Toniolo and Linder, Equation (13)

Average Error: 32.8 → 25.5

Time: 44.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -9.944037691838278 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \sqrt[3]{U - U*} \cdot \left(n \cdot \left(\left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.1144092498290265 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \frac{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \left(U - U*\right) \cdot \left(\sqrt[3]{n} \cdot \left(\left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1136681 = 2.0;
        double r1136682 = n;
        double r1136683 = r1136681 * r1136682;
        double r1136684 = U;
        double r1136685 = r1136683 * r1136684;
        double r1136686 = t;
        double r1136687 = l;
        double r1136688 = r1136687 * r1136687;
        double r1136689 = Om;
        double r1136690 = r1136688 / r1136689;
        double r1136691 = r1136681 * r1136690;
        double r1136692 = r1136686 - r1136691;
        double r1136693 = r1136687 / r1136689;
        double r1136694 = pow(r1136693, r1136681);
        double r1136695 = r1136682 * r1136694;
        double r1136696 = U_;
        double r1136697 = r1136684 - r1136696;
        double r1136698 = r1136695 * r1136697;
        double r1136699 = r1136692 - r1136698;
        double r1136700 = r1136685 * r1136699;
        double r1136701 = sqrt(r1136700);
        return r1136701;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1136702 = n;
        double r1136703 = -9.944037691838278e-75;
        bool r1136704 = r1136702 <= r1136703;
        double r1136705 = 2.0;
        double r1136706 = r1136705 * r1136702;
        double r1136707 = U;
        double r1136708 = r1136706 * r1136707;
        double r1136709 = t;
        double r1136710 = l;
        double r1136711 = Om;
        double r1136712 = r1136710 / r1136711;
        double r1136713 = r1136710 * r1136712;
        double r1136714 = r1136705 * r1136713;
        double r1136715 = r1136709 - r1136714;
        double r1136716 = U_;
        double r1136717 = r1136707 - r1136716;
        double r1136718 = cbrt(r1136717);
        double r1136719 = r1136718 * r1136712;
        double r1136720 = r1136719 * r1136719;
        double r1136721 = r1136702 * r1136720;
        double r1136722 = r1136718 * r1136721;
        double r1136723 = r1136715 - r1136722;
        double r1136724 = r1136708 * r1136723;
        double r1136725 = sqrt(r1136724);
        double r1136726 = -1.1144092498290265e-301;
        bool r1136727 = r1136702 <= r1136726;
        double r1136728 = r1136712 * r1136705;
        double r1136729 = r1136713 * r1136702;
        double r1136730 = r1136729 * r1136717;
        double r1136731 = r1136730 / r1136711;
        double r1136732 = fma(r1136710, r1136728, r1136731);
        double r1136733 = r1136709 - r1136732;
        double r1136734 = r1136707 * r1136733;
        double r1136735 = r1136734 * r1136706;
        double r1136736 = sqrt(r1136735);
        double r1136737 = cbrt(r1136702);
        double r1136738 = r1136737 * r1136712;
        double r1136739 = r1136738 * r1136738;
        double r1136740 = r1136737 * r1136739;
        double r1136741 = r1136717 * r1136740;
        double r1136742 = fma(r1136710, r1136728, r1136741);
        double r1136743 = r1136709 - r1136742;
        double r1136744 = r1136707 * r1136743;
        double r1136745 = sqrt(r1136744);
        double r1136746 = sqrt(r1136706);
        double r1136747 = r1136745 * r1136746;
        double r1136748 = r1136727 ? r1136736 : r1136747;
        double r1136749 = r1136704 ? r1136725 : r1136748;
        return r1136749;
}

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 3 regimes

2. if n < -9.944037691838278e-75

   1. Initial program 30.7

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

   3. Applied *-un-lft-identity 30.7

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

   4. Applied times-frac 28.2

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

   5. Simplified 28.2

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

   7. Applied add-cube-cbrt 28.2

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

   8. Applied associate-*r* 28.2

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

   9. Simplified 27.1

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

   if -9.944037691838278e-75 < n < -1.1144092498290265e-301

   1. Initial program 34.9

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

   3. Applied *-un-lft-identity 34.9

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

   4. Applied times-frac 32.4

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

   5. Simplified 32.4

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

   7. Applied associate-*l* 31.2

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

   8. Simplified 31.2

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

   10. Applied associate-*r/ 31.2

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

   11. Applied associate-*l/ 30.2

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

   12. Applied associate-*l/ 30.1

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

   if -1.1144092498290265e-301 < n

   1. Initial program 32.9

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

   3. Applied *-un-lft-identity 32.9

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

   4. Applied times-frac 30.3

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

   5. Simplified 30.3

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

   7. Applied associate-*l* 29.9

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

   8. Simplified 29.9

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

   10. Applied add-cube-cbrt 30.0

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

   11. Applied associate-*r* 30.0

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

   12. Simplified 29.2

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

   14. Applied sqrt-prod 22.6

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

4. Final simplification 25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9.944037691838278 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \sqrt[3]{U - U*} \cdot \left(n \cdot \left(\left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{U - U*} \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.1144092498290265 \cdot 10^{-301}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \frac{\left(\left(\ell \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)}{Om}\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot \left(t - \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot 2, \left(U - U*\right) \cdot \left(\sqrt[3]{n} \cdot \left(\left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right) \cdot \left(\sqrt[3]{n} \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)} \cdot \sqrt{2 \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))