Average Error: 0.4 → 0.4
Time: 33.5s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1}{{\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5} \cdot 6} \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right) + 0.5\]
\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{1}{{\left(\frac{1}{{\left(\log u1\right)}^{1} \cdot {-2}^{1}}\right)}^{0.5} \cdot 6} \cdot \cos \left(\left(\pi \cdot 2\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r2263909 = 1.0;
        double r2263910 = 6.0;
        double r2263911 = r2263909 / r2263910;
        double r2263912 = -2.0;
        double r2263913 = u1;
        double r2263914 = log(r2263913);
        double r2263915 = r2263912 * r2263914;
        double r2263916 = 0.5;
        double r2263917 = pow(r2263915, r2263916);
        double r2263918 = r2263911 * r2263917;
        double r2263919 = 2.0;
        double r2263920 = atan2(1.0, 0.0);
        double r2263921 = r2263919 * r2263920;
        double r2263922 = u2;
        double r2263923 = r2263921 * r2263922;
        double r2263924 = cos(r2263923);
        double r2263925 = r2263918 * r2263924;
        double r2263926 = r2263925 + r2263916;
        return r2263926;
}
↓
double f(double u1, double u2) {
        double r2263927 = 1.0;
        double r2263928 = 1.0;
        double r2263929 = u1;
        double r2263930 = log(r2263929);
        double r2263931 = pow(r2263930, r2263927);
        double r2263932 = -2.0;
        double r2263933 = pow(r2263932, r2263927);
        double r2263934 = r2263931 * r2263933;
        double r2263935 = r2263928 / r2263934;
        double r2263936 = 0.5;
        double r2263937 = pow(r2263935, r2263936);
        double r2263938 = 6.0;
        double r2263939 = r2263937 * r2263938;
        double r2263940 = r2263927 / r2263939;
        double r2263941 = atan2(1.0, 0.0);
        double r2263942 = 2.0;
        double r2263943 = r2263941 * r2263942;
        double r2263944 = u2;
        double r2263945 = r2263943 * r2263944;
        double r2263946 = cos(r2263945);
        double r2263947 = r2263940 * r2263946;
        double r2263948 = r2263947 + r2263936;
        return r2263948;
}

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 associate-/l*0.3

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

  6. Taylor expanded around 0 0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))