Toniolo and Linder, Equation (13)

Average Error: 33.7 → 18.2

Time: 5.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\left(\left|\sqrt[3]{\sqrt[3]{U}}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{U}} \cdot \left(t \cdot n - \frac{\ell}{\frac{Om}{n}} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 3.808352790811895 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)} \cdot \sqrt{\sqrt[3]{U} \cdot \left(t \cdot n + \left(\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \left(-n\right)\right) \cdot \left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)\right) \cdot \left(2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2142789 = 2.0;
        double r2142790 = n;
        double r2142791 = r2142789 * r2142790;
        double r2142792 = U;
        double r2142793 = r2142791 * r2142792;
        double r2142794 = t;
        double r2142795 = l;
        double r2142796 = r2142795 * r2142795;
        double r2142797 = Om;
        double r2142798 = r2142796 / r2142797;
        double r2142799 = r2142789 * r2142798;
        double r2142800 = r2142794 - r2142799;
        double r2142801 = r2142795 / r2142797;
        double r2142802 = pow(r2142801, r2142789);
        double r2142803 = r2142790 * r2142802;
        double r2142804 = U_;
        double r2142805 = r2142792 - r2142804;
        double r2142806 = r2142803 * r2142805;
        double r2142807 = r2142800 - r2142806;
        double r2142808 = r2142793 * r2142807;
        double r2142809 = sqrt(r2142808);
        return r2142809;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2142810 = 2.0;
        double r2142811 = n;
        double r2142812 = r2142810 * r2142811;
        double r2142813 = U;
        double r2142814 = r2142812 * r2142813;
        double r2142815 = t;
        double r2142816 = l;
        double r2142817 = r2142816 * r2142816;
        double r2142818 = Om;
        double r2142819 = r2142817 / r2142818;
        double r2142820 = r2142819 * r2142810;
        double r2142821 = r2142815 - r2142820;
        double r2142822 = r2142816 / r2142818;
        double r2142823 = pow(r2142822, r2142810);
        double r2142824 = r2142811 * r2142823;
        double r2142825 = U_;
        double r2142826 = r2142813 - r2142825;
        double r2142827 = r2142824 * r2142826;
        double r2142828 = r2142821 - r2142827;
        double r2142829 = r2142814 * r2142828;
        double r2142830 = sqrt(r2142829);
        double r2142831 = 0.0;
        bool r2142832 = r2142830 <= r2142831;
        double r2142833 = cbrt(r2142813);
        double r2142834 = cbrt(r2142833);
        double r2142835 = fabs(r2142834);
        double r2142836 = r2142815 * r2142811;
        double r2142837 = r2142818 / r2142811;
        double r2142838 = r2142816 / r2142837;
        double r2142839 = r2142810 * r2142816;
        double r2142840 = r2142825 - r2142813;
        double r2142841 = r2142840 * r2142838;
        double r2142842 = r2142839 - r2142841;
        double r2142843 = r2142838 * r2142842;
        double r2142844 = r2142836 - r2142843;
        double r2142845 = r2142834 * r2142844;
        double r2142846 = sqrt(r2142845);
        double r2142847 = r2142835 * r2142846;
        double r2142848 = r2142833 * r2142833;
        double r2142849 = r2142810 * r2142848;
        double r2142850 = sqrt(r2142849);
        double r2142851 = r2142847 * r2142850;
        double r2142852 = 3.808352790811895e+149;
        bool r2142853 = r2142830 <= r2142852;
        double r2142854 = cbrt(r2142822);
        double r2142855 = -r2142811;
        double r2142856 = r2142854 * r2142855;
        double r2142857 = r2142854 * r2142854;
        double r2142858 = r2142856 * r2142857;
        double r2142859 = r2142811 * r2142822;
        double r2142860 = r2142859 * r2142840;
        double r2142861 = r2142839 - r2142860;
        double r2142862 = r2142858 * r2142861;
        double r2142863 = r2142836 + r2142862;
        double r2142864 = r2142833 * r2142863;
        double r2142865 = sqrt(r2142864);
        double r2142866 = r2142850 * r2142865;
        double r2142867 = r2142853 ? r2142830 : r2142866;
        double r2142868 = r2142832 ? r2142851 : r2142867;
        return r2142868;
}

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 (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 0.0

   1. Initial program 56.9

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

   2. Simplified 39.7

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

   4. Applied add-cube-cbrt 39.9

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

   5. Applied associate-*l* 39.9

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

   6. Applied associate-*r* 39.9

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

   7. Applied sqrt-prod 21.6

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

   9. Applied sub-neg 21.6

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

   10. Applied distribute-lft-in 21.6

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

   11. Simplified 21.4

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

   13. Applied add-cube-cbrt 21.6

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

   14. Applied associate-*l* 21.6

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

   15. Applied sqrt-prod 17.6

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

   16. Simplified 17.6

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

   17. Simplified 18.8

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

   if 0.0 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 3.808352790811895e+149

   1. Initial program 1.9

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

   if 3.808352790811895e+149 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 60.0

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

   2. Simplified 51.6

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

   4. Applied add-cube-cbrt 51.7

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

   5. Applied associate-*l* 51.7

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

   6. Applied associate-*r* 51.7

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

   7. Applied sqrt-prod 47.4

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

   9. Applied sub-neg 47.4

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

   10. Applied distribute-lft-in 47.4

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

   11. Simplified 36.0

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

   13. Applied add-cube-cbrt 36.0

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

   14. Applied associate-*l* 36.0

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\left(\left|\sqrt[3]{\sqrt[3]{U}}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{U}} \cdot \left(t \cdot n - \frac{\ell}{\frac{Om}{n}} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)}\right) \cdot \sqrt{2 \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 3.808352790811895 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)} \cdot \sqrt{\sqrt[3]{U} \cdot \left(t \cdot n + \left(\left(\sqrt[3]{\frac{\ell}{Om}} \cdot \left(-n\right)\right) \cdot \left(\sqrt[3]{\frac{\ell}{Om}} \cdot \sqrt[3]{\frac{\ell}{Om}}\right)\right) \cdot \left(2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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*))))))