Toniolo and Linder, Equation (10+)

Average Error: 32.7 → 21.8

Time: 18.9s

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.69064108425965218 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 1.31245642580265899 \cdot 10^{38}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r150094 = 2.0;
        double r150095 = t;
        double r150096 = 3.0;
        double r150097 = pow(r150095, r150096);
        double r150098 = l;
        double r150099 = r150098 * r150098;
        double r150100 = r150097 / r150099;
        double r150101 = k;
        double r150102 = sin(r150101);
        double r150103 = r150100 * r150102;
        double r150104 = tan(r150101);
        double r150105 = r150103 * r150104;
        double r150106 = 1.0;
        double r150107 = r150101 / r150095;
        double r150108 = pow(r150107, r150094);
        double r150109 = r150106 + r150108;
        double r150110 = r150109 + r150106;
        double r150111 = r150105 * r150110;
        double r150112 = r150094 / r150111;
        return r150112;
}

↓

double f(double t, double l, double k) {
        double r150113 = t;
        double r150114 = -2.690641084259652e-62;
        bool r150115 = r150113 <= r150114;
        double r150116 = 2.0;
        double r150117 = cbrt(r150116);
        double r150118 = r150117 * r150117;
        double r150119 = sqrt(r150118);
        double r150120 = cbrt(r150113);
        double r150121 = 3.0;
        double r150122 = pow(r150120, r150121);
        double r150123 = r150119 / r150122;
        double r150124 = sqrt(r150117);
        double r150125 = r150124 / r150122;
        double r150126 = sqrt(r150116);
        double r150127 = k;
        double r150128 = sin(r150127);
        double r150129 = r150122 * r150128;
        double r150130 = r150126 / r150129;
        double r150131 = l;
        double r150132 = r150130 * r150131;
        double r150133 = r150125 * r150132;
        double r150134 = r150123 * r150133;
        double r150135 = tan(r150127);
        double r150136 = r150134 / r150135;
        double r150137 = 2.0;
        double r150138 = 1.0;
        double r150139 = r150127 / r150113;
        double r150140 = pow(r150139, r150116);
        double r150141 = fma(r150137, r150138, r150140);
        double r150142 = r150131 / r150141;
        double r150143 = r150136 * r150142;
        double r150144 = 1.312456425802659e+38;
        bool r150145 = r150113 <= r150144;
        double r150146 = r150120 * r150120;
        double r150147 = pow(r150146, r150121);
        double r150148 = r150126 / r150147;
        double r150149 = cbrt(r150135);
        double r150150 = r150149 * r150149;
        double r150151 = r150148 / r150150;
        double r150152 = r150132 / r150149;
        double r150153 = r150152 * r150142;
        double r150154 = r150151 * r150153;
        double r150155 = r150121 / r150137;
        double r150156 = pow(r150146, r150155);
        double r150157 = r150119 / r150156;
        double r150158 = r150124 / r150156;
        double r150159 = r150158 * r150132;
        double r150160 = r150157 * r150159;
        double r150161 = r150160 / r150135;
        double r150162 = r150161 * r150142;
        double r150163 = r150145 ? r150154 : r150162;
        double r150164 = r150115 ? r150143 : r150163;
        return r150164;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -2.690641084259652e-62

   1. Initial program 22.5

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 22.3

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 22.3

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 21.2

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 17.6

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 16.2

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 16.5

      \[\leadsto \frac{\frac{2}{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied unpow-prod-down 16.5

       \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   11. Applied associate-*l* 14.9

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 14.9

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied times-frac 14.7

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   15. Applied associate-*l* 13.0

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   16. Using strategy rm

   17. Applied unpow-prod-down 13.0

       \[\leadsto \frac{\frac{\sqrt{2}}{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   18. Applied add-cube-cbrt 13.0

       \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   19. Applied sqrt-prod 13.0

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \sqrt{\sqrt[3]{2}}}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   20. Applied times-frac 12.8

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   21. Applied associate-*l* 10.2

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   if -2.690641084259652e-62 < t < 1.312456425802659e+38

   1. Initial program 49.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 51.0

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 51.0

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 50.3

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 49.7

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 48.5

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 48.7

      \[\leadsto \frac{\frac{2}{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied unpow-prod-down 48.7

       \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   11. Applied associate-*l* 48.6

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 48.7

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied times-frac 48.7

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   15. Applied associate-*l* 44.5

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   16. Using strategy rm

   17. Applied add-cube-cbrt 44.6

       \[\leadsto \frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}\right) \cdot \sqrt[3]{\tan k}}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   18. Applied times-frac 44.6

       \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}}\right)} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   19. Applied associate-*l* 41.8

       \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   if 1.312456425802659e+38 < t

   1. Initial program 21.9

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 21.0

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 21.0

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   5. Applied times-frac 20.1

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

   6. Applied associate-*r* 16.2

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell}{1}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]

   7. Simplified 15.7

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \ell}{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 15.9

      \[\leadsto \frac{\frac{2}{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   10. Applied unpow-prod-down 15.9

       \[\leadsto \frac{\frac{2}{\color{blue}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)} \cdot \sin k} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   11. Applied associate-*l* 14.3

       \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)}} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt 14.3

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   14. Applied times-frac 14.1

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}\right)} \cdot \ell}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   15. Applied associate-*l* 11.9

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
   16. Using strategy rm

   17. Applied sqr-pow 11.9

       \[\leadsto \frac{\frac{\sqrt{2}}{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   18. Applied add-cube-cbrt 11.9

       \[\leadsto \frac{\frac{\sqrt{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   19. Applied sqrt-prod 11.9

       \[\leadsto \frac{\frac{\color{blue}{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}} \cdot \sqrt{\sqrt[3]{2}}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   20. Applied times-frac 11.6

       \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

   21. Applied associate-*l* 8.5

       \[\leadsto \frac{\color{blue}{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 21.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.69064108425965218 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 1.31245642580265899 \cdot 10^{38}:\\ \;\;\;\;\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}} \cdot \left(\frac{\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell}{\sqrt[3]{\tan k}} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt[3]{2} \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{\sqrt[3]{2}}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))