Average Error: 0.4 → 0.3
Time: 11.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) - 0.5 \cdot 0.5}{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) - 0.5}\]
\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
↓
\frac{\left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) \cdot \left(\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\right) - 0.5 \cdot 0.5}{\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) - 0.5}
double f(double u1, double u2) {
        double r71912 = 1.0;
        double r71913 = 6.0;
        double r71914 = r71912 / r71913;
        double r71915 = -2.0;
        double r71916 = u1;
        double r71917 = log(r71916);
        double r71918 = r71915 * r71917;
        double r71919 = 0.5;
        double r71920 = pow(r71918, r71919);
        double r71921 = r71914 * r71920;
        double r71922 = 2.0;
        double r71923 = atan2(1.0, 0.0);
        double r71924 = r71922 * r71923;
        double r71925 = u2;
        double r71926 = r71924 * r71925;
        double r71927 = cos(r71926);
        double r71928 = r71921 * r71927;
        double r71929 = r71928 + r71919;
        return r71929;
}
↓
double f(double u1, double u2) {
        double r71930 = 1.0;
        double r71931 = -2.0;
        double r71932 = u1;
        double r71933 = log(r71932);
        double r71934 = r71931 * r71933;
        double r71935 = 0.5;
        double r71936 = pow(r71934, r71935);
        double r71937 = r71930 * r71936;
        double r71938 = 6.0;
        double r71939 = r71937 / r71938;
        double r71940 = 2.0;
        double r71941 = atan2(1.0, 0.0);
        double r71942 = r71940 * r71941;
        double r71943 = u2;
        double r71944 = r71942 * r71943;
        double r71945 = cos(r71944);
        double r71946 = r71939 * r71945;
        double r71947 = r71946 * r71946;
        double r71948 = r71935 * r71935;
        double r71949 = r71947 - r71948;
        double r71950 = r71946 - r71935;
        double r71951 = r71949 / r71950;
        return r71951;
}

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 associate-*l/0.3

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

  5. Applied flip-+0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 
(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))