Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 16.6

Time: 54.3s

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}\;\ell \le -3.47268414959450911810120694131919832031 \cdot 10^{93}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\left(\frac{{\left(\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)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \le -1.964840897459657610063482074102697351606 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(2 \cdot \frac{\frac{\left(\left(\sin k \cdot \sin k\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k} + \frac{\sin k \cdot \sin k}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(k \cdot k\right)}}\right)}\\ \mathbf{elif}\;\ell \le 1.669027633961708515782368133914968232382 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{elif}\;\ell \le 1.603933152635543696136198375797401755732 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(2 \cdot \left(\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right) + \frac{\left(k \cdot k\right) \cdot \left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{elif}\;\ell \le 1706592248229888262687313362944:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\left(\frac{{\left(\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)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \le 5.881209467023334016168620526048555243238 \cdot 10^{150}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(2 \cdot \frac{\frac{\left(\left(\sin k \cdot \sin k\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k} + \frac{\sin k \cdot \sin k}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(k \cdot k\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r6135302 = 2.0;
        double r6135303 = t;
        double r6135304 = 3.0;
        double r6135305 = pow(r6135303, r6135304);
        double r6135306 = l;
        double r6135307 = r6135306 * r6135306;
        double r6135308 = r6135305 / r6135307;
        double r6135309 = k;
        double r6135310 = sin(r6135309);
        double r6135311 = r6135308 * r6135310;
        double r6135312 = tan(r6135309);
        double r6135313 = r6135311 * r6135312;
        double r6135314 = 1.0;
        double r6135315 = r6135309 / r6135303;
        double r6135316 = pow(r6135315, r6135302);
        double r6135317 = r6135314 + r6135316;
        double r6135318 = r6135317 + r6135314;
        double r6135319 = r6135313 * r6135318;
        double r6135320 = r6135302 / r6135319;
        return r6135320;
}

↓

double f(double t, double l, double k) {
        double r6135321 = l;
        double r6135322 = -3.472684149594509e+93;
        bool r6135323 = r6135321 <= r6135322;
        double r6135324 = 2.0;
        double r6135325 = t;
        double r6135326 = cbrt(r6135325);
        double r6135327 = 3.0;
        double r6135328 = pow(r6135326, r6135327);
        double r6135329 = r6135328 / r6135321;
        double r6135330 = k;
        double r6135331 = sin(r6135330);
        double r6135332 = r6135329 * r6135331;
        double r6135333 = r6135329 * r6135332;
        double r6135334 = tan(r6135330);
        double r6135335 = r6135333 * r6135334;
        double r6135336 = r6135328 * r6135335;
        double r6135337 = 1.0;
        double r6135338 = r6135330 / r6135325;
        double r6135339 = pow(r6135338, r6135324);
        double r6135340 = r6135337 + r6135339;
        double r6135341 = r6135340 + r6135337;
        double r6135342 = r6135336 * r6135341;
        double r6135343 = r6135324 / r6135342;
        double r6135344 = -1.9648408974596576e-109;
        bool r6135345 = r6135321 <= r6135344;
        double r6135346 = 1.0;
        double r6135347 = -1.0;
        double r6135348 = pow(r6135347, r6135327);
        double r6135349 = r6135346 / r6135348;
        double r6135350 = pow(r6135349, r6135337);
        double r6135351 = r6135331 * r6135331;
        double r6135352 = cbrt(r6135347);
        double r6135353 = 9.0;
        double r6135354 = pow(r6135352, r6135353);
        double r6135355 = r6135351 * r6135354;
        double r6135356 = r6135325 * r6135325;
        double r6135357 = r6135356 * r6135325;
        double r6135358 = r6135355 * r6135357;
        double r6135359 = r6135321 * r6135321;
        double r6135360 = r6135358 / r6135359;
        double r6135361 = cos(r6135330);
        double r6135362 = r6135360 / r6135361;
        double r6135363 = r6135324 * r6135362;
        double r6135364 = r6135359 * r6135361;
        double r6135365 = r6135325 * r6135354;
        double r6135366 = r6135330 * r6135330;
        double r6135367 = r6135365 * r6135366;
        double r6135368 = r6135364 / r6135367;
        double r6135369 = r6135351 / r6135368;
        double r6135370 = r6135363 + r6135369;
        double r6135371 = r6135350 * r6135370;
        double r6135372 = r6135324 / r6135371;
        double r6135373 = 1.6690276339617085e-161;
        bool r6135374 = r6135321 <= r6135373;
        double r6135375 = r6135328 * r6135333;
        double r6135376 = r6135334 * r6135341;
        double r6135377 = r6135375 * r6135376;
        double r6135378 = r6135324 / r6135377;
        double r6135379 = 1.6039331526355437e-11;
        bool r6135380 = r6135321 <= r6135379;
        double r6135381 = r6135357 * r6135354;
        double r6135382 = r6135381 / r6135361;
        double r6135383 = r6135331 / r6135321;
        double r6135384 = r6135383 * r6135383;
        double r6135385 = r6135382 * r6135384;
        double r6135386 = r6135324 * r6135385;
        double r6135387 = r6135366 * r6135365;
        double r6135388 = r6135387 / r6135361;
        double r6135389 = r6135388 * r6135384;
        double r6135390 = r6135386 + r6135389;
        double r6135391 = r6135350 * r6135390;
        double r6135392 = r6135324 / r6135391;
        double r6135393 = 1.7065922482298883e+30;
        bool r6135394 = r6135321 <= r6135393;
        double r6135395 = 5.881209467023334e+150;
        bool r6135396 = r6135321 <= r6135395;
        double r6135397 = r6135396 ? r6135372 : r6135378;
        double r6135398 = r6135394 ? r6135343 : r6135397;
        double r6135399 = r6135380 ? r6135392 : r6135398;
        double r6135400 = r6135374 ? r6135378 : r6135399;
        double r6135401 = r6135345 ? r6135372 : r6135400;
        double r6135402 = r6135323 ? r6135343 : r6135401;
        return r6135402;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if l < -3.472684149594509e+93 or 1.6039331526355437e-11 < l < 1.7065922482298883e+30

   1. Initial program 48.9

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

   3. Applied add-cube-cbrt 49.0

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

   4. Applied unpow-prod-down 49.0

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

   5. Applied times-frac 37.4

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

   6. Applied associate-*l* 36.8

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

   8. Applied *-un-lft-identity 36.8

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

   9. Applied unpow-prod-down 36.8

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

   10. Applied times-frac 26.3

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

   11. Simplified 26.3

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

   13. Applied associate-*l* 25.8

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

   15. Applied associate-*l* 27.9

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

   if -3.472684149594509e+93 < l < -1.9648408974596576e-109 or 1.7065922482298883e+30 < l < 5.881209467023334e+150

   1. Initial program 29.0

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

   3. Applied add-cube-cbrt 29.3

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

   4. Applied unpow-prod-down 29.3

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

   5. Applied times-frac 26.2

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

   6. Applied associate-*l* 24.2

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

   8. Applied *-un-lft-identity 24.2

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

   9. Applied unpow-prod-down 24.2

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

   10. Applied times-frac 22.5

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

   11. Simplified 22.5

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

   12. Taylor expanded around -inf 21.0

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

   13. Simplified 20.7

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

   if -1.9648408974596576e-109 < l < 1.6690276339617085e-161 or 5.881209467023334e+150 < l

   1. Initial program 32.6

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

   3. Applied add-cube-cbrt 32.6

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

   4. Applied unpow-prod-down 32.6

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

   5. Applied times-frac 23.7

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

   6. Applied associate-*l* 21.6

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

   8. Applied *-un-lft-identity 21.6

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

   9. Applied unpow-prod-down 21.6

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

   10. Applied times-frac 13.2

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

   11. Simplified 13.2

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

   13. Applied associate-*l* 13.3

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

   15. Applied associate-*l* 12.9

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

   if 1.6690276339617085e-161 < l < 1.6039331526355437e-11

   1. Initial program 23.9

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

   3. Applied add-cube-cbrt 24.1

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

   4. Applied unpow-prod-down 24.1

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

   5. Applied times-frac 21.3

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

   6. Applied associate-*l* 18.4

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

   8. Applied *-un-lft-identity 18.4

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

   9. Applied unpow-prod-down 18.4

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

   10. Applied times-frac 17.9

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

   11. Simplified 17.9

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

   13. Applied associate-*l* 17.5

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

   14. Taylor expanded around -inf 14.5

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

   15. Simplified 7.6

       \[\leadsto \frac{2}{\color{blue}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(2 \cdot \left(\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right) + \frac{\left(k \cdot k\right) \cdot \left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 16.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.47268414959450911810120694131919832031 \cdot 10^{93}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\left(\frac{{\left(\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)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \le -1.964840897459657610063482074102697351606 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(2 \cdot \frac{\frac{\left(\left(\sin k \cdot \sin k\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k} + \frac{\sin k \cdot \sin k}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(k \cdot k\right)}}\right)}\\ \mathbf{elif}\;\ell \le 1.669027633961708515782368133914968232382 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{elif}\;\ell \le 1.603933152635543696136198375797401755732 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(2 \cdot \left(\frac{\left(\left(t \cdot t\right) \cdot t\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right) + \frac{\left(k \cdot k\right) \cdot \left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right)}{\cos k} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right)\right)}\\ \mathbf{elif}\;\ell \le 1706592248229888262687313362944:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\left(\frac{{\left(\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)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \le 5.881209467023334016168620526048555243238 \cdot 10^{150}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \left(2 \cdot \frac{\frac{\left(\left(\sin k \cdot \sin k\right) \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(\left(t \cdot t\right) \cdot t\right)}{\ell \cdot \ell}}{\cos k} + \frac{\sin k \cdot \sin k}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(t \cdot {\left(\sqrt[3]{-1}\right)}^{9}\right) \cdot \left(k \cdot k\right)}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))