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 r71921 = 1.0;
        double r71922 = 6.0;
        double r71923 = r71921 / r71922;
        double r71924 = -2.0;
        double r71925 = u1;
        double r71926 = log(r71925);
        double r71927 = r71924 * r71926;
        double r71928 = 0.5;
        double r71929 = pow(r71927, r71928);
        double r71930 = r71923 * r71929;
        double r71931 = 2.0;
        double r71932 = atan2(1.0, 0.0);
        double r71933 = r71931 * r71932;
        double r71934 = u2;
        double r71935 = r71933 * r71934;
        double r71936 = cos(r71935);
        double r71937 = r71930 * r71936;
        double r71938 = r71937 + r71928;
        return r71938;
}
↓
double f(double u1, double u2) {
        double r71939 = 1.0;
        double r71940 = -2.0;
        double r71941 = u1;
        double r71942 = log(r71941);
        double r71943 = r71940 * r71942;
        double r71944 = 0.5;
        double r71945 = pow(r71943, r71944);
        double r71946 = 6.0;
        double r71947 = r71945 / r71946;
        double r71948 = r71939 * r71947;
        double r71949 = 2.0;
        double r71950 = atan2(1.0, 0.0);
        double r71951 = r71949 * r71950;
        double r71952 = u2;
        double r71953 = r71951 * r71952;
        double r71954 = cos(r71953);
        double r71955 = r71948 * r71954;
        double r71956 = r71955 + r71944;
        return r71956;
}

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 2020065 
(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))