Average Error: 48.5 → 10.4
Time: 28.7s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.1514252442459826 \cdot 10^{-102} \lor \neg \left(k \le 7.02952442766517907 \cdot 10^{-84} \lor \neg \left(k \le 1.90701078281367166 \cdot 10^{146}\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]
\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}\;k \le -1.1514252442459826 \cdot 10^{-102} \lor \neg \left(k \le 7.02952442766517907 \cdot 10^{-84} \lor \neg \left(k \le 1.90701078281367166 \cdot 10^{146}\right)\right):\\
\;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\

\end{array}
double f(double t, double l, double k) {
        double r94467 = 2.0;
        double r94468 = t;
        double r94469 = 3.0;
        double r94470 = pow(r94468, r94469);
        double r94471 = l;
        double r94472 = r94471 * r94471;
        double r94473 = r94470 / r94472;
        double r94474 = k;
        double r94475 = sin(r94474);
        double r94476 = r94473 * r94475;
        double r94477 = tan(r94474);
        double r94478 = r94476 * r94477;
        double r94479 = 1.0;
        double r94480 = r94474 / r94468;
        double r94481 = pow(r94480, r94467);
        double r94482 = r94479 + r94481;
        double r94483 = r94482 - r94479;
        double r94484 = r94478 * r94483;
        double r94485 = r94467 / r94484;
        return r94485;
}
↓
double f(double t, double l, double k) {
        double r94486 = k;
        double r94487 = -1.1514252442459826e-102;
        bool r94488 = r94486 <= r94487;
        double r94489 = 7.029524427665179e-84;
        bool r94490 = r94486 <= r94489;
        double r94491 = 1.9070107828136717e+146;
        bool r94492 = r94486 <= r94491;
        double r94493 = !r94492;
        bool r94494 = r94490 || r94493;
        double r94495 = !r94494;
        bool r94496 = r94488 || r94495;
        double r94497 = 2.0;
        double r94498 = 1.0;
        double r94499 = cbrt(r94498);
        double r94500 = r94499 * r94499;
        double r94501 = pow(r94486, r94497);
        double r94502 = r94500 / r94501;
        double r94503 = 1.0;
        double r94504 = pow(r94502, r94503);
        double r94505 = t;
        double r94506 = pow(r94505, r94503);
        double r94507 = r94499 / r94506;
        double r94508 = pow(r94507, r94503);
        double r94509 = cos(r94486);
        double r94510 = l;
        double r94511 = r94509 * r94510;
        double r94512 = r94508 * r94511;
        double r94513 = r94504 * r94512;
        double r94514 = sin(r94486);
        double r94515 = fabs(r94514);
        double r94516 = r94515 / r94510;
        double r94517 = r94515 * r94516;
        double r94518 = r94513 / r94517;
        double r94519 = r94497 * r94518;
        double r94520 = 2.0;
        double r94521 = r94497 / r94520;
        double r94522 = pow(r94486, r94521);
        double r94523 = r94522 * r94506;
        double r94524 = r94522 * r94523;
        double r94525 = r94498 / r94524;
        double r94526 = pow(r94525, r94503);
        double r94527 = r94526 * r94511;
        double r94528 = r94527 / r94517;
        double r94529 = r94497 * r94528;
        double r94530 = r94496 ? r94519 : r94529;
        return r94530;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < -1.1514252442459826e-102 or 7.029524427665179e-84 < k < 1.9070107828136717e+146

    1. Initial program 49.0

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

    2. Simplified39.7

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

    3. Taylor expanded around inf 18.8

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

    5. Applied add-sqr-sqrt18.8

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

    6. Applied times-frac18.8

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

    7. Simplified18.8

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

    8. Simplified18.0

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

    10. Applied frac-times17.6

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

    11. Applied associate-*r/11.7

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

    13. Applied add-cube-cbrt11.7

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

    14. Applied times-frac11.4

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

    15. Applied unpow-prod-down11.4

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

    16. Applied associate-*l*8.4

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

    if -1.1514252442459826e-102 < k < 7.029524427665179e-84 or 1.9070107828136717e+146 < k

    1. Initial program 47.5

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

    2. Simplified43.4

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

    3. Taylor expanded around inf 30.7

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

    5. Applied add-sqr-sqrt30.7

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

    6. Applied times-frac30.7

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

    7. Simplified30.7

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

    8. Simplified28.6

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

    10. Applied frac-times26.8

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

    11. Applied associate-*r/24.7

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

    13. Applied sqr-pow24.7

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

    14. Applied associate-*l*14.6

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.1514252442459826 \cdot 10^{-102} \lor \neg \left(k \le 7.02952442766517907 \cdot 10^{-84} \lor \neg \left(k \le 1.90701078281367166 \cdot 10^{146}\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))