Average Error: 47.8 → 17.6
Time: 59.1s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le 1.167555341999574553698834286211832380644 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos k}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]
\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 1.167555341999574553698834286211832380644 \cdot 10^{154}:\\
\;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos k}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\\

\end{array}
double f(double t, double l, double k) {
        double r101691 = 2.0;
        double r101692 = t;
        double r101693 = 3.0;
        double r101694 = pow(r101692, r101693);
        double r101695 = l;
        double r101696 = r101695 * r101695;
        double r101697 = r101694 / r101696;
        double r101698 = k;
        double r101699 = sin(r101698);
        double r101700 = r101697 * r101699;
        double r101701 = tan(r101698);
        double r101702 = r101700 * r101701;
        double r101703 = 1.0;
        double r101704 = r101698 / r101692;
        double r101705 = pow(r101704, r101691);
        double r101706 = r101703 + r101705;
        double r101707 = r101706 - r101703;
        double r101708 = r101702 * r101707;
        double r101709 = r101691 / r101708;
        return r101709;
}
↓
double f(double t, double l, double k) {
        double r101710 = l;
        double r101711 = 1.1675553419995746e+154;
        bool r101712 = r101710 <= r101711;
        double r101713 = 2.0;
        double r101714 = 1.0;
        double r101715 = k;
        double r101716 = 2.0;
        double r101717 = r101713 / r101716;
        double r101718 = pow(r101715, r101717);
        double r101719 = r101714 / r101718;
        double r101720 = 1.0;
        double r101721 = pow(r101719, r101720);
        double r101722 = pow(r101710, r101716);
        double r101723 = cos(r101715);
        double r101724 = r101722 * r101723;
        double r101725 = sin(r101715);
        double r101726 = pow(r101725, r101716);
        double r101727 = r101724 / r101726;
        double r101728 = r101721 * r101727;
        double r101729 = t;
        double r101730 = pow(r101729, r101720);
        double r101731 = r101714 / r101730;
        double r101732 = pow(r101731, r101720);
        double r101733 = r101728 * r101732;
        double r101734 = r101733 * r101721;
        double r101735 = r101713 * r101734;
        double r101736 = 3.0;
        double r101737 = pow(r101729, r101736);
        double r101738 = r101737 / r101710;
        double r101739 = r101738 / r101710;
        double r101740 = r101739 * r101726;
        double r101741 = r101713 / r101740;
        double r101742 = r101715 / r101729;
        double r101743 = pow(r101742, r101717);
        double r101744 = r101741 / r101743;
        double r101745 = r101723 / r101743;
        double r101746 = r101744 * r101745;
        double r101747 = r101712 ? r101735 : r101746;
        return r101747;
}

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 < 1.1675553419995746e+154

    1. Initial program 46.5

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

    2. Simplified38.2

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

    3. Taylor expanded around inf 18.5

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

    5. Applied sqr-pow18.5

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

    6. Applied associate-*l*16.2

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

    8. Applied add-cube-cbrt16.2

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

    9. Applied times-frac16.0

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

    10. Applied unpow-prod-down16.0

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

    11. Applied associate-*l*14.4

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

    12. Simplified14.4

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

    14. Applied add-sqr-sqrt14.4

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

    15. Applied times-frac14.3

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

    16. Applied unpow-prod-down14.3

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

    17. Applied associate-*r*14.8

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

    18. Simplified14.8

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

    if 1.1675553419995746e+154 < l

    1. Initial program 64.0

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

    2. Simplified64.0

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

    4. Applied sqr-pow64.0

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

    5. Applied tan-quot64.0

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

    6. Applied associate-*r/64.0

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

    7. Applied associate-/r/64.0

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

    8. Applied times-frac64.0

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

    9. Simplified51.1

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

  4. Final simplification17.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 1.167555341999574553698834286211832380644 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\cos k}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))))