Average Error: 0.4 → 0.3
Time: 10.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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 r60771 = 1.0;
        double r60772 = 6.0;
        double r60773 = r60771 / r60772;
        double r60774 = -2.0;
        double r60775 = u1;
        double r60776 = log(r60775);
        double r60777 = r60774 * r60776;
        double r60778 = 0.5;
        double r60779 = pow(r60777, r60778);
        double r60780 = r60773 * r60779;
        double r60781 = 2.0;
        double r60782 = atan2(1.0, 0.0);
        double r60783 = r60781 * r60782;
        double r60784 = u2;
        double r60785 = r60783 * r60784;
        double r60786 = cos(r60785);
        double r60787 = r60780 * r60786;
        double r60788 = r60787 + r60778;
        return r60788;
}
↓
double f(double u1, double u2) {
        double r60789 = 1.0;
        double r60790 = -2.0;
        double r60791 = u1;
        double r60792 = log(r60791);
        double r60793 = r60790 * r60792;
        double r60794 = 0.5;
        double r60795 = pow(r60793, r60794);
        double r60796 = 6.0;
        double r60797 = r60795 / r60796;
        double r60798 = r60789 * r60797;
        double r60799 = 2.0;
        double r60800 = atan2(1.0, 0.0);
        double r60801 = r60799 * r60800;
        double r60802 = u2;
        double r60803 = r60801 * r60802;
        double r60804 = cos(r60803);
        double r60805 = r60798 * r60804;
        double r60806 = r60805 + r60794;
        return r60806;
}

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 2020021 
(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))