Toniolo and Linder, Equation (10-)

Average Error: 47.1 → 8.3

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -4.041273932463307 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{t} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \frac{k}{t}\right)\right)\right)}}{-\sin k}}{\tan k}\\ \mathbf{elif}\;k \le -2.3396485923337978 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}}{-\sin k} \cdot \frac{\left(\ell \cdot \left(\ell \cdot t\right)\right) \cdot t}{\tan k}\\ \mathbf{elif}\;k \le -3.375675962257107 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{-2}{\frac{\frac{\frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{t}}{\ell} \cdot t}{t}}}{-\sin k} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;k \le -1.032730037883888 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-\sin k}\\ \mathbf{elif}\;k \le 9.656245993958174 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \frac{\frac{\frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{t}}{\ell}}{t}}}{-\sin k} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;k \le 1.345868103112779 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\left(\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{k}{t}\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r1829068 = 2.0;
        double r1829069 = t;
        double r1829070 = 3.0;
        double r1829071 = pow(r1829069, r1829070);
        double r1829072 = l;
        double r1829073 = r1829072 * r1829072;
        double r1829074 = r1829071 / r1829073;
        double r1829075 = k;
        double r1829076 = sin(r1829075);
        double r1829077 = r1829074 * r1829076;
        double r1829078 = tan(r1829075);
        double r1829079 = r1829077 * r1829078;
        double r1829080 = 1.0;
        double r1829081 = r1829075 / r1829069;
        double r1829082 = pow(r1829081, r1829068);
        double r1829083 = r1829080 + r1829082;
        double r1829084 = r1829083 - r1829080;
        double r1829085 = r1829079 * r1829084;
        double r1829086 = r1829068 / r1829085;
        return r1829086;
}

↓

double f(double t, double l, double k) {
        double r1829087 = k;
        double r1829088 = -4.041273932463307e+247;
        bool r1829089 = r1829087 <= r1829088;
        double r1829090 = -2.0;
        double r1829091 = t;
        double r1829092 = r1829087 / r1829091;
        double r1829093 = l;
        double r1829094 = r1829091 / r1829093;
        double r1829095 = r1829094 * r1829092;
        double r1829096 = r1829094 * r1829095;
        double r1829097 = r1829091 * r1829096;
        double r1829098 = r1829092 * r1829097;
        double r1829099 = r1829090 / r1829098;
        double r1829100 = sin(r1829087);
        double r1829101 = -r1829100;
        double r1829102 = r1829099 / r1829101;
        double r1829103 = tan(r1829087);
        double r1829104 = r1829102 / r1829103;
        double r1829105 = -2.3396485923337978e+231;
        bool r1829106 = r1829087 <= r1829105;
        double r1829107 = r1829091 * r1829087;
        double r1829108 = r1829091 * r1829107;
        double r1829109 = r1829108 * r1829087;
        double r1829110 = r1829091 * r1829109;
        double r1829111 = r1829090 / r1829110;
        double r1829112 = r1829111 / r1829101;
        double r1829113 = r1829093 * r1829091;
        double r1829114 = r1829093 * r1829113;
        double r1829115 = r1829114 * r1829091;
        double r1829116 = r1829115 / r1829103;
        double r1829117 = r1829112 * r1829116;
        double r1829118 = -3.375675962257107e+164;
        bool r1829119 = r1829087 <= r1829118;
        double r1829120 = r1829107 * r1829107;
        double r1829121 = r1829120 / r1829091;
        double r1829122 = r1829121 / r1829093;
        double r1829123 = r1829122 * r1829091;
        double r1829124 = r1829123 / r1829091;
        double r1829125 = r1829090 / r1829124;
        double r1829126 = r1829125 / r1829101;
        double r1829127 = r1829093 / r1829103;
        double r1829128 = r1829126 * r1829127;
        double r1829129 = -1.032730037883888e-155;
        bool r1829130 = r1829087 <= r1829129;
        double r1829131 = r1829087 * r1829087;
        double r1829132 = r1829131 / r1829093;
        double r1829133 = r1829091 * r1829132;
        double r1829134 = r1829090 / r1829133;
        double r1829135 = r1829134 / r1829101;
        double r1829136 = r1829127 * r1829135;
        double r1829137 = 9.656245993958174e-126;
        bool r1829138 = r1829087 <= r1829137;
        double r1829139 = r1829122 / r1829091;
        double r1829140 = r1829091 * r1829139;
        double r1829141 = r1829090 / r1829140;
        double r1829142 = r1829141 / r1829101;
        double r1829143 = r1829142 * r1829127;
        double r1829144 = 1.345868103112779e+154;
        bool r1829145 = r1829087 <= r1829144;
        double r1829146 = r1829087 / r1829093;
        double r1829147 = r1829094 * r1829146;
        double r1829148 = r1829147 * r1829092;
        double r1829149 = r1829148 * r1829091;
        double r1829150 = r1829090 / r1829149;
        double r1829151 = r1829101 * r1829103;
        double r1829152 = r1829150 / r1829151;
        double r1829153 = r1829145 ? r1829136 : r1829152;
        double r1829154 = r1829138 ? r1829143 : r1829153;
        double r1829155 = r1829130 ? r1829136 : r1829154;
        double r1829156 = r1829119 ? r1829128 : r1829155;
        double r1829157 = r1829106 ? r1829117 : r1829156;
        double r1829158 = r1829089 ? r1829104 : r1829157;
        return r1829158;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 6 regimes

2. if k < -4.041273932463307e+247

   1. Initial program 34.5

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

   2. Simplified 17.7

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

   4. Applied frac-2neg 17.7

      \[\leadsto \color{blue}{\frac{-\frac{2}{\left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}}{-\sin k \cdot \tan k}}\]

   5. Simplified 12.4

      \[\leadsto \frac{\color{blue}{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}}{-\sin k \cdot \tan k}\]
   6. Using strategy rm

   7. Applied distribute-lft-neg-in 12.4

      \[\leadsto \frac{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}{\color{blue}{\left(-\sin k\right) \cdot \tan k}}\]

   8. Applied associate-/r* 12.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}{-\sin k}}{\tan k}}\]

   if -4.041273932463307e+247 < k < -2.3396485923337978e+231

   1. Initial program 39.4

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

   2. Simplified 20.1

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

   4. Applied frac-2neg 20.1

      \[\leadsto \color{blue}{\frac{-\frac{2}{\left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}}{-\sin k \cdot \tan k}}\]

   5. Simplified 16.4

      \[\leadsto \frac{\color{blue}{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}}{-\sin k \cdot \tan k}\]
   6. Using strategy rm

   7. Applied associate-*r* 16.1

      \[\leadsto \frac{\frac{-2}{\color{blue}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}}{-\sin k \cdot \tan k}\]
   8. Using strategy rm

   9. Applied distribute-lft-neg-in 16.1

      \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\color{blue}{\left(-\sin k\right) \cdot \tan k}}\]

   10. Applied frac-times 24.9

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\color{blue}{\frac{t \cdot k}{\ell \cdot t}} \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   11. Applied frac-times 30.1

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \color{blue}{\frac{\left(t \cdot k\right) \cdot t}{\left(\ell \cdot t\right) \cdot \ell}}\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   12. Applied frac-times 27.6

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{k \cdot \left(\left(t \cdot k\right) \cdot t\right)}{t \cdot \left(\left(\ell \cdot t\right) \cdot \ell\right)}} \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   13. Applied associate-*l/ 26.8

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\left(k \cdot \left(\left(t \cdot k\right) \cdot t\right)\right) \cdot t}{t \cdot \left(\left(\ell \cdot t\right) \cdot \ell\right)}}}}{\left(-\sin k\right) \cdot \tan k}\]

   14. Applied associate-/r/ 26.9

       \[\leadsto \frac{\color{blue}{\frac{-2}{\left(k \cdot \left(\left(t \cdot k\right) \cdot t\right)\right) \cdot t} \cdot \left(t \cdot \left(\left(\ell \cdot t\right) \cdot \ell\right)\right)}}{\left(-\sin k\right) \cdot \tan k}\]

   15. Applied times-frac 26.9

       \[\leadsto \color{blue}{\frac{\frac{-2}{\left(k \cdot \left(\left(t \cdot k\right) \cdot t\right)\right) \cdot t}}{-\sin k} \cdot \frac{t \cdot \left(\left(\ell \cdot t\right) \cdot \ell\right)}{\tan k}}\]

   if -2.3396485923337978e+231 < k < -3.375675962257107e+164

   1. Initial program 42.2

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

   2. Simplified 16.9

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

   4. Applied frac-2neg 16.9

      \[\leadsto \color{blue}{\frac{-\frac{2}{\left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}}{-\sin k \cdot \tan k}}\]

   5. Simplified 13.1

      \[\leadsto \frac{\color{blue}{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}}{-\sin k \cdot \tan k}\]
   6. Using strategy rm

   7. Applied associate-*r* 12.4

      \[\leadsto \frac{\frac{-2}{\color{blue}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}}{-\sin k \cdot \tan k}\]
   8. Using strategy rm

   9. Applied distribute-lft-neg-in 12.4

      \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\color{blue}{\left(-\sin k\right) \cdot \tan k}}\]

   10. Applied associate-*l/ 12.4

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\color{blue}{\frac{t \cdot \frac{k}{t}}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   11. Applied associate-*l/ 12.4

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \color{blue}{\frac{\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}}{\ell}}\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   12. Applied associate-*r/ 12.6

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)}{\ell}} \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   13. Applied associate-*l/ 18.5

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}{\ell}}}}{\left(-\sin k\right) \cdot \tan k}\]

   14. Applied associate-/r/ 18.5

       \[\leadsto \frac{\color{blue}{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t} \cdot \ell}}{\left(-\sin k\right) \cdot \tan k}\]

   15. Applied times-frac 18.5

       \[\leadsto \color{blue}{\frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}}\]
   16. Using strategy rm

   17. Applied associate-*r/ 20.2

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\color{blue}{\frac{t \cdot k}{t}} \cdot \frac{t}{\ell}\right)\right) \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   18. Applied associate-*l/ 20.2

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \color{blue}{\frac{\left(t \cdot k\right) \cdot \frac{t}{\ell}}{t}}\right) \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   19. Applied associate-*r/ 20.2

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\frac{k}{t} \cdot \left(\left(t \cdot k\right) \cdot \frac{t}{\ell}\right)}{t}} \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   20. Applied associate-*l/ 20.4

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\left(\frac{k}{t} \cdot \left(\left(t \cdot k\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}{t}}}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   21. Simplified 14.0

       \[\leadsto \frac{\frac{-2}{\frac{\color{blue}{\frac{\frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{t}}{\ell} \cdot t}}{t}}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   if -3.375675962257107e+164 < k < -1.032730037883888e-155 or 9.656245993958174e-126 < k < 1.345868103112779e+154

   1. Initial program 51.9

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

   2. Simplified 21.6

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

   4. Applied frac-2neg 21.6

      \[\leadsto \color{blue}{\frac{-\frac{2}{\left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}}{-\sin k \cdot \tan k}}\]

   5. Simplified 16.2

      \[\leadsto \frac{\color{blue}{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}}{-\sin k \cdot \tan k}\]
   6. Using strategy rm

   7. Applied associate-*r* 16.4

      \[\leadsto \frac{\frac{-2}{\color{blue}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}}{-\sin k \cdot \tan k}\]
   8. Using strategy rm

   9. Applied distribute-lft-neg-in 16.4

      \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\color{blue}{\left(-\sin k\right) \cdot \tan k}}\]

   10. Applied associate-*l/ 16.4

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\color{blue}{\frac{t \cdot \frac{k}{t}}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   11. Applied associate-*l/ 16.4

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \color{blue}{\frac{\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}}{\ell}}\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   12. Applied associate-*r/ 14.8

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)}{\ell}} \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   13. Applied associate-*l/ 10.0

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}{\ell}}}}{\left(-\sin k\right) \cdot \tan k}\]

   14. Applied associate-/r/ 9.8

       \[\leadsto \frac{\color{blue}{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t} \cdot \ell}}{\left(-\sin k\right) \cdot \tan k}\]

   15. Applied times-frac 9.6

       \[\leadsto \color{blue}{\frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}}\]

   16. Taylor expanded around 0 3.4

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{{k}^{2}}{\ell}} \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   17. Simplified 3.4

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{k \cdot k}{\ell}} \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   if -1.032730037883888e-155 < k < 9.656245993958174e-126

   1. Initial program 62.1

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

   2. Simplified 53.5

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

   4. Applied frac-2neg 53.5

      \[\leadsto \color{blue}{\frac{-\frac{2}{\left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}}{-\sin k \cdot \tan k}}\]

   5. Simplified 54.7

      \[\leadsto \frac{\color{blue}{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}}{-\sin k \cdot \tan k}\]
   6. Using strategy rm

   7. Applied associate-*r* 53.9

      \[\leadsto \frac{\frac{-2}{\color{blue}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}}{-\sin k \cdot \tan k}\]
   8. Using strategy rm

   9. Applied distribute-lft-neg-in 53.9

      \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\color{blue}{\left(-\sin k\right) \cdot \tan k}}\]

   10. Applied associate-*l/ 53.9

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\color{blue}{\frac{t \cdot \frac{k}{t}}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   11. Applied associate-*l/ 53.9

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \color{blue}{\frac{\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}}{\ell}}\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   12. Applied associate-*r/ 52.3

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)}{\ell}} \cdot t}}{\left(-\sin k\right) \cdot \tan k}\]

   13. Applied associate-*l/ 51.2

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}{\ell}}}}{\left(-\sin k\right) \cdot \tan k}\]

   14. Applied associate-/r/ 51.2

       \[\leadsto \frac{\color{blue}{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t} \cdot \ell}}{\left(-\sin k\right) \cdot \tan k}\]

   15. Applied times-frac 50.0

       \[\leadsto \color{blue}{\frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\left(t \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}}\]
   16. Using strategy rm

   17. Applied associate-*r/ 50.0

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\color{blue}{\frac{t \cdot k}{t}} \cdot \frac{t}{\ell}\right)\right) \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   18. Applied associate-*l/ 50.2

       \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \color{blue}{\frac{\left(t \cdot k\right) \cdot \frac{t}{\ell}}{t}}\right) \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   19. Applied associate-*r/ 50.2

       \[\leadsto \frac{\frac{-2}{\color{blue}{\frac{\frac{k}{t} \cdot \left(\left(t \cdot k\right) \cdot \frac{t}{\ell}\right)}{t}} \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   20. Simplified 15.5

       \[\leadsto \frac{\frac{-2}{\frac{\color{blue}{\frac{\frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{t}}{\ell}}}{t} \cdot t}}{-\sin k} \cdot \frac{\ell}{\tan k}\]

   if 1.345868103112779e+154 < k

   1. Initial program 38.4

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

   2. Simplified 18.3

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

   4. Applied frac-2neg 18.3

      \[\leadsto \color{blue}{\frac{-\frac{2}{\left(\frac{k}{t} \cdot \left(\frac{k}{t} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\frac{t}{\ell} \cdot t\right)}}{-\sin k \cdot \tan k}}\]

   5. Simplified 13.8

      \[\leadsto \frac{\color{blue}{\frac{-2}{\frac{k}{t} \cdot \left(\left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right) \cdot t\right)}}}{-\sin k \cdot \tan k}\]
   6. Using strategy rm

   7. Applied associate-*r* 13.5

      \[\leadsto \frac{\frac{-2}{\color{blue}{\left(\frac{k}{t} \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{t}\right) \cdot \frac{t}{\ell}\right)\right) \cdot t}}}{-\sin k \cdot \tan k}\]

   8. Taylor expanded around 0 13.5

      \[\leadsto \frac{\frac{-2}{\left(\frac{k}{t} \cdot \left(\color{blue}{\frac{k}{\ell}} \cdot \frac{t}{\ell}\right)\right) \cdot t}}{-\sin k \cdot \tan k}\]
3. Recombined 6 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.041273932463307 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{\frac{-2}{\frac{k}{t} \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \frac{k}{t}\right)\right)\right)}}{-\sin k}}{\tan k}\\ \mathbf{elif}\;k \le -2.3396485923337978 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \left(\left(t \cdot \left(t \cdot k\right)\right) \cdot k\right)}}{-\sin k} \cdot \frac{\left(\ell \cdot \left(\ell \cdot t\right)\right) \cdot t}{\tan k}\\ \mathbf{elif}\;k \le -3.375675962257107 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{-2}{\frac{\frac{\frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{t}}{\ell} \cdot t}{t}}}{-\sin k} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;k \le -1.032730037883888 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-\sin k}\\ \mathbf{elif}\;k \le 9.656245993958174 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{-2}{t \cdot \frac{\frac{\frac{\left(t \cdot k\right) \cdot \left(t \cdot k\right)}{t}}{\ell}}{t}}}{-\sin k} \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;k \le 1.345868103112779 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{-2}{t \cdot \frac{k \cdot k}{\ell}}}{-\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{\left(\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{k}{t}\right) \cdot t}}{\left(-\sin k\right) \cdot \tan k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))