Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 15.8

Time: 47.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.5661868334908373 \cdot 10^{164}:\\ \;\;\;\;\frac{2}{\left(\left(\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 \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}\\ \mathbf{elif}\;t \le -3.4658746301197293 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{\left(2 \cdot 3\right)} \cdot \sin k}{\ell} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}\\ \mathbf{elif}\;t \le -5.7602960310594033 \cdot 10^{-234}:\\ \;\;\;\;\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}\;t \le 1.0756150083791892 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \le 3.4417764551753214 \cdot 10^{144}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{\left(2 \cdot 3\right)} \cdot \sin k}{\ell} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{\sqrt{t}}\right)}^{3} \cdot \left({\left(\sqrt[3]{\sqrt{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)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r118572 = 2.0;
        double r118573 = t;
        double r118574 = 3.0;
        double r118575 = pow(r118573, r118574);
        double r118576 = l;
        double r118577 = r118576 * r118576;
        double r118578 = r118575 / r118577;
        double r118579 = k;
        double r118580 = sin(r118579);
        double r118581 = r118578 * r118580;
        double r118582 = tan(r118579);
        double r118583 = r118581 * r118582;
        double r118584 = 1.0;
        double r118585 = r118579 / r118573;
        double r118586 = pow(r118585, r118572);
        double r118587 = r118584 + r118586;
        double r118588 = r118587 + r118584;
        double r118589 = r118583 * r118588;
        double r118590 = r118572 / r118589;
        return r118590;
}

↓

double f(double t, double l, double k) {
        double r118591 = t;
        double r118592 = -1.5661868334908373e+164;
        bool r118593 = r118591 <= r118592;
        double r118594 = 2.0;
        double r118595 = cbrt(r118591);
        double r118596 = 3.0;
        double r118597 = pow(r118595, r118596);
        double r118598 = l;
        double r118599 = r118597 / r118598;
        double r118600 = k;
        double r118601 = sin(r118600);
        double r118602 = r118599 * r118601;
        double r118603 = r118599 * r118602;
        double r118604 = r118597 * r118603;
        double r118605 = tan(r118600);
        double r118606 = r118604 * r118605;
        double r118607 = 1.0;
        double r118608 = r118600 / r118591;
        double r118609 = pow(r118608, r118594);
        double r118610 = r118607 + r118609;
        double r118611 = r118610 + r118607;
        double r118612 = sqrt(r118611);
        double r118613 = r118606 * r118612;
        double r118614 = r118613 * r118612;
        double r118615 = r118594 / r118614;
        double r118616 = -3.4658746301197293e-18;
        bool r118617 = r118591 <= r118616;
        double r118618 = 2.0;
        double r118619 = r118618 * r118596;
        double r118620 = pow(r118595, r118619);
        double r118621 = r118620 * r118601;
        double r118622 = r118621 / r118598;
        double r118623 = r118622 * r118597;
        double r118624 = r118605 * r118611;
        double r118625 = r118623 * r118624;
        double r118626 = r118625 / r118598;
        double r118627 = r118594 / r118626;
        double r118628 = -5.760296031059403e-234;
        bool r118629 = r118591 <= r118628;
        double r118630 = r118604 * r118624;
        double r118631 = r118594 / r118630;
        double r118632 = 1.0756150083791892e-101;
        bool r118633 = r118591 <= r118632;
        double r118634 = pow(r118601, r118618);
        double r118635 = pow(r118600, r118618);
        double r118636 = r118634 * r118635;
        double r118637 = r118591 * r118636;
        double r118638 = cos(r118600);
        double r118639 = pow(r118598, r118618);
        double r118640 = r118638 * r118639;
        double r118641 = r118637 / r118640;
        double r118642 = 3.0;
        double r118643 = pow(r118591, r118642);
        double r118644 = r118643 * r118634;
        double r118645 = r118639 * r118638;
        double r118646 = r118644 / r118645;
        double r118647 = r118594 * r118646;
        double r118648 = r118641 + r118647;
        double r118649 = r118594 / r118648;
        double r118650 = 3.4417764551753214e+144;
        bool r118651 = r118591 <= r118650;
        double r118652 = sqrt(r118591);
        double r118653 = cbrt(r118652);
        double r118654 = pow(r118653, r118596);
        double r118655 = r118654 * r118603;
        double r118656 = r118654 * r118655;
        double r118657 = r118656 * r118605;
        double r118658 = r118657 * r118611;
        double r118659 = r118594 / r118658;
        double r118660 = r118651 ? r118627 : r118659;
        double r118661 = r118633 ? r118649 : r118660;
        double r118662 = r118629 ? r118631 : r118661;
        double r118663 = r118617 ? r118627 : r118662;
        double r118664 = r118593 ? r118615 : r118663;
        return r118664;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 5 regimes

2. if t < -1.5661868334908373e+164

   1. Initial program 21.6

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

   3. Applied add-cube-cbrt 21.6

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

   4. Applied unpow-prod-down 21.6

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

   5. Applied times-frac 17.4

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

   6. Applied associate-*l* 17.4

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

   8. Applied *-un-lft-identity 17.4

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

   9. Applied unpow-prod-down 17.4

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

   10. Applied times-frac 6.5

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

   11. Simplified 6.5

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

   13. Applied associate-*l* 4.8

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

   15. Applied add-sqr-sqrt 4.8

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

   16. Applied associate-*r* 4.8

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

   if -1.5661868334908373e+164 < t < -3.4658746301197293e-18 or 1.0756150083791892e-101 < t < 3.4417764551753214e+144

   1. Initial program 24.2

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

   3. Applied add-cube-cbrt 24.5

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

   4. Applied unpow-prod-down 24.5

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

   5. Applied times-frac 16.4

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

   6. Applied associate-*l* 12.1

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

   8. Applied *-un-lft-identity 12.1

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

   9. Applied unpow-prod-down 12.1

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

   10. Applied times-frac 11.6

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

   11. Simplified 11.6

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

   13. Applied associate-*l* 12.4

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

   15. Applied associate-*l/ 12.5

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

   16. Applied associate-*r/ 12.5

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

   17. Applied associate-*r/ 11.7

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

   18. Applied associate-*l/ 10.4

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

   19. Applied associate-*l/ 10.2

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

   20. Simplified 10.2

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

   if -3.4658746301197293e-18 < t < -5.760296031059403e-234

   1. Initial program 46.6

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

   3. Applied add-cube-cbrt 46.7

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

   4. Applied unpow-prod-down 46.7

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

   5. Applied times-frac 36.0

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

   6. Applied associate-*l* 34.8

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

   8. Applied *-un-lft-identity 34.8

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

   9. Applied unpow-prod-down 34.8

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

   10. Applied times-frac 30.6

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

   11. Simplified 30.6

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

   13. Applied associate-*l* 30.8

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

   15. Applied associate-*l* 29.5

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

   if -5.760296031059403e-234 < t < 1.0756150083791892e-101

   1. Initial program 63.3

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

   2. Taylor expanded around inf 42.3

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

   if 3.4417764551753214e+144 < t

   1. Initial program 21.8

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

   3. Applied add-cube-cbrt 21.8

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

   4. Applied unpow-prod-down 21.8

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

   5. Applied times-frac 16.5

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

   6. Applied associate-*l* 16.2

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

   8. Applied *-un-lft-identity 16.2

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

   9. Applied unpow-prod-down 16.2

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

   10. Applied times-frac 6.4

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

   11. Simplified 6.4

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

   13. Applied associate-*l* 4.8

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

   15. Applied add-sqr-sqrt 4.8

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

   16. Applied cbrt-prod 4.8

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

   17. Applied unpow-prod-down 4.8

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

   18. Applied associate-*l* 4.8

       \[\leadsto \frac{2}{\left(\color{blue}{\left({\left(\sqrt[3]{\sqrt{t}}\right)}^{3} \cdot \left({\left(\sqrt[3]{\sqrt{t}}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.5661868334908373 \cdot 10^{164}:\\ \;\;\;\;\frac{2}{\left(\left(\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 \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}\right) \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}\\ \mathbf{elif}\;t \le -3.4658746301197293 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{\left(2 \cdot 3\right)} \cdot \sin k}{\ell} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}\\ \mathbf{elif}\;t \le -5.7602960310594033 \cdot 10^{-234}:\\ \;\;\;\;\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}\;t \le 1.0756150083791892 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \le 3.4417764551753214 \cdot 10^{144}:\\ \;\;\;\;\frac{2}{\frac{\left(\frac{{\left(\sqrt[3]{t}\right)}^{\left(2 \cdot 3\right)} \cdot \sin k}{\ell} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{\sqrt{t}}\right)}^{3} \cdot \left({\left(\sqrt[3]{\sqrt{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)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))))