Average Error: 48.1 → 12.6
Time: 1.2m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.211150141668747897121385502175922469413 \cdot 10^{296}:\\ \;\;\;\;\frac{\left(2 \cdot \cos k\right) \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}}{\ell}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{\sqrt[3]{{t}^{3}}}{\ell}}}{\sin k \cdot \tan k}\\ \end{array}\]
\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}\;\ell \cdot \ell \le 4.211150141668747897121385502175922469413 \cdot 10^{296}:\\
\;\;\;\;\frac{\left(2 \cdot \cos k\right) \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}}{\ell}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{\sqrt[3]{{t}^{3}}}{\ell}}}{\sin k \cdot \tan k}\\

\end{array}
double f(double t, double l, double k) {
        double r8546529 = 2.0;
        double r8546530 = t;
        double r8546531 = 3.0;
        double r8546532 = pow(r8546530, r8546531);
        double r8546533 = l;
        double r8546534 = r8546533 * r8546533;
        double r8546535 = r8546532 / r8546534;
        double r8546536 = k;
        double r8546537 = sin(r8546536);
        double r8546538 = r8546535 * r8546537;
        double r8546539 = tan(r8546536);
        double r8546540 = r8546538 * r8546539;
        double r8546541 = 1.0;
        double r8546542 = r8546536 / r8546530;
        double r8546543 = pow(r8546542, r8546529);
        double r8546544 = r8546541 + r8546543;
        double r8546545 = r8546544 - r8546541;
        double r8546546 = r8546540 * r8546545;
        double r8546547 = r8546529 / r8546546;
        return r8546547;
}
↓
double f(double t, double l, double k) {
        double r8546548 = l;
        double r8546549 = r8546548 * r8546548;
        double r8546550 = 4.211150141668748e+296;
        bool r8546551 = r8546549 <= r8546550;
        double r8546552 = 2.0;
        double r8546553 = k;
        double r8546554 = cos(r8546553);
        double r8546555 = r8546552 * r8546554;
        double r8546556 = 1.0;
        double r8546557 = t;
        double r8546558 = 1.0;
        double r8546559 = pow(r8546557, r8546558);
        double r8546560 = 2.0;
        double r8546561 = r8546552 / r8546560;
        double r8546562 = pow(r8546553, r8546561);
        double r8546563 = r8546559 * r8546562;
        double r8546564 = r8546556 / r8546563;
        double r8546565 = pow(r8546564, r8546558);
        double r8546566 = r8546555 * r8546565;
        double r8546567 = sin(r8546553);
        double r8546568 = r8546567 / r8546548;
        double r8546569 = r8546568 * r8546568;
        double r8546570 = r8546566 / r8546569;
        double r8546571 = r8546556 / r8546562;
        double r8546572 = pow(r8546571, r8546558);
        double r8546573 = r8546570 * r8546572;
        double r8546574 = 3.0;
        double r8546575 = pow(r8546557, r8546574);
        double r8546576 = cbrt(r8546575);
        double r8546577 = r8546576 * r8546576;
        double r8546578 = r8546577 / r8546548;
        double r8546579 = r8546556 / r8546578;
        double r8546580 = r8546553 / r8546557;
        double r8546581 = pow(r8546580, r8546552);
        double r8546582 = r8546579 / r8546581;
        double r8546583 = r8546576 / r8546548;
        double r8546584 = r8546552 / r8546583;
        double r8546585 = tan(r8546553);
        double r8546586 = r8546567 * r8546585;
        double r8546587 = r8546584 / r8546586;
        double r8546588 = r8546582 * r8546587;
        double r8546589 = r8546551 ? r8546573 : r8546588;
        return r8546589;
}

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 (* l l) < 4.211150141668748e+296

    1. Initial program 45.2

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

    2. Simplified37.4

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

    3. Taylor expanded around inf 14.9

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

    4. Simplified12.5

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

    6. Applied sqr-pow12.5

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

    7. Applied associate-*r*8.2

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

    9. Applied *-un-lft-identity8.2

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

    10. Applied times-frac8.0

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

    11. Applied unpow-prod-down8.0

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

    12. Applied associate-*r*5.5

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

    14. Applied associate-*r/5.5

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

    15. Applied associate-*l/5.4

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

    if 4.211150141668748e+296 < (* l l)

    1. Initial program 63.5

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

    2. Simplified63.2

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

    4. Applied add-cube-cbrt63.2

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

    5. Applied times-frac53.8

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

    6. Applied *-un-lft-identity53.8

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

    7. Applied times-frac53.8

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

    8. Applied times-frac50.2

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

  4. Final simplification12.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.211150141668747897121385502175922469413 \cdot 10^{296}:\\ \;\;\;\;\frac{\left(2 \cdot \cos k\right) \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}}{\ell}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\frac{2}{\frac{\sqrt[3]{{t}^{3}}}{\ell}}}{\sin k \cdot \tan k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))))