Average Error: 0.4 → 0.3
Time: 9.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(2 \cdot \log \left(\frac{1}{u1}\right)\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\frac{1 \cdot {\left(2 \cdot \log \left(\frac{1}{u1}\right)\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r61892 = 1.0;
        double r61893 = 6.0;
        double r61894 = r61892 / r61893;
        double r61895 = -2.0;
        double r61896 = u1;
        double r61897 = log(r61896);
        double r61898 = r61895 * r61897;
        double r61899 = 0.5;
        double r61900 = pow(r61898, r61899);
        double r61901 = r61894 * r61900;
        double r61902 = 2.0;
        double r61903 = atan2(1.0, 0.0);
        double r61904 = r61902 * r61903;
        double r61905 = u2;
        double r61906 = r61904 * r61905;
        double r61907 = cos(r61906);
        double r61908 = r61901 * r61907;
        double r61909 = r61908 + r61899;
        return r61909;
}
↓
double f(double u1, double u2) {
        double r61910 = 1.0;
        double r61911 = 2.0;
        double r61912 = 1.0;
        double r61913 = u1;
        double r61914 = r61912 / r61913;
        double r61915 = log(r61914);
        double r61916 = r61911 * r61915;
        double r61917 = 0.5;
        double r61918 = pow(r61916, r61917);
        double r61919 = r61910 * r61918;
        double r61920 = 6.0;
        double r61921 = r61919 / r61920;
        double r61922 = atan2(1.0, 0.0);
        double r61923 = r61911 * r61922;
        double r61924 = u2;
        double r61925 = r61923 * r61924;
        double r61926 = cos(r61925);
        double r61927 = r61921 * r61926;
        double r61928 = r61927 + r61917;
        return r61928;
}

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 associate-*l/0.3

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

  4. Taylor expanded around inf 0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020060 
(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))