Average Error: 0.4 → 0.3
Time: 26.1s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5\]
\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(\left({\left(\log u1 \cdot -2\right)}^{0.5} \cdot \sqrt{\frac{1}{6}}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + 0.5
double f(double u1, double u2) {
        double r867957 = 1.0;
        double r867958 = 6.0;
        double r867959 = r867957 / r867958;
        double r867960 = -2.0;
        double r867961 = u1;
        double r867962 = log(r867961);
        double r867963 = r867960 * r867962;
        double r867964 = 0.5;
        double r867965 = pow(r867963, r867964);
        double r867966 = r867959 * r867965;
        double r867967 = 2.0;
        double r867968 = atan2(1.0, 0.0);
        double r867969 = r867967 * r867968;
        double r867970 = u2;
        double r867971 = r867969 * r867970;
        double r867972 = cos(r867971);
        double r867973 = r867966 * r867972;
        double r867974 = r867973 + r867964;
        return r867974;
}
↓
double f(double u1, double u2) {
        double r867975 = u1;
        double r867976 = log(r867975);
        double r867977 = -2.0;
        double r867978 = r867976 * r867977;
        double r867979 = 0.5;
        double r867980 = pow(r867978, r867979);
        double r867981 = 0.16666666666666666;
        double r867982 = sqrt(r867981);
        double r867983 = r867980 * r867982;
        double r867984 = r867983 * r867982;
        double r867985 = 2.0;
        double r867986 = atan2(1.0, 0.0);
        double r867987 = u2;
        double r867988 = r867986 * r867987;
        double r867989 = r867985 * r867988;
        double r867990 = cos(r867989);
        double r867991 = r867984 * r867990;
        double r867992 = r867991 + r867979;
        return r867992;
}

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. Simplified0.4

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

  4. Applied add-sqr-sqrt0.4

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019154 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0 u1 1) (<= 0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))