = pow(r73677, r73670);
        double r73679 = r73673 * r73678;
        double r73680 = r73671 * r73679;
        double r73681 = 0.5;
        double r73682 = pow(r73680, r73681);
        double r73683 = 0.16666666666666666;
        double r73684 = sqrt(r73683);
        double r73685 = 2.0;
        double r73686 = pow(r73684, r73685);
        double r73687 = r73682 * r73686;
        double r73688 = 2.0;
        double r73689 = atan2(1.0, 0.0);
        double r73690 = r73688 * r73689;
        double r73691 = u2;
        double r73692 = r73690 * r73691;
        double r73693 = cos(r73692);
        double r73694 = r73687 * r73693;
        double r73695 = r73694 + r73681;
        return r73695;
}

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. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\left({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot {\left(\sqrt{0.1666666666666666574148081281236954964697}\right)}^{2}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  6. Final simplification0.4

    \[\leadsto \left({\left({-1}^{1} \cdot \left({-2}^{1} \cdot {\left(\log \left(\frac{1}{u1}\right)\right)}^{1}\right)\right)}^{0.5} \cdot {\left(\sqrt{0.1666666666666666574148081281236954964697}\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2020001 
(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))