Average Error: 0.4 → 0.3
Time: 30.8s
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 r78607 = 1.0;
        double r78608 = 6.0;
        double r78609 = r78607 / r78608;
        double r78610 = -2.0;
        double r78611 = u1;
        double r78612 = log(r78611);
        double r78613 = r78610 * r78612;
        double r78614 = 0.5;
        double r78615 = pow(r78613, r78614);
        double r78616 = r78609 * r78615;
        double r78617 = 2.0;
        double r78618 = atan2(1.0, 0.0);
        double r78619 = r78617 * r78618;
        double r78620 = u2;
        double r78621 = r78619 * r78620;
        double r78622 = cos(r78621);
        double r78623 = r78616 * r78622;
        double r78624 = r78623 + r78614;
        return r78624;
}
↓
double f(double u1, double u2) {
        double r78625 = 1.0;
        double r78626 = 6.0;
        double r78627 = r78625 / r78626;
        double r78628 = sqrt(r78627);
        double r78629 = -2.0;
        double r78630 = u1;
        double r78631 = log(r78630);
        double r78632 = r78629 * r78631;
        double r78633 = 0.5;
        double r78634 = pow(r78632, r78633);
        double r78635 = r78628 * r78634;
        double r78636 = r78628 * r78635;
        double r78637 = 2.0;
        double r78638 = atan2(1.0, 0.0);
        double r78639 = r78637 * r78638;
        double r78640 = u2;
        double r78641 = r78639 * r78640;
        double r78642 = cos(r78641);
        double r78643 = r78636 * r78642;
        double r78644 = r78643 + r78633;
        return r78644;
}

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))