Average Error: 0.4 → 0.3
Time: 11.2s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r73898 = 1.0;
        double r73899 = 6.0;
        double r73900 = r73898 / r73899;
        double r73901 = -2.0;
        double r73902 = u1;
        double r73903 = log(r73902);
        double r73904 = r73901 * r73903;
        double r73905 = 0.5;
        double r73906 = pow(r73904, r73905);
        double r73907 = r73900 * r73906;
        double r73908 = 2.0;
        double r73909 = atan2(1.0, 0.0);
        double r73910 = r73908 * r73909;
        double r73911 = u2;
        double r73912 = r73910 * r73911;
        double r73913 = cos(r73912);
        double r73914 = r73907 * r73913;
        double r73915 = r73914 + r73905;
        return r73915;
}
↓
double f(double u1, double u2) {
        double r73916 = 1.0;
        double r73917 = -2.0;
        double r73918 = u1;
        double r73919 = log(r73918);
        double r73920 = r73917 * r73919;
        double r73921 = 0.5;
        double r73922 = pow(r73920, r73921);
        double r73923 = 6.0;
        double r73924 = r73922 / r73923;
        double r73925 = r73916 * r73924;
        double r73926 = 2.0;
        double r73927 = atan2(1.0, 0.0);
        double r73928 = r73926 * r73927;
        double r73929 = u2;
        double r73930 = r73928 * r73929;
        double r73931 = cos(r73930);
        double r73932 = r73925 * r73931;
        double r73933 = r73932 + r73921;
        return r73933;
}

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 div-inv0.4

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

  4. Applied associate-*l*0.4

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020003 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (<= 0.0 u1 1) (<= 0.0 u2 1))
  (+ (* (* (/ 1 6) (pow (* -2 (log u1)) 0.5)) (cos (* (* 2 PI) u2))) 0.5))