Average Error: 0.4 → 0.3
Time: 13.4s
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 r81819 = 1.0;
        double r81820 = 6.0;
        double r81821 = r81819 / r81820;
        double r81822 = -2.0;
        double r81823 = u1;
        double r81824 = log(r81823);
        double r81825 = r81822 * r81824;
        double r81826 = 0.5;
        double r81827 = pow(r81825, r81826);
        double r81828 = r81821 * r81827;
        double r81829 = 2.0;
        double r81830 = atan2(1.0, 0.0);
        double r81831 = r81829 * r81830;
        double r81832 = u2;
        double r81833 = r81831 * r81832;
        double r81834 = cos(r81833);
        double r81835 = r81828 * r81834;
        double r81836 = r81835 + r81826;
        return r81836;
}
↓
double f(double u1, double u2) {
        double r81837 = 1.0;
        double r81838 = -2.0;
        double r81839 = u1;
        double r81840 = log(r81839);
        double r81841 = r81838 * r81840;
        double r81842 = 0.5;
        double r81843 = pow(r81841, r81842);
        double r81844 = 6.0;
        double r81845 = r81843 / r81844;
        double r81846 = r81837 * r81845;
        double r81847 = 2.0;
        double r81848 = atan2(1.0, 0.0);
        double r81849 = r81847 * r81848;
        double r81850 = u2;
        double r81851 = r81849 * r81850;
        double r81852 = cos(r81851);
        double r81853 = r81846 * r81852;
        double r81854 = r81853 + r81842;
        return r81854;
}

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 2019322 
(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))