Average Error: 0.4 → 0.3
Time: 2.8m
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6} + 0.5\]
\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{{\left(\log u1 \cdot -2\right)}^{0.5} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)}{6} + 0.5
double f(double u1, double u2) {
        double r21383932 = 1.0;
        double r21383933 = 6.0;
        double r21383934 = r21383932 / r21383933;
        double r21383935 = -2.0;
        double r21383936 = u1;
        double r21383937 = log(r21383936);
        double r21383938 = r21383935 * r21383937;
        double r21383939 = 0.5;
        double r21383940 = pow(r21383938, r21383939);
        double r21383941 = r21383934 * r21383940;
        double r21383942 = 2.0;
        double r21383943 = atan2(1.0, 0.0);
        double r21383944 = r21383942 * r21383943;
        double r21383945 = u2;
        double r21383946 = r21383944 * r21383945;
        double r21383947 = cos(r21383946);
        double r21383948 = r21383941 * r21383947;
        double r21383949 = r21383948 + r21383939;
        return r21383949;
}
↓
double f(double u1, double u2) {
        double r21383950 = u1;
        double r21383951 = log(r21383950);
        double r21383952 = -2.0;
        double r21383953 = r21383951 * r21383952;
        double r21383954 = 0.5;
        double r21383955 = pow(r21383953, r21383954);
        double r21383956 = u2;
        double r21383957 = 2.0;
        double r21383958 = atan2(1.0, 0.0);
        double r21383959 = r21383957 * r21383958;
        double r21383960 = r21383956 * r21383959;
        double r21383961 = cos(r21383960);
        double r21383962 = r21383955 * r21383961;
        double r21383963 = 6.0;
        double r21383964 = r21383962 / r21383963;
        double r21383965 = r21383964 + r21383954;
        return r21383965;
}

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

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

  4. Applied associate-*l/0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019107 
(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))