Toniolo and Linder, Equation (13)

Average Error: 34.7 → 29.4

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.44834988828803289 \cdot 10^{196}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(U - U*\right) \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r268782 = 2.0;
        double r268783 = n;
        double r268784 = r268782 * r268783;
        double r268785 = U;
        double r268786 = r268784 * r268785;
        double r268787 = t;
        double r268788 = l;
        double r268789 = r268788 * r268788;
        double r268790 = Om;
        double r268791 = r268789 / r268790;
        double r268792 = r268782 * r268791;
        double r268793 = r268787 - r268792;
        double r268794 = r268788 / r268790;
        double r268795 = pow(r268794, r268782);
        double r268796 = r268783 * r268795;
        double r268797 = U_;
        double r268798 = r268785 - r268797;
        double r268799 = r268796 * r268798;
        double r268800 = r268793 - r268799;
        double r268801 = r268786 * r268800;
        double r268802 = sqrt(r268801);
        return r268802;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r268803 = t;
        double r268804 = 1.4483498882880329e+196;
        bool r268805 = r268803 <= r268804;
        double r268806 = 2.0;
        double r268807 = n;
        double r268808 = r268806 * r268807;
        double r268809 = l;
        double r268810 = Om;
        double r268811 = r268809 / r268810;
        double r268812 = r268809 * r268811;
        double r268813 = U;
        double r268814 = U_;
        double r268815 = r268813 - r268814;
        double r268816 = 2.0;
        double r268817 = r268806 / r268816;
        double r268818 = pow(r268811, r268817);
        double r268819 = r268807 * r268818;
        double r268820 = r268815 * r268819;
        double r268821 = r268820 * r268818;
        double r268822 = fma(r268812, r268806, r268821);
        double r268823 = r268803 - r268822;
        double r268824 = r268823 * r268813;
        double r268825 = r268808 * r268824;
        double r268826 = sqrt(r268825);
        double r268827 = r268808 * r268813;
        double r268828 = sqrt(r268827);
        double r268829 = r268815 * r268807;
        double r268830 = pow(r268811, r268806);
        double r268831 = r268829 * r268830;
        double r268832 = fma(r268812, r268806, r268831);
        double r268833 = r268803 - r268832;
        double r268834 = sqrt(r268833);
        double r268835 = r268828 * r268834;
        double r268836 = r268805 ? r268826 : r268835;
        return r268836;
}

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 < 1.4483498882880329e+196

   1. Initial program 34.0

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

   3. Applied *-un-lft-identity 34.0

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

   4. Applied times-frac 31.1

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

   5. Simplified 31.1

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

   7. Applied associate-*l* 31.1

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

   8. Simplified 33.4

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

   10. Applied sqr-pow 33.4

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

   11. Applied associate-*r* 32.4

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

   13. Applied associate-*l* 29.7

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

   if 1.4483498882880329e+196 < t

   1. Initial program 41.0

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

   3. Applied *-un-lft-identity 41.0

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

   4. Applied times-frac 37.7

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

   5. Simplified 37.7

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

   7. Applied sqrt-prod 24.4

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

   8. Simplified 27.4

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

4. Final simplification 29.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.44834988828803289 \cdot 10^{196}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(U - U*\right) \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))