Average Error: 32.8 → 2.6
Time: 42.4s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -5.516036138020374502266814377566367529757 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right) \cdot t}{\cos k} \cdot 2 + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}}{\sin k}}}{2}}\\ \mathbf{elif}\;k \le 3.049035263999028012390937801482074624326 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{\frac{\left(\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k}}{\sin k}} + 2 \cdot \frac{\frac{t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \left(t \cdot \sin k\right)\right)}{\ell}}{\cos k}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right) \cdot t}{\cos k} \cdot 2 + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}}{\sin k}}}{2}}\\ \end{array}\]
\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 -5.516036138020374502266814377566367529757 \cdot 10^{-90}:\\
\;\;\;\;\frac{1}{\frac{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right) \cdot t}{\cos k} \cdot 2 + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}}{\sin k}}}{2}}\\

\mathbf{elif}\;k \le 3.049035263999028012390937801482074624326 \cdot 10^{-50}:\\
\;\;\;\;\frac{1}{\frac{\frac{t}{\frac{\left(\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k}}{\sin k}} + 2 \cdot \frac{\frac{t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \left(t \cdot \sin k\right)\right)}{\ell}}{\cos k}}{2}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r3383373 = 2.0;
        double r3383374 = t;
        double r3383375 = 3.0;
        double r3383376 = pow(r3383374, r3383375);
        double r3383377 = l;
        double r3383378 = r3383377 * r3383377;
        double r3383379 = r3383376 / r3383378;
        double r3383380 = k;
        double r3383381 = sin(r3383380);
        double r3383382 = r3383379 * r3383381;
        double r3383383 = tan(r3383380);
        double r3383384 = r3383382 * r3383383;
        double r3383385 = 1.0;
        double r3383386 = r3383380 / r3383374;
        double r3383387 = pow(r3383386, r3383373);
        double r3383388 = r3383385 + r3383387;
        double r3383389 = r3383388 + r3383385;
        double r3383390 = r3383384 * r3383389;
        double r3383391 = r3383373 / r3383390;
        return r3383391;
}
↓
double f(double t, double l, double k) {
        double r3383392 = k;
        double r3383393 = -5.5160361380203745e-90;
        bool r3383394 = r3383392 <= r3383393;
        double r3383395 = 1.0;
        double r3383396 = sin(r3383392);
        double r3383397 = l;
        double r3383398 = r3383396 / r3383397;
        double r3383399 = t;
        double r3383400 = r3383398 * r3383399;
        double r3383401 = r3383400 * r3383400;
        double r3383402 = r3383401 * r3383399;
        double r3383403 = cos(r3383392);
        double r3383404 = r3383402 / r3383403;
        double r3383405 = 2.0;
        double r3383406 = r3383404 * r3383405;
        double r3383407 = r3383397 / r3383392;
        double r3383408 = r3383399 / r3383407;
        double r3383409 = r3383403 / r3383396;
        double r3383410 = r3383409 * r3383407;
        double r3383411 = r3383410 / r3383396;
        double r3383412 = r3383408 / r3383411;
        double r3383413 = r3383406 + r3383412;
        double r3383414 = r3383413 / r3383405;
        double r3383415 = r3383395 / r3383414;
        double r3383416 = 3.049035263999028e-50;
        bool r3383417 = r3383392 <= r3383416;
        double r3383418 = r3383410 * r3383407;
        double r3383419 = r3383418 / r3383396;
        double r3383420 = r3383399 / r3383419;
        double r3383421 = r3383399 * r3383396;
        double r3383422 = r3383421 / r3383397;
        double r3383423 = r3383422 * r3383421;
        double r3383424 = r3383399 * r3383423;
        double r3383425 = r3383424 / r3383397;
        double r3383426 = r3383425 / r3383403;
        double r3383427 = r3383405 * r3383426;
        double r3383428 = r3383420 + r3383427;
        double r3383429 = r3383428 / r3383405;
        double r3383430 = r3383395 / r3383429;
        double r3383431 = r3383417 ? r3383430 : r3383415;
        double r3383432 = r3383394 ? r3383415 : r3383431;
        return r3383432;
}

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 < -5.5160361380203745e-90 or 3.049035263999028e-50 < k

    1. Initial program 32.0

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

    3. Applied add-cube-cbrt32.1

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

    4. Applied unpow-prod-down32.1

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

    5. Applied times-frac24.7

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

    6. Taylor expanded around inf 24.1

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

    7. Simplified16.8

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

    9. Applied clear-num16.8

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

    10. Simplified5.4

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

    12. Applied *-un-lft-identity5.4

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

    13. Applied times-frac5.2

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

    14. Applied associate-/r*1.1

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

    15. Simplified1.1

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

    if -5.5160361380203745e-90 < k < 3.049035263999028e-50

    1. Initial program 35.1

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

    3. Applied add-cube-cbrt35.3

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

    4. Applied unpow-prod-down35.3

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

    5. Applied times-frac29.6

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

    6. Taylor expanded around inf 53.1

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

    7. Simplified28.1

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

    9. Applied clear-num28.1

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

    10. Simplified10.5

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

    12. Applied associate-*r/11.1

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

    13. Applied associate-*r/13.3

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

    14. Applied associate-*r/14.5

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

    15. Simplified7.0

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -5.516036138020374502266814377566367529757 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\frac{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right) \cdot t}{\cos k} \cdot 2 + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}}{\sin k}}}{2}}\\ \mathbf{elif}\;k \le 3.049035263999028012390937801482074624326 \cdot 10^{-50}:\\ \;\;\;\;\frac{1}{\frac{\frac{t}{\frac{\left(\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}\right) \cdot \frac{\ell}{k}}{\sin k}} + 2 \cdot \frac{\frac{t \cdot \left(\frac{t \cdot \sin k}{\ell} \cdot \left(t \cdot \sin k\right)\right)}{\ell}}{\cos k}}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{\left(\left(\frac{\sin k}{\ell} \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot t\right)\right) \cdot t}{\cos k} \cdot 2 + \frac{\frac{t}{\frac{\ell}{k}}}{\frac{\frac{\cos k}{\sin k} \cdot \frac{\ell}{k}}{\sin k}}}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))