Average Error: 0.4 → 0.3
Time: 10.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(2 \cdot \left(u2 \cdot \pi\right)\right) + 0.5\]
\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(2 \cdot \left(u2 \cdot \pi\right)\right) + 0.5
double f(double u1, double u2) {
        double r65829 = 1.0;
        double r65830 = 6.0;
        double r65831 = r65829 / r65830;
        double r65832 = -2.0;
        double r65833 = u1;
        double r65834 = log(r65833);
        double r65835 = r65832 * r65834;
        double r65836 = 0.5;
        double r65837 = pow(r65835, r65836);
        double r65838 = r65831 * r65837;
        double r65839 = 2.0;
        double r65840 = atan2(1.0, 0.0);
        double r65841 = r65839 * r65840;
        double r65842 = u2;
        double r65843 = r65841 * r65842;
        double r65844 = cos(r65843);
        double r65845 = r65838 * r65844;
        double r65846 = r65845 + r65836;
        return r65846;
}
↓
double f(double u1, double u2) {
        double r65847 = 1.0;
        double r65848 = -2.0;
        double r65849 = u1;
        double r65850 = log(r65849);
        double r65851 = r65848 * r65850;
        double r65852 = 0.5;
        double r65853 = pow(r65851, r65852);
        double r65854 = 6.0;
        double r65855 = r65853 / r65854;
        double r65856 = r65847 * r65855;
        double r65857 = 2.0;
        double r65858 = u2;
        double r65859 = atan2(1.0, 0.0);
        double r65860 = r65858 * r65859;
        double r65861 = r65857 * r65860;
        double r65862 = cos(r65861);
        double r65863 = r65856 * r65862;
        double r65864 = r65863 + r65852;
        return r65864;
}

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. Taylor expanded around 0 0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020027 
(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))