Average Error: 0.4 → 0.3
Time: 30.6s
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 r78828 = 1.0;
        double r78829 = 6.0;
        double r78830 = r78828 / r78829;
        double r78831 = -2.0;
        double r78832 = u1;
        double r78833 = log(r78832);
        double r78834 = r78831 * r78833;
        double r78835 = 0.5;
        double r78836 = pow(r78834, r78835);
        double r78837 = r78830 * r78836;
        double r78838 = 2.0;
        double r78839 = atan2(1.0, 0.0);
        double r78840 = r78838 * r78839;
        double r78841 = u2;
        double r78842 = r78840 * r78841;
        double r78843 = cos(r78842);
        double r78844 = r78837 * r78843;
        double r78845 = r78844 + r78835;
        return r78845;
}
↓
double f(double u1, double u2) {
        double r78846 = 1.0;
        double r78847 = -2.0;
        double r78848 = u1;
        double r78849 = log(r78848);
        double r78850 = r78847 * r78849;
        double r78851 = 0.5;
        double r78852 = pow(r78850, r78851);
        double r78853 = 6.0;
        double r78854 = r78852 / r78853;
        double r78855 = r78846 * r78854;
        double r78856 = 2.0;
        double r78857 = atan2(1.0, 0.0);
        double r78858 = r78856 * r78857;
        double r78859 = u2;
        double r78860 = r78858 * r78859;
        double r78861 = cos(r78860);
        double r78862 = r78855 * r78861;
        double r78863 = r78862 + r78851;
        return r78863;
}

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 2019198 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))