Average Error: 0.4 → 0.3
Time: 27.7s
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 r810970 = 1.0;
        double r810971 = 6.0;
        double r810972 = r810970 / r810971;
        double r810973 = -2.0;
        double r810974 = u1;
        double r810975 = log(r810974);
        double r810976 = r810973 * r810975;
        double r810977 = 0.5;
        double r810978 = pow(r810976, r810977);
        double r810979 = r810972 * r810978;
        double r810980 = 2.0;
        double r810981 = atan2(1.0, 0.0);
        double r810982 = r810980 * r810981;
        double r810983 = u2;
        double r810984 = r810982 * r810983;
        double r810985 = cos(r810984);
        double r810986 = r810979 * r810985;
        double r810987 = r810986 + r810977;
        return r810987;
}
↓
double f(double u1, double u2) {
        double r810988 = u1;
        double r810989 = log(r810988);
        double r810990 = -2.0;
        double r810991 = r810989 * r810990;
        double r810992 = 0.5;
        double r810993 = pow(r810991, r810992);
        double r810994 = 0.16666666666666666;
        double r810995 = sqrt(r810994);
        double r810996 = r810993 * r810995;
        double r810997 = r810996 * r810995;
        double r810998 = 2.0;
        double r810999 = atan2(1.0, 0.0);
        double r811000 = u2;
        double r811001 = r810999 * r811000;
        double r811002 = r810998 * r811001;
        double r811003 = cos(r811002);
        double r811004 = r810997 * r811003;
        double r811005 = r811004 + r810992;
        return r811005;
}

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 2019156 
(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))