Average Error: 32.7 → 16.9
Time: 56.6s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.932296528705668984693986702426810627746 \cdot 10^{146}:\\ \;\;\;\;\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right)\right) \cdot \tan k\right)}\\ \mathbf{elif}\;k \le -2.465489873106864392898978142261571099549 \cdot 10^{69}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;k \le -3.515615117987834914814812597504544001232 \cdot 10^{-226}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \le 496413.6209539859555661678314208984375:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{t}^{\left(\frac{1}{3} \cdot 3\right)}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \le 2.402994973162251042133283519939024529294 \cdot 10^{139}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;k \le 3.506733478446760175749683528136847068451 \cdot 10^{202}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \le 7.387345331331856574424040723290335999974 \cdot 10^{224}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}\\ \end{array}\]
\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 -1.932296528705668984693986702426810627746 \cdot 10^{146}:\\
\;\;\;\;\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right)\right) \cdot \tan k\right)}\\

\mathbf{elif}\;k \le -2.465489873106864392898978142261571099549 \cdot 10^{69}:\\
\;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\

\mathbf{elif}\;k \le -3.515615117987834914814812597504544001232 \cdot 10^{-226}:\\
\;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\

\mathbf{elif}\;k \le 496413.6209539859555661678314208984375:\\
\;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{t}^{\left(\frac{1}{3} \cdot 3\right)}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\

\mathbf{elif}\;k \le 2.402994973162251042133283519939024529294 \cdot 10^{139}:\\
\;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\

\mathbf{elif}\;k \le 3.506733478446760175749683528136847068451 \cdot 10^{202}:\\
\;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\

\mathbf{elif}\;k \le 7.387345331331856574424040723290335999974 \cdot 10^{224}:\\
\;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r124163 = 2.0;
        double r124164 = t;
        double r124165 = 3.0;
        double r124166 = pow(r124164, r124165);
        double r124167 = l;
        double r124168 = r124167 * r124167;
        double r124169 = r124166 / r124168;
        double r124170 = k;
        double r124171 = sin(r124170);
        double r124172 = r124169 * r124171;
        double r124173 = tan(r124170);
        double r124174 = r124172 * r124173;
        double r124175 = 1.0;
        double r124176 = r124170 / r124164;
        double r124177 = pow(r124176, r124163);
        double r124178 = r124175 + r124177;
        double r124179 = r124178 + r124175;
        double r124180 = r124174 * r124179;
        double r124181 = r124163 / r124180;
        return r124181;
}
↓
double f(double t, double l, double k) {
        double r124182 = k;
        double r124183 = -1.932296528705669e+146;
        bool r124184 = r124182 <= r124183;
        double r124185 = 2.0;
        double r124186 = 1.0;
        double r124187 = t;
        double r124188 = r124182 / r124187;
        double r124189 = pow(r124188, r124185);
        double r124190 = r124186 + r124189;
        double r124191 = r124190 + r124186;
        double r124192 = cbrt(r124187);
        double r124193 = 3.0;
        double r124194 = pow(r124192, r124193);
        double r124195 = l;
        double r124196 = cbrt(r124195);
        double r124197 = r124196 * r124196;
        double r124198 = r124194 / r124197;
        double r124199 = r124194 / r124196;
        double r124200 = cbrt(r124192);
        double r124201 = r124200 * r124200;
        double r124202 = pow(r124201, r124193);
        double r124203 = pow(r124200, r124193);
        double r124204 = r124203 / r124195;
        double r124205 = sin(r124182);
        double r124206 = r124204 * r124205;
        double r124207 = r124202 * r124206;
        double r124208 = r124199 * r124207;
        double r124209 = r124198 * r124208;
        double r124210 = tan(r124182);
        double r124211 = r124209 * r124210;
        double r124212 = r124191 * r124211;
        double r124213 = r124185 / r124212;
        double r124214 = -2.4654898731068644e+69;
        bool r124215 = r124182 <= r124214;
        double r124216 = 1.0;
        double r124217 = -1.0;
        double r124218 = pow(r124217, r124193);
        double r124219 = r124216 / r124218;
        double r124220 = pow(r124219, r124186);
        double r124221 = cbrt(r124217);
        double r124222 = 9.0;
        double r124223 = pow(r124221, r124222);
        double r124224 = 3.0;
        double r124225 = pow(r124187, r124224);
        double r124226 = 2.0;
        double r124227 = pow(r124205, r124226);
        double r124228 = r124225 * r124227;
        double r124229 = r124223 * r124228;
        double r124230 = cos(r124182);
        double r124231 = pow(r124195, r124226);
        double r124232 = r124230 * r124231;
        double r124233 = r124229 / r124232;
        double r124234 = r124220 * r124233;
        double r124235 = r124185 * r124234;
        double r124236 = pow(r124182, r124226);
        double r124237 = r124236 * r124187;
        double r124238 = r124227 * r124237;
        double r124239 = r124223 * r124238;
        double r124240 = r124239 / r124232;
        double r124241 = r124220 * r124240;
        double r124242 = r124235 + r124241;
        double r124243 = r124185 / r124242;
        double r124244 = -3.515615117987835e-226;
        bool r124245 = r124182 <= r124244;
        double r124246 = r124194 / r124195;
        double r124247 = r124246 * r124205;
        double r124248 = r124199 * r124247;
        double r124249 = r124248 * r124210;
        double r124250 = r124198 * r124249;
        double r124251 = r124250 * r124191;
        double r124252 = r124185 / r124251;
        double r124253 = 496413.62095398596;
        bool r124254 = r124182 <= r124253;
        double r124255 = 0.3333333333333333;
        double r124256 = r124255 * r124193;
        double r124257 = pow(r124187, r124256);
        double r124258 = r124257 / r124195;
        double r124259 = r124258 * r124205;
        double r124260 = r124199 * r124259;
        double r124261 = r124198 * r124260;
        double r124262 = r124261 * r124210;
        double r124263 = r124262 * r124191;
        double r124264 = r124185 / r124263;
        double r124265 = 2.402994973162251e+139;
        bool r124266 = r124182 <= r124265;
        double r124267 = 3.50673347844676e+202;
        bool r124268 = r124182 <= r124267;
        double r124269 = r124198 * r124199;
        double r124270 = r124247 * r124210;
        double r124271 = r124269 * r124270;
        double r124272 = r124271 * r124191;
        double r124273 = r124185 / r124272;
        double r124274 = 7.3873453313318566e+224;
        bool r124275 = r124182 <= r124274;
        double r124276 = r124198 * r124248;
        double r124277 = r124276 * r124210;
        double r124278 = cbrt(r124191);
        double r124279 = r124278 * r124278;
        double r124280 = r124277 * r124279;
        double r124281 = r124280 * r124278;
        double r124282 = r124185 / r124281;
        double r124283 = r124275 ? r124243 : r124282;
        double r124284 = r124268 ? r124273 : r124283;
        double r124285 = r124266 ? r124243 : r124284;
        double r124286 = r124254 ? r124264 : r124285;
        double r124287 = r124245 ? r124252 : r124286;
        double r124288 = r124215 ? r124243 : r124287;
        double r124289 = r124184 ? r124213 : r124288;
        return r124289;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 6 regimes

  2. if k < -1.932296528705669e+146

    1. Initial program 34.9

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

    3. Applied add-cube-cbrt34.9

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

    4. Applied unpow-prod-down34.9

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

    5. Applied times-frac27.8

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

    6. Applied associate-*l*27.8

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

    8. Applied add-cube-cbrt27.8

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

    9. Applied unpow-prod-down27.8

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

    10. Applied times-frac22.4

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

    12. Applied associate-*l*22.4

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

    14. Applied *-un-lft-identity22.4

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

    15. Applied add-cube-cbrt22.4

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

    16. Applied unpow-prod-down22.4

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

    17. Applied times-frac22.4

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

    18. Applied associate-*l*22.4

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

    if -1.932296528705669e+146 < k < -2.4654898731068644e+69 or 496413.62095398596 < k < 2.402994973162251e+139 or 3.50673347844676e+202 < k < 7.3873453313318566e+224

    1. Initial program 32.7

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

    3. Applied add-cube-cbrt32.8

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

    4. Applied unpow-prod-down32.8

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

    5. Applied times-frac25.2

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

    6. Applied associate-*l*25.2

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

    7. Taylor expanded around -inf 20.4

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}}\]

    if -2.4654898731068644e+69 < k < -3.515615117987835e-226

    1. Initial program 30.4

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

    3. Applied add-cube-cbrt30.6

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

    4. Applied unpow-prod-down30.6

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

    5. Applied times-frac23.0

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

    6. Applied associate-*l*20.3

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

    8. Applied add-cube-cbrt20.3

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

    9. Applied unpow-prod-down20.3

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

    10. Applied times-frac13.4

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

    12. Applied associate-*l*12.8

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

    14. Applied associate-*l*9.9

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

    if -3.515615117987835e-226 < k < 496413.62095398596

    1. Initial program 34.2

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

    3. Applied add-cube-cbrt34.4

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

    4. Applied unpow-prod-down34.4

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

    5. Applied times-frac27.6

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

    6. Applied associate-*l*21.4

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

    8. Applied add-cube-cbrt21.4

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

    9. Applied unpow-prod-down21.4

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

    10. Applied times-frac16.3

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

    12. Applied associate-*l*13.0

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

    14. Applied pow1/339.1

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

    15. Applied pow-pow12.8

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

    if 2.402994973162251e+139 < k < 3.50673347844676e+202

    1. Initial program 30.6

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

    3. Applied add-cube-cbrt30.7

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

    4. Applied unpow-prod-down30.7

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

    5. Applied times-frac24.6

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

    6. Applied associate-*l*24.6

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

    8. Applied add-cube-cbrt24.6

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

    9. Applied unpow-prod-down24.6

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

    10. Applied times-frac19.4

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

    12. Applied associate-*l*19.4

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

    if 7.3873453313318566e+224 < k

    1. Initial program 32.0

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

    3. Applied add-cube-cbrt32.0

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

    4. Applied unpow-prod-down32.0

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

    5. Applied times-frac26.2

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

    6. Applied associate-*l*26.2

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

    8. Applied add-cube-cbrt26.2

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

    9. Applied unpow-prod-down26.2

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

    10. Applied times-frac21.3

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

    12. Applied associate-*l*21.3

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

    14. Applied add-cube-cbrt21.3

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

    15. Applied associate-*r*21.3

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.932296528705668984693986702426810627746 \cdot 10^{146}:\\ \;\;\;\;\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{\sqrt[3]{t}}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right)\right) \cdot \tan k\right)}\\ \mathbf{elif}\;k \le -2.465489873106864392898978142261571099549 \cdot 10^{69}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;k \le -3.515615117987834914814812597504544001232 \cdot 10^{-226}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \le 496413.6209539859555661678314208984375:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{t}^{\left(\frac{1}{3} \cdot 3\right)}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \le 2.402994973162251042133283519939024529294 \cdot 10^{139}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{elif}\;k \le 3.506733478446760175749683528136847068451 \cdot 10^{202}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;k \le 7.387345331331856574424040723290335999974 \cdot 10^{224}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \tan k\right) \cdot \left(\sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right)\right) \cdot \sqrt[3]{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))