Average Error: 0.4 → 0.3
Time: 11.1s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r65859 = 1.0;
        double r65860 = 6.0;
        double r65861 = r65859 / r65860;
        double r65862 = -2.0;
        double r65863 = u1;
        double r65864 = log(r65863);
        double r65865 = r65862 * r65864;
        double r65866 = 0.5;
        double r65867 = pow(r65865, r65866);
        double r65868 = r65861 * r65867;
        double r65869 = 2.0;
        double r65870 = atan2(1.0, 0.0);
        double r65871 = r65869 * r65870;
        double r65872 = u2;
        double r65873 = r65871 * r65872;
        double r65874 = cos(r65873);
        double r65875 = r65868 * r65874;
        double r65876 = r65875 + r65866;
        return r65876;
}
↓
double f(double u1, double u2) {
        double r65877 = 1.0;
        double r65878 = -2.0;
        double r65879 = u1;
        double r65880 = log(r65879);
        double r65881 = r65878 * r65880;
        double r65882 = 0.5;
        double r65883 = pow(r65881, r65882);
        double r65884 = 6.0;
        double r65885 = r65883 / r65884;
        double r65886 = r65877 * r65885;
        double r65887 = 2.0;
        double r65888 = atan2(1.0, 0.0);
        double r65889 = r65887 * r65888;
        double r65890 = u2;
        double r65891 = r65889 * r65890;
        double r65892 = cos(r65891);
        double r65893 = r65886 * r65892;
        double r65894 = r65893 + r65882;
        return r65894;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019353 
(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))