Average Error: 0.4 → 0.3
Time: 15.0s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\frac{1 \cdot {\left(-2 \cdot \log u1\right)}^{0.5}}{6} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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
↓
\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
double f(double u1, double u2) {
        double r67897 = 1.0;
        double r67898 = 6.0;
        double r67899 = r67897 / r67898;
        double r67900 = -2.0;
        double r67901 = u1;
        double r67902 = log(r67901);
        double r67903 = r67900 * r67902;
        double r67904 = 0.5;
        double r67905 = pow(r67903, r67904);
        double r67906 = r67899 * r67905;
        double r67907 = 2.0;
        double r67908 = atan2(1.0, 0.0);
        double r67909 = r67907 * r67908;
        double r67910 = u2;
        double r67911 = r67909 * r67910;
        double r67912 = cos(r67911);
        double r67913 = r67906 * r67912;
        double r67914 = r67913 + r67904;
        return r67914;
}
↓
double f(double u1, double u2) {
        double r67915 = 1.0;
        double r67916 = -2.0;
        double r67917 = u1;
        double r67918 = log(r67917);
        double r67919 = r67916 * r67918;
        double r67920 = 0.5;
        double r67921 = pow(r67919, r67920);
        double r67922 = r67915 * r67921;
        double r67923 = 6.0;
        double r67924 = r67922 / r67923;
        double r67925 = 2.0;
        double r67926 = atan2(1.0, 0.0);
        double r67927 = r67925 * r67926;
        double r67928 = u2;
        double r67929 = r67927 * r67928;
        double r67930 = cos(r67929);
        double r67931 = r67924 * r67930;
        double r67932 = r67931 + r67920;
        return r67932;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(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))