Average Error: 0.4 → 0.4
Time: 33.6s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[0.5 + \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)\]
\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
↓
0.5 + \left(\left(\sqrt{\frac{1}{6}} \cdot {\left(\log u1 \cdot -2\right)}^{0.5}\right) \cdot \sqrt{\frac{1}{6}}\right) \cdot \cos \left(\left(\pi \cdot u2\right) \cdot 2\right)
double f(double u1, double u2) {
        double r1920604 = 1.0;
        double r1920605 = 6.0;
        double r1920606 = r1920604 / r1920605;
        double r1920607 = -2.0;
        double r1920608 = u1;
        double r1920609 = log(r1920608);
        double r1920610 = r1920607 * r1920609;
        double r1920611 = 0.5;
        double r1920612 = pow(r1920610, r1920611);
        double r1920613 = r1920606 * r1920612;
        double r1920614 = 2.0;
        double r1920615 = atan2(1.0, 0.0);
        double r1920616 = r1920614 * r1920615;
        double r1920617 = u2;
        double r1920618 = r1920616 * r1920617;
        double r1920619 = cos(r1920618);
        double r1920620 = r1920613 * r1920619;
        double r1920621 = r1920620 + r1920611;
        return r1920621;
}
↓
double f(double u1, double u2) {
        double r1920622 = 0.5;
        double r1920623 = 1.0;
        double r1920624 = 6.0;
        double r1920625 = r1920623 / r1920624;
        double r1920626 = sqrt(r1920625);
        double r1920627 = u1;
        double r1920628 = log(r1920627);
        double r1920629 = -2.0;
        double r1920630 = r1920628 * r1920629;
        double r1920631 = pow(r1920630, r1920622);
        double r1920632 = r1920626 * r1920631;
        double r1920633 = r1920632 * r1920626;
        double r1920634 = atan2(1.0, 0.0);
        double r1920635 = u2;
        double r1920636 = r1920634 * r1920635;
        double r1920637 = 2.0;
        double r1920638 = r1920636 * r1920637;
        double r1920639 = cos(r1920638);
        double r1920640 = r1920633 * r1920639;
        double r1920641 = r1920622 + r1920640;
        return r1920641;
}

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 add-sqr-sqrt0.4

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

  4. Applied associate-*l*0.4

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

  5. Taylor expanded around 0 0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))