Average Error: 0.4 → 0.3
Time: 17.7s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(\sqrt{\frac{1}{6}} \cdot \left(\sqrt{\frac{1}{6}} \cdot {\left(-2 \cdot \log u1\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + \frac{1}{2}\]
\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)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + \frac{1}{2}
double f(double u1, double u2) {
        double r72635 = 1.0;
        double r72636 = 6.0;
        double r72637 = r72635 / r72636;
        double r72638 = -2.0;
        double r72639 = u1;
        double r72640 = log(r72639);
        double r72641 = r72638 * r72640;
        double r72642 = 0.5;
        double r72643 = pow(r72641, r72642);
        double r72644 = r72637 * r72643;
        double r72645 = 2.0;
        double r72646 = atan2(1.0, 0.0);
        double r72647 = r72645 * r72646;
        double r72648 = u2;
        double r72649 = r72647 * r72648;
        double r72650 = cos(r72649);
        double r72651 = r72644 * r72650;
        double r72652 = r72651 + r72642;
        return r72652;
}
↓
double f(double u1, double u2) {
        double r72653 = 1.0;
        double r72654 = 6.0;
        double r72655 = r72653 / r72654;
        double r72656 = sqrt(r72655);
        double r72657 = -2.0;
        double r72658 = u1;
        double r72659 = log(r72658);
        double r72660 = r72657 * r72659;
        double r72661 = 2.0;
        double r72662 = r72653 / r72661;
        double r72663 = pow(r72660, r72662);
        double r72664 = r72656 * r72663;
        double r72665 = r72656 * r72664;
        double r72666 = atan2(1.0, 0.0);
        double r72667 = r72661 * r72666;
        double r72668 = u2;
        double r72669 = r72667 * r72668;
        double r72670 = cos(r72669);
        double r72671 = r72665 * r72670;
        double r72672 = r72671 + r72662;
        return r72672;
}

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. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019304 
(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))