Average Error: 0.4 → 0.3
Time: 11.6s
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 r74685 = 1.0;
        double r74686 = 6.0;
        double r74687 = r74685 / r74686;
        double r74688 = -2.0;
        double r74689 = u1;
        double r74690 = log(r74689);
        double r74691 = r74688 * r74690;
        double r74692 = 0.5;
        double r74693 = pow(r74691, r74692);
        double r74694 = r74687 * r74693;
        double r74695 = 2.0;
        double r74696 = atan2(1.0, 0.0);
        double r74697 = r74695 * r74696;
        double r74698 = u2;
        double r74699 = r74697 * r74698;
        double r74700 = cos(r74699);
        double r74701 = r74694 * r74700;
        double r74702 = r74701 + r74692;
        return r74702;
}
↓
double f(double u1, double u2) {
        double r74703 = 1.0;
        double r74704 = -2.0;
        double r74705 = u1;
        double r74706 = log(r74705);
        double r74707 = r74704 * r74706;
        double r74708 = 0.5;
        double r74709 = pow(r74707, r74708);
        double r74710 = 6.0;
        double r74711 = r74709 / r74710;
        double r74712 = r74703 * r74711;
        double r74713 = 2.0;
        double r74714 = atan2(1.0, 0.0);
        double r74715 = r74713 * r74714;
        double r74716 = u2;
        double r74717 = r74715 * r74716;
        double r74718 = cos(r74717);
        double r74719 = r74712 * r74718;
        double r74720 = r74719 + r74708;
        return r74720;
}

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 2020056 
(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))