Toniolo and Linder, Equation (13)

Average Error: 33.1 → 27.5

Time: 51.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -5.889660218949 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}\right) - \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\right) \cdot \sqrt[3]{U - U*}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, \left(\frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}} \cdot 0 - \frac{\frac{U - U*}{\frac{Om}{\ell}}}{\frac{\frac{Om}{\ell}}{n}}\right) + t\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2234810 = 2.0;
        double r2234811 = n;
        double r2234812 = r2234810 * r2234811;
        double r2234813 = U;
        double r2234814 = r2234812 * r2234813;
        double r2234815 = t;
        double r2234816 = l;
        double r2234817 = r2234816 * r2234816;
        double r2234818 = Om;
        double r2234819 = r2234817 / r2234818;
        double r2234820 = r2234810 * r2234819;
        double r2234821 = r2234815 - r2234820;
        double r2234822 = r2234816 / r2234818;
        double r2234823 = pow(r2234822, r2234810);
        double r2234824 = r2234811 * r2234823;
        double r2234825 = U_;
        double r2234826 = r2234813 - r2234825;
        double r2234827 = r2234824 * r2234826;
        double r2234828 = r2234821 - r2234827;
        double r2234829 = r2234814 * r2234828;
        double r2234830 = sqrt(r2234829);
        return r2234830;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2234831 = U;
        double r2234832 = -5.889660218949e-311;
        bool r2234833 = r2234831 <= r2234832;
        double r2234834 = 2.0;
        double r2234835 = n;
        double r2234836 = r2234834 * r2234835;
        double r2234837 = r2234836 * r2234831;
        double r2234838 = t;
        double r2234839 = l;
        double r2234840 = Om;
        double r2234841 = cbrt(r2234840);
        double r2234842 = r2234839 / r2234841;
        double r2234843 = r2234842 * r2234842;
        double r2234844 = r2234843 / r2234841;
        double r2234845 = r2234834 * r2234844;
        double r2234846 = r2234838 - r2234845;
        double r2234847 = U_;
        double r2234848 = r2234831 - r2234847;
        double r2234849 = cbrt(r2234848);
        double r2234850 = r2234849 * r2234849;
        double r2234851 = r2234840 / r2234839;
        double r2234852 = r2234851 * r2234851;
        double r2234853 = r2234835 / r2234852;
        double r2234854 = r2234850 * r2234853;
        double r2234855 = r2234854 * r2234849;
        double r2234856 = r2234846 - r2234855;
        double r2234857 = r2234837 * r2234856;
        double r2234858 = sqrt(r2234857);
        double r2234859 = sqrt(r2234831);
        double r2234860 = -2.0;
        double r2234861 = 0.0;
        double r2234862 = r2234844 * r2234861;
        double r2234863 = r2234848 / r2234851;
        double r2234864 = r2234851 / r2234835;
        double r2234865 = r2234863 / r2234864;
        double r2234866 = r2234862 - r2234865;
        double r2234867 = r2234866 + r2234838;
        double r2234868 = fma(r2234860, r2234844, r2234867);
        double r2234869 = r2234836 * r2234868;
        double r2234870 = sqrt(r2234869);
        double r2234871 = r2234859 * r2234870;
        double r2234872 = r2234833 ? r2234858 : r2234871;
        return r2234872;
}

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 U < -5.889660218949e-311

   1. Initial program 33.4

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

   3. Applied add-cube-cbrt 33.5

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

   4. Applied associate-/r* 33.5

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

   5. Simplified 31.5

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

   6. Taylor expanded around 0 37.2

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

   7. Simplified 31.5

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

   9. Applied add-cube-cbrt 31.5

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

   10. Applied associate-*r* 31.5

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

   if -5.889660218949e-311 < U

   1. Initial program 32.7

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

   3. Applied add-cube-cbrt 32.8

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

   4. Applied associate-/r* 32.8

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

   5. Simplified 30.6

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

   6. Taylor expanded around 0 36.1

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

   7. Simplified 30.5

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

   9. Applied add-sqr-sqrt 46.2

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

   10. Applied prod-diff 46.2

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

   11. Applied associate--l+ 46.2

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

   12. Applied distribute-lft-in 46.2

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

   13. Simplified 31.3

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

   14. Simplified 30.4

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

   16. Applied distribute-lft-out 30.4

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

   17. Applied sqrt-prod 23.5

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

   18. Simplified 23.5

       \[\leadsto \sqrt{U} \cdot \color{blue}{\sqrt{\mathsf{fma}\left(-2, \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, t + \left(\frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}} \cdot 0 - \frac{\frac{U - U*}{\frac{Om}{\ell}}}{\frac{\frac{Om}{\ell}}{n}}\right)\right) \cdot \left(n \cdot 2\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 27.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -5.889660218949 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}\right) - \left(\left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right) \cdot \frac{n}{\frac{Om}{\ell} \cdot \frac{Om}{\ell}}\right) \cdot \sqrt[3]{U - U*}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \mathsf{fma}\left(-2, \frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}}, \left(\frac{\frac{\ell}{\sqrt[3]{Om}} \cdot \frac{\ell}{\sqrt[3]{Om}}}{\sqrt[3]{Om}} \cdot 0 - \frac{\frac{U - U*}{\frac{Om}{\ell}}}{\frac{\frac{Om}{\ell}}{n}}\right) + t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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*))))))