Toniolo and Linder, Equation (13)

Average Error: 34.6 → 29.3

Time: 46.0s

Precision: 64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le 3.683386903089245746265762362490297308203 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r84616 = 2.0;
        double r84617 = n;
        double r84618 = r84616 * r84617;
        double r84619 = U;
        double r84620 = r84618 * r84619;
        double r84621 = t;
        double r84622 = l;
        double r84623 = r84622 * r84622;
        double r84624 = Om;
        double r84625 = r84623 / r84624;
        double r84626 = r84616 * r84625;
        double r84627 = r84621 - r84626;
        double r84628 = r84622 / r84624;
        double r84629 = pow(r84628, r84616);
        double r84630 = r84617 * r84629;
        double r84631 = U_;
        double r84632 = r84619 - r84631;
        double r84633 = r84630 * r84632;
        double r84634 = r84627 - r84633;
        double r84635 = r84620 * r84634;
        double r84636 = sqrt(r84635);
        return r84636;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r84637 = t;
        double r84638 = 3.683386903089246e+135;
        bool r84639 = r84637 <= r84638;
        double r84640 = 2.0;
        double r84641 = l;
        double r84642 = Om;
        double r84643 = r84641 / r84642;
        double r84644 = r84641 * r84643;
        double r84645 = n;
        double r84646 = cbrt(r84645);
        double r84647 = r84646 * r84646;
        double r84648 = 2.0;
        double r84649 = r84640 / r84648;
        double r84650 = pow(r84643, r84649);
        double r84651 = r84646 * r84650;
        double r84652 = r84647 * r84651;
        double r84653 = U;
        double r84654 = U_;
        double r84655 = r84653 - r84654;
        double r84656 = r84655 * r84650;
        double r84657 = r84652 * r84656;
        double r84658 = fma(r84640, r84644, r84657);
        double r84659 = r84637 - r84658;
        double r84660 = r84640 * r84645;
        double r84661 = r84660 * r84653;
        double r84662 = r84659 * r84661;
        double r84663 = sqrt(r84662);
        double r84664 = r84648 * r84649;
        double r84665 = pow(r84643, r84664);
        double r84666 = r84645 * r84665;
        double r84667 = r84655 * r84666;
        double r84668 = fma(r84640, r84644, r84667);
        double r84669 = r84637 - r84668;
        double r84670 = sqrt(r84669);
        double r84671 = sqrt(r84661);
        double r84672 = r84670 * r84671;
        double r84673 = r84639 ? r84663 : r84672;
        return r84673;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 2 regimes

2. if t < 3.683386903089246e+135

   1. Initial program 34.1

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 34.1

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 34.1

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   5. Applied times-frac 31.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{1} \cdot \frac{\ell}{Om}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   6. Simplified 31.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\ell} \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   7. Using strategy rm

   8. Applied sqr-pow 31.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   9. Applied associate-*r* 30.4

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 30.5

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(\color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   12. Applied associate-*l* 30.5

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   13. Using strategy rm

   14. Applied associate-*l* 30.4

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   15. Simplified 30.4

       \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \color{blue}{\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   if 3.683386903089246e+135 < t

   1. Initial program 37.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   2. Simplified 37.8

      \[\leadsto \color{blue}{\sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 37.8

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{1 \cdot Om}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   5. Applied times-frac 35.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{1} \cdot \frac{\ell}{Om}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   6. Simplified 35.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \color{blue}{\ell} \cdot \frac{\ell}{Om}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   7. Using strategy rm

   8. Applied sqr-pow 35.2

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]

   9. Applied associate-*r* 34.6

      \[\leadsto \sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\]
   10. Using strategy rm

   11. Applied sqrt-prod 22.6

       \[\leadsto \color{blue}{\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}}\]

   12. Simplified 23.1

       \[\leadsto \color{blue}{\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.683386903089245746265762362490297308203 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(2, \ell \cdot \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))