Average Error: 0.4 → 0.3
Time: 1.7m
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 r4715907 = 1.0;
        double r4715908 = 6.0;
        double r4715909 = r4715907 / r4715908;
        double r4715910 = -2.0;
        double r4715911 = u1;
        double r4715912 = log(r4715911);
        double r4715913 = r4715910 * r4715912;
        double r4715914 = 0.5;
        double r4715915 = pow(r4715913, r4715914);
        double r4715916 = r4715909 * r4715915;
        double r4715917 = 2.0;
        double r4715918 = atan2(1.0, 0.0);
        double r4715919 = r4715917 * r4715918;
        double r4715920 = u2;
        double r4715921 = r4715919 * r4715920;
        double r4715922 = cos(r4715921);
        double r4715923 = r4715916 * r4715922;
        double r4715924 = r4715923 + r4715914;
        return r4715924;
}
↓
double f(double u1, double u2) {
        double r4715925 = u1;
        double r4715926 = log(r4715925);
        double r4715927 = -2.0;
        double r4715928 = r4715926 * r4715927;
        double r4715929 = 0.5;
        double r4715930 = pow(r4715928, r4715929);
        double r4715931 = 0.16666666666666666;
        double r4715932 = sqrt(r4715931);
        double r4715933 = r4715930 * r4715932;
        double r4715934 = r4715933 * r4715932;
        double r4715935 = 2.0;
        double r4715936 = atan2(1.0, 0.0);
        double r4715937 = u2;
        double r4715938 = r4715936 * r4715937;
        double r4715939 = r4715935 * r4715938;
        double r4715940 = cos(r4715939);
        double r4715941 = r4715934 * r4715940;
        double r4715942 = r4715941 + r4715929;
        return r4715942;
}

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 2019158 
(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))