Toniolo and Linder, Equation (13)

Average Error: 33.7 → 18.7

Time: 4.7m

Precision: 64

Math

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

↓

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

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r11191707 = 2.0;
        double r11191708 = n;
        double r11191709 = r11191707 * r11191708;
        double r11191710 = U;
        double r11191711 = r11191709 * r11191710;
        double r11191712 = t;
        double r11191713 = l;
        double r11191714 = r11191713 * r11191713;
        double r11191715 = Om;
        double r11191716 = r11191714 / r11191715;
        double r11191717 = r11191707 * r11191716;
        double r11191718 = r11191712 - r11191717;
        double r11191719 = r11191713 / r11191715;
        double r11191720 = pow(r11191719, r11191707);
        double r11191721 = r11191708 * r11191720;
        double r11191722 = U_;
        double r11191723 = r11191710 - r11191722;
        double r11191724 = r11191721 * r11191723;
        double r11191725 = r11191718 - r11191724;
        double r11191726 = r11191711 * r11191725;
        double r11191727 = sqrt(r11191726);
        return r11191727;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r11191728 = U;
        double r11191729 = n;
        double r11191730 = cbrt(r11191729);
        double r11191731 = r11191728 * r11191730;
        double r11191732 = cbrt(r11191731);
        double r11191733 = r11191732 * r11191732;
        double r11191734 = sqrt(r11191733);
        double r11191735 = t;
        double r11191736 = 2.0;
        double r11191737 = l;
        double r11191738 = Om;
        double r11191739 = r11191738 / r11191737;
        double r11191740 = r11191737 / r11191739;
        double r11191741 = r11191736 * r11191740;
        double r11191742 = r11191735 - r11191741;
        double r11191743 = U_;
        double r11191744 = r11191728 - r11191743;
        double r11191745 = r11191737 / r11191738;
        double r11191746 = r11191745 * r11191745;
        double r11191747 = r11191729 * r11191746;
        double r11191748 = r11191744 * r11191747;
        double r11191749 = r11191742 - r11191748;
        double r11191750 = cbrt(r11191732);
        double r11191751 = r11191749 * r11191750;
        double r11191752 = sqrt(r11191751);
        double r11191753 = cbrt(r11191733);
        double r11191754 = sqrt(r11191753);
        double r11191755 = r11191752 * r11191754;
        double r11191756 = r11191734 * r11191755;
        double r11191757 = r11191730 * r11191730;
        double r11191758 = r11191757 * r11191736;
        double r11191759 = sqrt(r11191758);
        double r11191760 = r11191756 * r11191759;
        return r11191760;
}

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. Initial program 33.7

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

3. Applied add-cube-cbrt 34.0

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

4. Applied associate-*r* 34.0

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

5. Applied associate-*l* 34.0

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

6. Applied associate-*l* 32.8

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

7. Applied sqrt-prod 28.3

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

8. Simplified 28.3

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

10. Applied add-cube-cbrt 28.4

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

11. Applied associate-*l* 28.4

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

12. Applied sqrt-prod 23.4

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

14. Applied associate-/l* 20.4

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

16. Applied add-cube-cbrt 20.4

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

17. Applied cbrt-prod 20.4

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

18. Applied associate-*l* 20.4

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

19. Applied sqrt-prod 18.7

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

20. Final simplification 18.7

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

Reproduce

herbie shell --seed 2019158 +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*))))))