Average Error: 0.4 → 0.3
Time: 11.5s
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 r67825 = 1.0;
        double r67826 = 6.0;
        double r67827 = r67825 / r67826;
        double r67828 = -2.0;
        double r67829 = u1;
        double r67830 = log(r67829);
        double r67831 = r67828 * r67830;
        double r67832 = 0.5;
        double r67833 = pow(r67831, r67832);
        double r67834 = r67827 * r67833;
        double r67835 = 2.0;
        double r67836 = atan2(1.0, 0.0);
        double r67837 = r67835 * r67836;
        double r67838 = u2;
        double r67839 = r67837 * r67838;
        double r67840 = cos(r67839);
        double r67841 = r67834 * r67840;
        double r67842 = r67841 + r67832;
        return r67842;
}
↓
double f(double u1, double u2) {
        double r67843 = 1.0;
        double r67844 = -2.0;
        double r67845 = u1;
        double r67846 = log(r67845);
        double r67847 = r67844 * r67846;
        double r67848 = 0.5;
        double r67849 = pow(r67847, r67848);
        double r67850 = 6.0;
        double r67851 = r67849 / r67850;
        double r67852 = r67843 * r67851;
        double r67853 = 2.0;
        double r67854 = atan2(1.0, 0.0);
        double r67855 = r67853 * r67854;
        double r67856 = u2;
        double r67857 = r67855 * r67856;
        double r67858 = cos(r67857);
        double r67859 = r67852 * r67858;
        double r67860 = r67859 + r67848;
        return r67860;
}

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 2019354 
(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))