Average Error: 0.4 → 0.3
Time: 34.8s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\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(\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
double f(double u1, double u2) {
        double r73607 = 1.0;
        double r73608 = 6.0;
        double r73609 = r73607 / r73608;
        double r73610 = -2.0;
        double r73611 = u1;
        double r73612 = log(r73611);
        double r73613 = r73610 * r73612;
        double r73614 = 0.5;
        double r73615 = pow(r73613, r73614);
        double r73616 = r73609 * r73615;
        double r73617 = 2.0;
        double r73618 = atan2(1.0, 0.0);
        double r73619 = r73617 * r73618;
        double r73620 = u2;
        double r73621 = r73619 * r73620;
        double r73622 = cos(r73621);
        double r73623 = r73616 * r73622;
        double r73624 = r73623 + r73614;
        return r73624;
}
↓
double f(double u1, double u2) {
        double r73625 = 1.0;
        double r73626 = 6.0;
        double r73627 = r73625 / r73626;
        double r73628 = sqrt(r73627);
        double r73629 = -2.0;
        double r73630 = u1;
        double r73631 = log(r73630);
        double r73632 = r73629 * r73631;
        double r73633 = 0.5;
        double r73634 = pow(r73632, r73633);
        double r73635 = r73628 * r73634;
        double r73636 = r73628 * r73635;
        double r73637 = 2.0;
        double r73638 = atan2(1.0, 0.0);
        double r73639 = r73637 * r73638;
        double r73640 = u2;
        double r73641 = r73639 * r73640;
        double r73642 = cos(r73641);
        double r73643 = r73636 * r73642;
        double r73644 = r73643 + r73633;
        return r73644;
}

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*0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 
(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))