Average Error: 31.7 → 11.5
Time: 4.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.0143648348712567 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}} \cdot \frac{\frac{\ell}{\sin k \cdot t}}{t}\\ \mathbf{elif}\;\ell \le 8.796141127304192 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{t}}{\frac{\sin k}{\ell} \cdot \left(t \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)}}{\frac{\sin k}{\ell} \cdot \left(t \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)} \cdot \left(\cos k \cdot \frac{\sqrt{2}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}} \cdot \frac{\frac{\ell}{\sin k \cdot t}}{t}\\ \end{array}\]
\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 \le -1.0143648348712567 \cdot 10^{+22}:\\
\;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}} \cdot \frac{\frac{\ell}{\sin k \cdot t}}{t}\\

\mathbf{elif}\;\ell \le 8.796141127304192 \cdot 10^{+67}:\\
\;\;\;\;\frac{\frac{\frac{\sqrt{2}}{t}}{\frac{\sin k}{\ell} \cdot \left(t \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)}}{\frac{\sin k}{\ell} \cdot \left(t \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)} \cdot \left(\cos k \cdot \frac{\sqrt{2}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}} \cdot \frac{\frac{\ell}{\sin k \cdot t}}{t}\\

\end{array}
double f(double t, double l, double k) {
        double r36794598 = 2.0;
        double r36794599 = t;
        double r36794600 = 3.0;
        double r36794601 = pow(r36794599, r36794600);
        double r36794602 = l;
        double r36794603 = r36794602 * r36794602;
        double r36794604 = r36794601 / r36794603;
        double r36794605 = k;
        double r36794606 = sin(r36794605);
        double r36794607 = r36794604 * r36794606;
        double r36794608 = tan(r36794605);
        double r36794609 = r36794607 * r36794608;
        double r36794610 = 1.0;
        double r36794611 = r36794605 / r36794599;
        double r36794612 = pow(r36794611, r36794598);
        double r36794613 = r36794610 + r36794612;
        double r36794614 = r36794613 + r36794610;
        double r36794615 = r36794609 * r36794614;
        double r36794616 = r36794598 / r36794615;
        return r36794616;
}
↓
double f(double t, double l, double k) {
        double r36794617 = l;
        double r36794618 = -1.0143648348712567e+22;
        bool r36794619 = r36794617 <= r36794618;
        double r36794620 = 2.0;
        double r36794621 = k;
        double r36794622 = t;
        double r36794623 = r36794621 / r36794622;
        double r36794624 = r36794623 * r36794623;
        double r36794625 = r36794624 + r36794620;
        double r36794626 = r36794620 / r36794625;
        double r36794627 = sin(r36794621);
        double r36794628 = r36794617 / r36794622;
        double r36794629 = r36794627 / r36794628;
        double r36794630 = cos(r36794621);
        double r36794631 = r36794629 / r36794630;
        double r36794632 = r36794626 / r36794631;
        double r36794633 = r36794627 * r36794622;
        double r36794634 = r36794617 / r36794633;
        double r36794635 = r36794634 / r36794622;
        double r36794636 = r36794632 * r36794635;
        double r36794637 = 8.796141127304192e+67;
        bool r36794638 = r36794617 <= r36794637;
        double r36794639 = sqrt(r36794620);
        double r36794640 = r36794639 / r36794622;
        double r36794641 = r36794627 / r36794617;
        double r36794642 = cbrt(r36794625);
        double r36794643 = r36794622 * r36794642;
        double r36794644 = r36794641 * r36794643;
        double r36794645 = r36794640 / r36794644;
        double r36794646 = r36794645 / r36794644;
        double r36794647 = r36794639 / r36794642;
        double r36794648 = r36794630 * r36794647;
        double r36794649 = r36794646 * r36794648;
        double r36794650 = r36794638 ? r36794649 : r36794636;
        double r36794651 = r36794619 ? r36794636 : r36794650;
        return r36794651;
}

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 < -1.0143648348712567e+22 or 8.796141127304192e+67 < l

    1. Initial program 47.8

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

    2. Simplified29.3

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

    4. Applied tan-quot29.3

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

    5. Applied associate-*r/29.3

      \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\color{blue}{\frac{\sin k \cdot \sin k}{\cos k}} \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}\]

    6. Applied associate-*l/29.3

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

    7. Simplified21.0

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

    9. Applied *-un-lft-identity21.0

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

    10. Applied times-frac21.0

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

    11. Applied *-un-lft-identity21.0

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

    12. Applied *-un-lft-identity21.0

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

    13. Applied times-frac21.0

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

    14. Applied times-frac19.0

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

    15. Simplified18.9

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\sin k \cdot t}}{t}} \cdot \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}}\]

    if -1.0143648348712567e+22 < l < 8.796141127304192e+67

    1. Initial program 23.7

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

    2. Simplified21.7

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

    4. Applied tan-quot21.7

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

    5. Applied associate-*r/21.7

      \[\leadsto \frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\color{blue}{\frac{\sin k \cdot \sin k}{\cos k}} \cdot \frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}\]

    6. Applied associate-*l/21.7

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

    7. Simplified10.9

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

    9. Applied div-inv10.9

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

    10. Applied add-cube-cbrt11.1

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

    11. Applied add-sqr-sqrt11.0

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

    12. Applied times-frac11.1

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

    13. Applied times-frac11.1

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

    14. Simplified7.9

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

    15. Simplified7.9

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

  4. Final simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.0143648348712567 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}} \cdot \frac{\frac{\ell}{\sin k \cdot t}}{t}\\ \mathbf{elif}\;\ell \le 8.796141127304192 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{2}}{t}}{\frac{\sin k}{\ell} \cdot \left(t \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)}}{\frac{\sin k}{\ell} \cdot \left(t \cdot \sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}\right)} \cdot \left(\cos k \cdot \frac{\sqrt{2}}{\sqrt[3]{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{t} \cdot \frac{k}{t} + 2}}{\frac{\frac{\sin k}{\frac{\ell}{t}}}{\cos k}} \cdot \frac{\frac{\ell}{\sin k \cdot t}}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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))))