Average Error: 0.4 → 0.3
Time: 33.7s
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 r1293924 = 1.0;
        double r1293925 = 6.0;
        double r1293926 = r1293924 / r1293925;
        double r1293927 = -2.0;
        double r1293928 = u1;
        double r1293929 = log(r1293928);
        double r1293930 = r1293927 * r1293929;
        double r1293931 = 0.5;
        double r1293932 = pow(r1293930, r1293931);
        double r1293933 = r1293926 * r1293932;
        double r1293934 = 2.0;
        double r1293935 = atan2(1.0, 0.0);
        double r1293936 = r1293934 * r1293935;
        double r1293937 = u2;
        double r1293938 = r1293936 * r1293937;
        double r1293939 = cos(r1293938);
        double r1293940 = r1293933 * r1293939;
        double r1293941 = r1293940 + r1293931;
        return r1293941;
}
↓
double f(double u1, double u2) {
        double r1293942 = u1;
        double r1293943 = log(r1293942);
        double r1293944 = -2.0;
        double r1293945 = r1293943 * r1293944;
        double r1293946 = 0.5;
        double r1293947 = pow(r1293945, r1293946);
        double r1293948 = 0.16666666666666666;
        double r1293949 = sqrt(r1293948);
        double r1293950 = r1293947 * r1293949;
        double r1293951 = r1293950 * r1293949;
        double r1293952 = 2.0;
        double r1293953 = atan2(1.0, 0.0);
        double r1293954 = u2;
        double r1293955 = r1293953 * r1293954;
        double r1293956 = r1293952 * r1293955;
        double r1293957 = cos(r1293956);
        double r1293958 = r1293951 * r1293957;
        double r1293959 = r1293958 + r1293946;
        return r1293959;
}

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 2019141 
(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))