Average Error: 0.4 → 0.3
Time: 25.6s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]
\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({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5
double f(double u1, double u2) {
        double r727995 = 1.0;
        double r727996 = 6.0;
        double r727997 = r727995 / r727996;
        double r727998 = -2.0;
        double r727999 = u1;
        double r728000 = log(r727999);
        double r728001 = r727998 * r728000;
        double r728002 = 0.5;
        double r728003 = pow(r728001, r728002);
        double r728004 = r727997 * r728003;
        double r728005 = 2.0;
        double r728006 = atan2(1.0, 0.0);
        double r728007 = r728005 * r728006;
        double r728008 = u2;
        double r728009 = r728007 * r728008;
        double r728010 = cos(r728009);
        double r728011 = r728004 * r728010;
        double r728012 = r728011 + r728002;
        return r728012;
}
↓
double f(double u1, double u2) {
        double r728013 = u1;
        double r728014 = log(r728013);
        double r728015 = -2.0;
        double r728016 = r728014 * r728015;
        double r728017 = 0.5;
        double r728018 = pow(r728016, r728017);
        double r728019 = 0.16666666666666666;
        double r728020 = sqrt(r728019);
        double r728021 = r728018 * r728020;
        double r728022 = r728021 * r728020;
        double r728023 = 2.0;
        double r728024 = atan2(1.0, 0.0);
        double r728025 = u2;
        double r728026 = r728024 * r728025;
        double r728027 = r728023 * r728026;
        double r728028 = cos(r728027);
        double r728029 = r728022 * r728028;
        double r728030 = r728029 + r728017;
        return r728030;
}

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

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 
(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))