Average Error: 0.4 → 0.3
Time: 41.6s
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 r2554886 = 1.0;
        double r2554887 = 6.0;
        double r2554888 = r2554886 / r2554887;
        double r2554889 = -2.0;
        double r2554890 = u1;
        double r2554891 = log(r2554890);
        double r2554892 = r2554889 * r2554891;
        double r2554893 = 0.5;
        double r2554894 = pow(r2554892, r2554893);
        double r2554895 = r2554888 * r2554894;
        double r2554896 = 2.0;
        double r2554897 = atan2(1.0, 0.0);
        double r2554898 = r2554896 * r2554897;
        double r2554899 = u2;
        double r2554900 = r2554898 * r2554899;
        double r2554901 = cos(r2554900);
        double r2554902 = r2554895 * r2554901;
        double r2554903 = r2554902 + r2554893;
        return r2554903;
}
↓
double f(double u1, double u2) {
        double r2554904 = u1;
        double r2554905 = log(r2554904);
        double r2554906 = -2.0;
        double r2554907 = r2554905 * r2554906;
        double r2554908 = 0.5;
        double r2554909 = pow(r2554907, r2554908);
        double r2554910 = 0.16666666666666666;
        double r2554911 = sqrt(r2554910);
        double r2554912 = r2554909 * r2554911;
        double r2554913 = r2554912 * r2554911;
        double r2554914 = 2.0;
        double r2554915 = atan2(1.0, 0.0);
        double r2554916 = u2;
        double r2554917 = r2554915 * r2554916;
        double r2554918 = r2554914 * r2554917;
        double r2554919 = cos(r2554918);
        double r2554920 = r2554913 * r2554919;
        double r2554921 = r2554920 + r2554908;
        return r2554921;
}

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 2019138 
(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))