Average Error: 0.4 → 0.4
Time: 27.5s
Precision: 64
\[0 \le u1 \le 1 \land 0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left({\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5\]
\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
↓
\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left({\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5
double f(double u1, double u2) {
        double r2416701 = 1.0;
        double r2416702 = 6.0;
        double r2416703 = r2416701 / r2416702;
        double r2416704 = -2.0;
        double r2416705 = u1;
        double r2416706 = log(r2416705);
        double r2416707 = r2416704 * r2416706;
        double r2416708 = 0.5;
        double r2416709 = pow(r2416707, r2416708);
        double r2416710 = r2416703 * r2416709;
        double r2416711 = 2.0;
        double r2416712 = atan2(1.0, 0.0);
        double r2416713 = r2416711 * r2416712;
        double r2416714 = u2;
        double r2416715 = r2416713 * r2416714;
        double r2416716 = cos(r2416715);
        double r2416717 = r2416710 * r2416716;
        double r2416718 = r2416717 + r2416708;
        return r2416718;
}
↓
double f(double u1, double u2) {
        double r2416719 = u2;
        double r2416720 = 2.0;
        double r2416721 = atan2(1.0, 0.0);
        double r2416722 = r2416720 * r2416721;
        double r2416723 = r2416719 * r2416722;
        double r2416724 = cos(r2416723);
        double r2416725 = -2.0;
        double r2416726 = 1.0;
        double r2416727 = pow(r2416725, r2416726);
        double r2416728 = u1;
        double r2416729 = log(r2416728);
        double r2416730 = pow(r2416729, r2416726);
        double r2416731 = r2416727 * r2416730;
        double r2416732 = 0.5;
        double r2416733 = pow(r2416731, r2416732);
        double r2416734 = 0.16666666666666666;
        double r2416735 = r2416733 * r2416734;
        double r2416736 = r2416724 * r2416735;
        double r2416737 = r2416736 + r2416732;
        return r2416737;
}

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. Simplified0.3

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

  5. Taylor expanded around 0 0.4

    \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  6. Final simplification0.4

    \[\leadsto \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \left({\left({-2}^{1.0} \cdot {\left(\log u1\right)}^{1.0}\right)}^{0.5} \cdot \frac{1}{6}\right) + 0.5\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(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))