Toniolo and Linder, Equation (10-)

Average Error: 48.8 → 11.0

Time: 2.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.872166269477434311777223808277429991508 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{\sqrt[3]{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}}{\frac{\frac{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}{\cos k}}{\sqrt[3]{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \cdot \ell \le 6.887437092972987609123285536923391190439 \cdot 10^{214}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\cos k}} \cdot \left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{1}{\sin k} \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 6.368954335372231118820204523405185907613 \cdot 10^{248}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\frac{\ell}{\sin k}}} \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{\tan k}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r8349406 = 2.0;
        double r8349407 = t;
        double r8349408 = 3.0;
        double r8349409 = pow(r8349407, r8349408);
        double r8349410 = l;
        double r8349411 = r8349410 * r8349410;
        double r8349412 = r8349409 / r8349411;
        double r8349413 = k;
        double r8349414 = sin(r8349413);
        double r8349415 = r8349412 * r8349414;
        double r8349416 = tan(r8349413);
        double r8349417 = r8349415 * r8349416;
        double r8349418 = 1.0;
        double r8349419 = r8349413 / r8349407;
        double r8349420 = pow(r8349419, r8349406);
        double r8349421 = r8349418 + r8349420;
        double r8349422 = r8349421 - r8349418;
        double r8349423 = r8349417 * r8349422;
        double r8349424 = r8349406 / r8349423;
        return r8349424;
}

↓

double f(double t, double l, double k) {
        double r8349425 = l;
        double r8349426 = r8349425 * r8349425;
        double r8349427 = 3.872166269477434e-306;
        bool r8349428 = r8349426 <= r8349427;
        double r8349429 = 1.0;
        double r8349430 = k;
        double r8349431 = 2.0;
        double r8349432 = 2.0;
        double r8349433 = r8349431 / r8349432;
        double r8349434 = pow(r8349430, r8349433);
        double r8349435 = r8349429 / r8349434;
        double r8349436 = 1.0;
        double r8349437 = pow(r8349435, r8349436);
        double r8349438 = cbrt(r8349437);
        double r8349439 = r8349438 * r8349438;
        double r8349440 = sin(r8349430);
        double r8349441 = r8349440 / r8349425;
        double r8349442 = r8349441 * r8349441;
        double r8349443 = cos(r8349430);
        double r8349444 = r8349442 / r8349443;
        double r8349445 = r8349444 / r8349438;
        double r8349446 = r8349439 / r8349445;
        double r8349447 = t;
        double r8349448 = pow(r8349447, r8349436);
        double r8349449 = r8349434 * r8349448;
        double r8349450 = r8349429 / r8349449;
        double r8349451 = pow(r8349450, r8349436);
        double r8349452 = r8349446 * r8349451;
        double r8349453 = r8349452 * r8349431;
        double r8349454 = 6.887437092972988e+214;
        bool r8349455 = r8349426 <= r8349454;
        double r8349456 = r8349440 / r8349443;
        double r8349457 = r8349437 / r8349456;
        double r8349458 = r8349429 / r8349448;
        double r8349459 = r8349458 / r8349434;
        double r8349460 = pow(r8349459, r8349436);
        double r8349461 = r8349429 / r8349440;
        double r8349462 = r8349461 * r8349426;
        double r8349463 = r8349460 * r8349462;
        double r8349464 = r8349457 * r8349463;
        double r8349465 = r8349431 * r8349464;
        double r8349466 = 6.368954335372231e+248;
        bool r8349467 = r8349426 <= r8349466;
        double r8349468 = r8349460 * r8349437;
        double r8349469 = r8349468 / r8349444;
        double r8349470 = r8349431 * r8349469;
        double r8349471 = r8349430 / r8349447;
        double r8349472 = pow(r8349471, r8349433);
        double r8349473 = r8349429 / r8349472;
        double r8349474 = cbrt(r8349447);
        double r8349475 = r8349474 * r8349474;
        double r8349476 = 3.0;
        double r8349477 = pow(r8349475, r8349476);
        double r8349478 = r8349425 / r8349440;
        double r8349479 = r8349477 / r8349478;
        double r8349480 = r8349473 / r8349479;
        double r8349481 = r8349431 / r8349472;
        double r8349482 = pow(r8349474, r8349476);
        double r8349483 = tan(r8349430);
        double r8349484 = r8349425 / r8349483;
        double r8349485 = r8349482 / r8349484;
        double r8349486 = r8349481 / r8349485;
        double r8349487 = r8349480 * r8349486;
        double r8349488 = r8349467 ? r8349470 : r8349487;
        double r8349489 = r8349455 ? r8349465 : r8349488;
        double r8349490 = r8349428 ? r8349453 : r8349489;
        return r8349490;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if (* l l) < 3.872166269477434e-306

   1. Initial program 46.7

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

   2. Simplified 38.0

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

   3. Taylor expanded around inf 20.1

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 20.1

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

   6. Applied associate-*r* 20.1

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

   8. Applied *-un-lft-identity 20.1

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

   9. Applied times-frac 20.1

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

   10. Applied unpow-prod-down 20.1

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

   11. Applied associate-*l* 20.1

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

   12. Simplified 20.1

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

   14. Applied add-cube-cbrt 20.1

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

   15. Applied associate-/l* 20.1

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

   16. Simplified 8.1

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

   if 3.872166269477434e-306 < (* l l) < 6.887437092972988e+214

   1. Initial program 45.3

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

   2. Simplified 35.4

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

   3. Taylor expanded around inf 8.9

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 8.9

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

   6. Applied associate-*r* 4.9

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

   8. Applied *-un-lft-identity 4.9

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

   9. Applied times-frac 4.6

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

   10. Applied unpow-prod-down 4.6

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

   11. Applied associate-*l* 3.6

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

   12. Simplified 3.6

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

   14. Applied times-frac 3.1

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

   15. Applied *-un-lft-identity 3.1

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

   16. Applied unpow-prod-down 3.1

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

   17. Applied *-un-lft-identity 3.1

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

   18. Applied times-frac 3.1

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

   19. Applied unpow-prod-down 3.1

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

   20. Applied times-frac 3.7

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

   21. Applied associate-*r* 2.5

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

   22. Simplified 2.4

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

   if 6.887437092972988e+214 < (* l l) < 6.368954335372231e+248

   1. Initial program 49.5

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

   2. Simplified 42.5

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

   3. Taylor expanded around inf 29.7

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {k}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 29.7

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

   6. Applied associate-*r* 22.4

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

   8. Applied *-un-lft-identity 22.4

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

   9. Applied times-frac 21.1

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

   10. Applied unpow-prod-down 21.1

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

   11. Applied associate-*l* 9.1

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

   12. Simplified 9.1

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

   14. Applied *-un-lft-identity 9.1

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

   15. Applied associate-*l* 9.1

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

   16. Simplified 20.7

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

   if 6.368954335372231e+248 < (* l l)

   1. Initial program 60.9

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

   2. Simplified 60.1

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

   4. Applied times-frac 60.1

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

   5. Applied add-cube-cbrt 60.1

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

   6. Applied unpow-prod-down 60.1

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

   7. Applied times-frac 46.3

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

   8. Applied sqr-pow 46.3

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

   9. Applied *-un-lft-identity 46.3

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

   10. Applied times-frac 46.1

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

   11. Applied times-frac 35.4

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 3.872166269477434311777223808277429991508 \cdot 10^{-306}:\\ \;\;\;\;\left(\frac{\sqrt[3]{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}}{\frac{\frac{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}{\cos k}}{\sqrt[3]{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \cdot \ell \le 6.887437092972987609123285536923391190439 \cdot 10^{214}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\cos k}} \cdot \left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{1}{\sin k} \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 6.368954335372231118820204523405185907613 \cdot 10^{248}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\frac{\ell}{\sin k}}} \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{\tan k}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))