Toniolo and Linder, Equation (10+)

Average Error: 31.9 → 18.7

Time: 35.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.484142728377253 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{\left(\left(\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}} \cdot \left(\frac{2}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\right)\\ \mathbf{elif}\;t \le 8.274207931760685 \cdot 10^{-176}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\log \left(e^{\sqrt[3]{\frac{\ell}{\sin k}}}\right)}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}} \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \ell\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r135543 = 2.0;
        double r135544 = t;
        double r135545 = 3.0;
        double r135546 = pow(r135544, r135545);
        double r135547 = l;
        double r135548 = r135547 * r135547;
        double r135549 = r135546 / r135548;
        double r135550 = k;
        double r135551 = sin(r135550);
        double r135552 = r135549 * r135551;
        double r135553 = tan(r135550);
        double r135554 = r135552 * r135553;
        double r135555 = 1.0;
        double r135556 = r135550 / r135544;
        double r135557 = pow(r135556, r135543);
        double r135558 = r135555 + r135557;
        double r135559 = r135558 + r135555;
        double r135560 = r135554 * r135559;
        double r135561 = r135543 / r135560;
        return r135561;
}

↓

double f(double t, double l, double k) {
        double r135562 = t;
        double r135563 = -2.484142728377253e-91;
        bool r135564 = r135562 <= r135563;
        double r135565 = 1.0;
        double r135566 = k;
        double r135567 = tan(r135566);
        double r135568 = cbrt(r135562);
        double r135569 = 3.0;
        double r135570 = pow(r135568, r135569);
        double r135571 = l;
        double r135572 = sin(r135566);
        double r135573 = r135571 / r135572;
        double r135574 = cbrt(r135573);
        double r135575 = r135570 / r135574;
        double r135576 = r135567 * r135575;
        double r135577 = r135576 * r135575;
        double r135578 = r135577 * r135575;
        double r135579 = r135565 / r135578;
        double r135580 = 2.0;
        double r135581 = 2.0;
        double r135582 = 1.0;
        double r135583 = r135566 / r135562;
        double r135584 = pow(r135583, r135580);
        double r135585 = fma(r135581, r135582, r135584);
        double r135586 = r135580 / r135585;
        double r135587 = r135586 * r135571;
        double r135588 = r135579 * r135587;
        double r135589 = 8.274207931760685e-176;
        bool r135590 = r135562 <= r135589;
        double r135591 = exp(r135574);
        double r135592 = log(r135591);
        double r135593 = r135570 / r135592;
        double r135594 = r135576 * r135593;
        double r135595 = r135594 * r135575;
        double r135596 = r135595 * r135585;
        double r135597 = r135580 / r135596;
        double r135598 = r135597 * r135571;
        double r135599 = r135575 * r135585;
        double r135600 = r135577 * r135599;
        double r135601 = r135580 / r135600;
        double r135602 = r135601 * r135571;
        double r135603 = r135590 ? r135598 : r135602;
        double r135604 = r135564 ? r135588 : r135603;
        return r135604;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -2.484142728377253e-91

   1. Initial program 22.6

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

   2. Simplified 16.3

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

   4. Applied associate-/l* 16.1

      \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell}{\sin k}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 16.3

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

   7. Applied add-cube-cbrt 16.5

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

   8. Applied unpow-prod-down 16.5

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

   9. Applied times-frac 13.8

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

   10. Applied associate-*r* 13.6

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

   12. Applied unpow-prod-down 13.6

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

   13. Applied times-frac 11.7

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

   14. Applied associate-*r* 11.5

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

   16. Applied *-un-lft-identity 11.5

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

   17. Applied times-frac 11.5

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

   18. Applied associate-*l* 11.3

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

   if -2.484142728377253e-91 < t < 8.274207931760685e-176

   1. Initial program 62.9

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

   2. Simplified 62.8

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

   4. Applied associate-/l* 62.6

      \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell}{\sin k}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 62.6

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

   7. Applied add-cube-cbrt 62.6

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

   8. Applied unpow-prod-down 62.6

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

   9. Applied times-frac 58.5

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

   10. Applied associate-*r* 58.5

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

   12. Applied unpow-prod-down 58.5

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

   13. Applied times-frac 56.5

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

   14. Applied associate-*r* 56.5

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

   16. Applied add-log-exp 43.5

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

   if 8.274207931760685e-176 < t

   1. Initial program 27.2

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

   2. Simplified 22.3

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

   4. Applied associate-/l* 22.0

      \[\leadsto \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell}{\sin k}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 22.1

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

   7. Applied add-cube-cbrt 22.3

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

   8. Applied unpow-prod-down 22.3

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

   9. Applied times-frac 18.6

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

   10. Applied associate-*r* 18.5

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

   12. Applied unpow-prod-down 18.5

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

   13. Applied times-frac 16.6

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

   14. Applied associate-*r* 16.4

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

   16. Applied associate-*l* 15.0

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

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.484142728377253 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{\left(\left(\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}} \cdot \left(\frac{2}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\right)\\ \mathbf{elif}\;t \le 8.274207931760685 \cdot 10^{-176}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\log \left(e^{\sqrt[3]{\frac{\ell}{\sin k}}}\right)}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\frac{\ell}{\sin k}}} \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \ell\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))