Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 16.8

Time: 2.3m

Precision: 64

Math

\[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;k \le -1.5834988826166857 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt[3]{2.0}}{\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0} \cdot \left(\frac{\sqrt[3]{2.0}}{\tan k} \cdot \frac{\sqrt[3]{2.0}}{\frac{\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right)}\right)\\ \mathbf{elif}\;k \le -5.124747410806296 \cdot 10^{-15}:\\ \;\;\;\;\frac{2.0}{\mathsf{fma}\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\left(t \cdot t\right) \cdot t}{\cos k}, 2.0, \frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{elif}\;k \le 1.5429975906335487 \cdot 10^{-37}:\\ \;\;\;\;\frac{2.0}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right) \cdot \tan k\right)\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\\ \mathbf{elif}\;k \le 4.2765351122852666 \cdot 10^{+17}:\\ \;\;\;\;\frac{2.0}{\mathsf{fma}\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\left(t \cdot t\right) \cdot t}{\cos k}, 2.0, \frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{elif}\;k \le 1.430501198242309 \cdot 10^{+100}:\\ \;\;\;\;\frac{2.0}{\left(\left(\tan k \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right)\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\\ \mathbf{elif}\;k \le 5.937994838209732 \cdot 10^{+182}:\\ \;\;\;\;\frac{2.0}{-\mathsf{fma}\left(\frac{\left(t \cdot t\right) \cdot t}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\sin k \cdot \sin k}} \cdot 2.0, {\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}, \frac{{\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}}{\ell \cdot \ell} \cdot \frac{t \cdot \left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3727977 = 2.0;
        double r3727978 = t;
        double r3727979 = 3.0;
        double r3727980 = pow(r3727978, r3727979);
        double r3727981 = l;
        double r3727982 = r3727981 * r3727981;
        double r3727983 = r3727980 / r3727982;
        double r3727984 = k;
        double r3727985 = sin(r3727984);
        double r3727986 = r3727983 * r3727985;
        double r3727987 = tan(r3727984);
        double r3727988 = r3727986 * r3727987;
        double r3727989 = 1.0;
        double r3727990 = r3727984 / r3727978;
        double r3727991 = pow(r3727990, r3727977);
        double r3727992 = r3727989 + r3727991;
        double r3727993 = r3727992 + r3727989;
        double r3727994 = r3727988 * r3727993;
        double r3727995 = r3727977 / r3727994;
        return r3727995;
}

↓

double f(double t, double l, double k) {
        double r3727996 = k;
        double r3727997 = -1.5834988826166857e+46;
        bool r3727998 = r3727996 <= r3727997;
        double r3727999 = 2.0;
        double r3728000 = cbrt(r3727999);
        double r3728001 = 1.0;
        double r3728002 = t;
        double r3728003 = r3727996 / r3728002;
        double r3728004 = pow(r3728003, r3727999);
        double r3728005 = r3728001 + r3728004;
        double r3728006 = r3728005 + r3728001;
        double r3728007 = r3728000 / r3728006;
        double r3728008 = tan(r3727996);
        double r3728009 = r3728000 / r3728008;
        double r3728010 = sin(r3727996);
        double r3728011 = cbrt(r3728002);
        double r3728012 = 3.0;
        double r3728013 = pow(r3728011, r3728012);
        double r3728014 = l;
        double r3728015 = r3728013 / r3728014;
        double r3728016 = r3728010 * r3728015;
        double r3728017 = cbrt(r3728014);
        double r3728018 = r3728016 / r3728017;
        double r3728019 = r3728013 / r3728017;
        double r3728020 = r3728019 * r3728019;
        double r3728021 = r3728018 * r3728020;
        double r3728022 = r3728000 / r3728021;
        double r3728023 = r3728009 * r3728022;
        double r3728024 = r3728007 * r3728023;
        double r3728025 = -5.124747410806296e-15;
        bool r3728026 = r3727996 <= r3728025;
        double r3728027 = r3728010 / r3728014;
        double r3728028 = r3728027 * r3728027;
        double r3728029 = r3728002 * r3728002;
        double r3728030 = r3728029 * r3728002;
        double r3728031 = cos(r3727996);
        double r3728032 = r3728030 / r3728031;
        double r3728033 = r3728028 * r3728032;
        double r3728034 = r3728002 / r3728014;
        double r3728035 = r3728034 / r3728014;
        double r3728036 = r3727996 * r3728010;
        double r3728037 = r3728036 * r3728036;
        double r3728038 = r3728037 / r3728031;
        double r3728039 = r3728035 * r3728038;
        double r3728040 = fma(r3728033, r3727999, r3728039);
        double r3728041 = r3727999 / r3728040;
        double r3728042 = 1.5429975906335487e-37;
        bool r3728043 = r3727996 <= r3728042;
        double r3728044 = r3728017 * r3728017;
        double r3728045 = r3728013 / r3728044;
        double r3728046 = r3728016 * r3728019;
        double r3728047 = r3728046 * r3728008;
        double r3728048 = r3728045 * r3728047;
        double r3728049 = r3728048 * r3728006;
        double r3728050 = r3727999 / r3728049;
        double r3728051 = 4.2765351122852666e+17;
        bool r3728052 = r3727996 <= r3728051;
        double r3728053 = 1.430501198242309e+100;
        bool r3728054 = r3727996 <= r3728053;
        double r3728055 = r3728008 * r3728016;
        double r3728056 = r3728019 * r3728045;
        double r3728057 = r3728055 * r3728056;
        double r3728058 = r3728057 * r3728006;
        double r3728059 = r3727999 / r3728058;
        double r3728060 = 5.937994838209732e+182;
        bool r3728061 = r3727996 <= r3728060;
        double r3728062 = r3728014 * r3728014;
        double r3728063 = r3728062 * r3728031;
        double r3728064 = r3728010 * r3728010;
        double r3728065 = r3728063 / r3728064;
        double r3728066 = r3728030 / r3728065;
        double r3728067 = r3728066 * r3727999;
        double r3728068 = 1.0;
        double r3728069 = -1.0;
        double r3728070 = pow(r3728069, r3728012);
        double r3728071 = r3728068 / r3728070;
        double r3728072 = pow(r3728071, r3728001);
        double r3728073 = r3728072 / r3728062;
        double r3728074 = r3728002 * r3728037;
        double r3728075 = r3728074 / r3728031;
        double r3728076 = r3728073 * r3728075;
        double r3728077 = fma(r3728067, r3728072, r3728076);
        double r3728078 = -r3728077;
        double r3728079 = r3727999 / r3728078;
        double r3728080 = 0.0;
        double r3728081 = r3728061 ? r3728079 : r3728080;
        double r3728082 = r3728054 ? r3728059 : r3728081;
        double r3728083 = r3728052 ? r3728041 : r3728082;
        double r3728084 = r3728043 ? r3728050 : r3728083;
        double r3728085 = r3728026 ? r3728041 : r3728084;
        double r3728086 = r3727998 ? r3728024 : r3728085;
        return r3728086;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 6 regimes

2. if k < -1.5834988826166857e+46

   1. Initial program 32.2

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 32.3

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   4. Applied unpow-prod-down 32.3

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   5. Applied times-frac 25.2

      \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   6. Applied associate-*l* 25.2

      \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 25.2

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   9. Applied unpow-prod-down 25.2

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   10. Applied times-frac 19.7

       \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   11. Using strategy rm

   12. Applied associate-*l* 19.7

       \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 19.7

       \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{2.0} \cdot \sqrt[3]{2.0}\right) \cdot \sqrt[3]{2.0}}}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   15. Applied times-frac 19.9

       \[\leadsto \color{blue}{\frac{\sqrt[3]{2.0} \cdot \sqrt[3]{2.0}}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k} \cdot \frac{\sqrt[3]{2.0}}{\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0}}\]

   16. Simplified 21.0

       \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{2.0}}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k}{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{2.0}}{\tan k}\right)} \cdot \frac{\sqrt[3]{2.0}}{\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0}\]

   if -1.5834988826166857e+46 < k < -5.124747410806296e-15 or 1.5429975906335487e-37 < k < 4.2765351122852666e+17

   1. Initial program 28.1

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   2. Taylor expanded around inf 19.0

      \[\leadsto \frac{2.0}{\color{blue}{\frac{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}{\cos k \cdot {\ell}^{2}} + 2.0 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{{\ell}^{2} \cdot \cos k}}}\]

   3. Simplified 18.3

      \[\leadsto \frac{2.0}{\color{blue}{\mathsf{fma}\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\left(t \cdot t\right) \cdot t}{\cos k}, 2.0, \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\cos k} \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\]

   if -5.124747410806296e-15 < k < 1.5429975906335487e-37

   1. Initial program 34.5

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 34.7

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   4. Applied unpow-prod-down 34.7

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   5. Applied times-frac 27.9

      \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   6. Applied associate-*l* 22.7

      \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 22.7

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   9. Applied unpow-prod-down 22.7

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   10. Applied times-frac 15.8

       \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   11. Using strategy rm

   12. Applied associate-*l* 12.9

       \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   13. Using strategy rm

   14. Applied associate-*l* 8.4

       \[\leadsto \frac{2.0}{\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right)} \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   if 4.2765351122852666e+17 < k < 1.430501198242309e+100

   1. Initial program 29.8

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 30.0

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   4. Applied unpow-prod-down 30.0

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   5. Applied times-frac 21.4

      \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   6. Applied associate-*l* 21.4

      \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 21.4

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   9. Applied unpow-prod-down 21.4

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   10. Applied times-frac 16.1

       \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   11. Using strategy rm

   12. Applied associate-*l* 16.1

       \[\leadsto \frac{2.0}{\color{blue}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right) \cdot \tan k\right)\right)} \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   if 1.430501198242309e+100 < k < 5.937994838209732e+182

   1. Initial program 33.9

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 33.9

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   4. Applied unpow-prod-down 33.9

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   5. Applied times-frac 27.5

      \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   6. Applied associate-*l* 27.5

      \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 27.5

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   9. Applied unpow-prod-down 27.5

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   10. Applied times-frac 22.0

       \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   11. Taylor expanded around -inf 24.6

       \[\leadsto \frac{2.0}{\color{blue}{-\left(2.0 \cdot \left(\frac{{t}^{3} \cdot \left({\left(\sqrt[3]{-1}\right)}^{6} \cdot {\left(\sin k\right)}^{2}\right)}{{\ell}^{2} \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}\right) + \frac{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}{{\ell}^{2} \cdot \cos k} \cdot {\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}\right)}}\]

   12. Simplified 24.5

       \[\leadsto \frac{2.0}{\color{blue}{-\mathsf{fma}\left(2.0 \cdot \frac{t \cdot \left(t \cdot t\right)}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(\sin k \cdot -1\right) \cdot \left(\sin k \cdot -1\right)}}, {\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}, \frac{t \cdot \left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right)}{\cos k} \cdot \frac{{\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}}{\ell \cdot \ell}\right)}}\]

   if 5.937994838209732e+182 < k

   1. Initial program 34.3

      \[\frac{2.0}{\left(\left(\frac{{t}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 34.3

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3.0}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   4. Applied unpow-prod-down 34.3

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   5. Applied times-frac 27.9

      \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   6. Applied associate-*l* 27.9

      \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 27.9

      \[\leadsto \frac{2.0}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3.0}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   9. Applied unpow-prod-down 27.9

      \[\leadsto \frac{2.0}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3.0} \cdot {\left(\sqrt[3]{t}\right)}^{3.0}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   10. Applied times-frac 23.1

       \[\leadsto \frac{2.0}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]
   11. Using strategy rm

   12. Applied associate-*l* 23.1

       \[\leadsto \frac{2.0}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell} \cdot \sin k\right)\right)\right)} \cdot \tan k\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\]

   13. Taylor expanded around inf 22.6

       \[\leadsto \color{blue}{0}\]
3. Recombined 6 regimes into one program.

4. Final simplification 16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.5834988826166857 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt[3]{2.0}}{\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0} \cdot \left(\frac{\sqrt[3]{2.0}}{\tan k} \cdot \frac{\sqrt[3]{2.0}}{\frac{\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right)}\right)\\ \mathbf{elif}\;k \le -5.124747410806296 \cdot 10^{-15}:\\ \;\;\;\;\frac{2.0}{\mathsf{fma}\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\left(t \cdot t\right) \cdot t}{\cos k}, 2.0, \frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{elif}\;k \le 1.5429975906335487 \cdot 10^{-37}:\\ \;\;\;\;\frac{2.0}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}}\right) \cdot \tan k\right)\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\\ \mathbf{elif}\;k \le 4.2765351122852666 \cdot 10^{+17}:\\ \;\;\;\;\frac{2.0}{\mathsf{fma}\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{\left(t \cdot t\right) \cdot t}{\cos k}, 2.0, \frac{\frac{t}{\ell}}{\ell} \cdot \frac{\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)}{\cos k}\right)}\\ \mathbf{elif}\;k \le 1.430501198242309 \cdot 10^{+100}:\\ \;\;\;\;\frac{2.0}{\left(\left(\tan k \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\ell}\right)\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3.0}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)\right) \cdot \left(\left(1.0 + {\left(\frac{k}{t}\right)}^{2.0}\right) + 1.0\right)}\\ \mathbf{elif}\;k \le 5.937994838209732 \cdot 10^{+182}:\\ \;\;\;\;\frac{2.0}{-\mathsf{fma}\left(\frac{\left(t \cdot t\right) \cdot t}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\sin k \cdot \sin k}} \cdot 2.0, {\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}, \frac{{\left(\frac{1}{{-1}^{3.0}}\right)}^{1.0}}{\ell \cdot \ell} \cdot \frac{t \cdot \left(\left(k \cdot \sin k\right) \cdot \left(k \cdot \sin k\right)\right)}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))