Average Error: 0.4 → 0.4
Time: 11.9s
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 r72631 = 1.0;
        double r72632 = 6.0;
        double r72633 = r72631 / r72632;
        double r72634 = -2.0;
        double r72635 = u1;
        double r72636 = log(r72635);
        double r72637 = r72634 * r72636;
        double r72638 = 0.5;
        double r72639 = pow(r72637, r72638);
        double r72640 = r72633 * r72639;
        double r72641 = 2.0;
        double r72642 = atan2(1.0, 0.0);
        double r72643 = r72641 * r72642;
        double r72644 = u2;
        double r72645 = r72643 * r72644;
        double r72646 = cos(r72645);
        double r72647 = r72640 * r72646;
        double r72648 = r72647 + r72638;
        return r72648;
}
↓
double f(double u1, double u2) {
        double r72649 = 1.0;
        double r72650 = 6.0;
        double r72651 = r72649 / r72650;
        double r72652 = sqrt(r72651);
        double r72653 = -2.0;
        double r72654 = u1;
        double r72655 = log(r72654);
        double r72656 = r72653 * r72655;
        double r72657 = 0.5;
        double r72658 = pow(r72656, r72657);
        double r72659 = r72652 * r72658;
        double r72660 = r72652 * r72659;
        double r72661 = 2.0;
        double r72662 = atan2(1.0, 0.0);
        double r72663 = r72661 * r72662;
        double r72664 = u2;
        double r72665 = r72663 * r72664;
        double r72666 = cos(r72665);
        double r72667 = r72660 * r72666;
        double r72668 = r72667 + r72657;
        return r72668;
}

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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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))