Average Error: 0.4 → 0.3
Time: 18.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) + 0.5\]
\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
↓
\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) + 0.5
double f(double u1, double u2) {
        double r75836 = 1.0;
        double r75837 = 6.0;
        double r75838 = r75836 / r75837;
        double r75839 = -2.0;
        double r75840 = u1;
        double r75841 = log(r75840);
        double r75842 = r75839 * r75841;
        double r75843 = 0.5;
        double r75844 = pow(r75842, r75843);
        double r75845 = r75838 * r75844;
        double r75846 = 2.0;
        double r75847 = atan2(1.0, 0.0);
        double r75848 = r75846 * r75847;
        double r75849 = u2;
        double r75850 = r75848 * r75849;
        double r75851 = cos(r75850);
        double r75852 = r75845 * r75851;
        double r75853 = r75852 + r75843;
        return r75853;
}
↓
double f(double u1, double u2) {
        double r75854 = 2.0;
        double r75855 = atan2(1.0, 0.0);
        double r75856 = r75854 * r75855;
        double r75857 = u2;
        double r75858 = r75856 * r75857;
        double r75859 = cos(r75858);
        double r75860 = 1.0;
        double r75861 = 6.0;
        double r75862 = r75860 / r75861;
        double r75863 = sqrt(r75862);
        double r75864 = -2.0;
        double r75865 = u1;
        double r75866 = log(r75865);
        double r75867 = r75864 * r75866;
        double r75868 = 0.5;
        double r75869 = pow(r75867, r75868);
        double r75870 = r75863 * r75869;
        double r75871 = r75870 * r75863;
        double r75872 = r75859 * r75871;
        double r75873 = r75872 + r75868;
        return r75873;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.3

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

  6. Applied *-un-lft-identity0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019351 
(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))