Toniolo and Linder, Equation (10+)

Average Error: 32.1 → 11.4

Time: 3.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -4.766677063238053 \cdot 10^{+76}:\\ \;\;\;\;\left(\sqrt[3]{2} \cdot \frac{\cos k}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}\right) \cdot \left(\frac{\frac{\sqrt[3]{2}}{\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)} \cdot \frac{\sin k \cdot t}{\ell}}\right)\\ \mathbf{elif}\;\ell \le 5.68850156613633 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{\frac{\frac{t}{\frac{\ell}{\sin k}}}{\sqrt[3]{\cos k}}} \cdot \left(\left(\left(\sqrt[3]{\cos k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t}\right) \cdot \left(\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t} \cdot \sqrt[3]{\cos k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{2} \cdot \frac{\cos k}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}\right) \cdot \left(\frac{\frac{\sqrt[3]{2}}{\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)} \cdot \frac{\sin k \cdot t}{\ell}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r26882363 = 2.0;
        double r26882364 = t;
        double r26882365 = 3.0;
        double r26882366 = pow(r26882364, r26882365);
        double r26882367 = l;
        double r26882368 = r26882367 * r26882367;
        double r26882369 = r26882366 / r26882368;
        double r26882370 = k;
        double r26882371 = sin(r26882370);
        double r26882372 = r26882369 * r26882371;
        double r26882373 = tan(r26882370);
        double r26882374 = r26882372 * r26882373;
        double r26882375 = 1.0;
        double r26882376 = r26882370 / r26882364;
        double r26882377 = pow(r26882376, r26882363);
        double r26882378 = r26882375 + r26882377;
        double r26882379 = r26882378 + r26882375;
        double r26882380 = r26882374 * r26882379;
        double r26882381 = r26882363 / r26882380;
        return r26882381;
}

↓

double f(double t, double l, double k) {
        double r26882382 = l;
        double r26882383 = -4.766677063238053e+76;
        bool r26882384 = r26882382 <= r26882383;
        double r26882385 = 2.0;
        double r26882386 = cbrt(r26882385);
        double r26882387 = k;
        double r26882388 = cos(r26882387);
        double r26882389 = t;
        double r26882390 = r26882387 / r26882389;
        double r26882391 = fma(r26882390, r26882390, r26882385);
        double r26882392 = cbrt(r26882391);
        double r26882393 = r26882388 / r26882392;
        double r26882394 = r26882386 * r26882393;
        double r26882395 = r26882389 / r26882382;
        double r26882396 = sin(r26882387);
        double r26882397 = r26882395 * r26882396;
        double r26882398 = r26882397 * r26882392;
        double r26882399 = r26882386 / r26882398;
        double r26882400 = r26882399 / r26882389;
        double r26882401 = r26882396 * r26882389;
        double r26882402 = r26882401 / r26882382;
        double r26882403 = r26882392 * r26882402;
        double r26882404 = r26882386 / r26882403;
        double r26882405 = r26882400 * r26882404;
        double r26882406 = r26882394 * r26882405;
        double r26882407 = 5.68850156613633e+78;
        bool r26882408 = r26882382 <= r26882407;
        double r26882409 = r26882386 / r26882392;
        double r26882410 = r26882382 / r26882396;
        double r26882411 = r26882389 / r26882410;
        double r26882412 = cbrt(r26882388);
        double r26882413 = r26882411 / r26882412;
        double r26882414 = r26882409 / r26882413;
        double r26882415 = r26882412 * r26882410;
        double r26882416 = r26882409 / r26882389;
        double r26882417 = r26882415 * r26882416;
        double r26882418 = r26882416 * r26882412;
        double r26882419 = r26882417 * r26882418;
        double r26882420 = r26882414 * r26882419;
        double r26882421 = r26882408 ? r26882420 : r26882406;
        double r26882422 = r26882384 ? r26882406 : r26882421;
        return r26882422;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if l < -4.766677063238053e+76 or 5.68850156613633e+78 < l

   1. Initial program 51.9

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

   2. Simplified 30.4

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

   4. Applied tan-quot 30.4

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

   5. Applied associate-*r/ 30.4

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

   6. Applied associate-*l/ 30.4

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

   7. Simplified 29.7

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

   9. Applied div-inv 29.7

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

   10. Applied add-cube-cbrt 30.0

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

   11. Applied add-cube-cbrt 29.8

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

   12. Applied times-frac 29.8

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

   13. Applied times-frac 29.8

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

   14. Simplified 20.1

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

   15. Simplified 20.1

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

   17. Applied associate-*l/ 20.0

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

   if -4.766677063238053e+76 < l < 5.68850156613633e+78

   1. Initial program 24.4

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

   2. Simplified 22.4

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

   4. Applied tan-quot 22.4

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

   5. Applied associate-*r/ 22.4

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

   6. Applied associate-*l/ 22.4

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

   7. Simplified 11.3

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

   9. Applied add-cube-cbrt 11.3

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

   10. Applied times-frac 11.3

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

   11. Applied add-cube-cbrt 11.5

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

   12. Applied add-cube-cbrt 11.3

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

   13. Applied times-frac 11.3

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

   14. Applied times-frac 9.8

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

   15. Simplified 8.0

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

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.766677063238053 \cdot 10^{+76}:\\ \;\;\;\;\left(\sqrt[3]{2} \cdot \frac{\cos k}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}\right) \cdot \left(\frac{\frac{\sqrt[3]{2}}{\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)} \cdot \frac{\sin k \cdot t}{\ell}}\right)\\ \mathbf{elif}\;\ell \le 5.68850156613633 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{\frac{\frac{t}{\frac{\ell}{\sin k}}}{\sqrt[3]{\cos k}}} \cdot \left(\left(\left(\sqrt[3]{\cos k} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t}\right) \cdot \left(\frac{\frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t} \cdot \sqrt[3]{\cos k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{2} \cdot \frac{\cos k}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}\right) \cdot \left(\frac{\frac{\sqrt[3]{2}}{\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{t} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)} \cdot \frac{\sin k \cdot t}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))