Average Error: 0.4 → 0.4
Time: 28.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(0.1666666666666666574148081281236954964697 \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\]
\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(0.1666666666666666574148081281236954964697 \cdot {\left({-2}^{1} \cdot {\left(\log u1\right)}^{1}\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)
double f(double u1, double u2) {
        double r76696 = 1.0;
        double r76697 = 6.0;
        double r76698 = r76696 / r76697;
        double r76699 = -2.0;
        double r76700 = u1;
        double r76701 = log(r76700);
        double r76702 = r76699 * r76701;
        double r76703 = 0.5;
        double r76704 = pow(r76702, r76703);
        double r76705 = r76698 * r76704;
        double r76706 = 2.0;
        double r76707 = atan2(1.0, 0.0);
        double r76708 = r76706 * r76707;
        double r76709 = u2;
        double r76710 = r76708 * r76709;
        double r76711 = cos(r76710);
        double r76712 = r76705 * r76711;
        double r76713 = r76712 + r76703;
        return r76713;
}
↓
double f(double u1, double u2) {
        double r76714 = 0.5;
        double r76715 = 0.16666666666666666;
        double r76716 = -2.0;
        double r76717 = 1.0;
        double r76718 = pow(r76716, r76717);
        double r76719 = u1;
        double r76720 = log(r76719);
        double r76721 = pow(r76720, r76717);
        double r76722 = r76718 * r76721;
        double r76723 = pow(r76722, r76714);
        double r76724 = r76715 * r76723;
        double r76725 = 2.0;
        double r76726 = atan2(1.0, 0.0);
        double r76727 = r76725 * r76726;
        double r76728 = u2;
        double r76729 = r76727 * r76728;
        double r76730 = cos(r76729);
        double r76731 = r76724 * r76730;
        double r76732 = r76714 + r76731;
        return r76732;
}

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 64.0

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

  6. Simplified0.4

    \[\leadsto \color{blue}{\left({\left({\left(0 + \log u1\right)}^{1} \cdot {-2}^{1}\right)}^{0.5} \cdot 0.1666666666666666574148081281236954964697\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019306 
(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))