Average Error: 31.9 → 3.5
Time: 49.5s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -3.2137469732307354 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot \frac{t}{\frac{\ell}{t}}}{\frac{\ell}{t}}\right), \left(\frac{k}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)\right)\right) \cdot \tan k}\\ \mathbf{elif}\;k \le 5.835318740221373 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{\frac{t}{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}}{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot \frac{t}{\frac{\ell}{t}}}{\frac{\ell}{t}}\right), \left(\frac{k}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)\right)\right) \cdot \tan k}\\ \end{array}\]
\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}\;k \le -3.2137469732307354 \cdot 10^{-128}:\\
\;\;\;\;\frac{2}{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot \frac{t}{\frac{\ell}{t}}}{\frac{\ell}{t}}\right), \left(\frac{k}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)\right)\right) \cdot \tan k}\\

\mathbf{elif}\;k \le 5.835318740221373 \cdot 10^{-87}:\\
\;\;\;\;\frac{\frac{2}{\tan k}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{\frac{t}{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}}{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot \frac{t}{\frac{\ell}{t}}}{\frac{\ell}{t}}\right), \left(\frac{k}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)\right)\right) \cdot \tan k}\\

\end{array}
double f(double t, double l, double k) {
        double r2558649 = 2.0;
        double r2558650 = t;
        double r2558651 = 3.0;
        double r2558652 = pow(r2558650, r2558651);
        double r2558653 = l;
        double r2558654 = r2558653 * r2558653;
        double r2558655 = r2558652 / r2558654;
        double r2558656 = k;
        double r2558657 = sin(r2558656);
        double r2558658 = r2558655 * r2558657;
        double r2558659 = tan(r2558656);
        double r2558660 = r2558658 * r2558659;
        double r2558661 = 1.0;
        double r2558662 = r2558656 / r2558650;
        double r2558663 = pow(r2558662, r2558649);
        double r2558664 = r2558661 + r2558663;
        double r2558665 = r2558664 + r2558661;
        double r2558666 = r2558660 * r2558665;
        double r2558667 = r2558649 / r2558666;
        return r2558667;
}
↓
double f(double t, double l, double k) {
        double r2558668 = k;
        double r2558669 = -3.2137469732307354e-128;
        bool r2558670 = r2558668 <= r2558669;
        double r2558671 = 2.0;
        double r2558672 = sin(r2558668);
        double r2558673 = t;
        double r2558674 = l;
        double r2558675 = r2558674 / r2558673;
        double r2558676 = r2558673 / r2558675;
        double r2558677 = r2558672 * r2558676;
        double r2558678 = r2558677 / r2558675;
        double r2558679 = r2558668 / r2558674;
        double r2558680 = r2558672 * r2558679;
        double r2558681 = r2558673 * r2558680;
        double r2558682 = r2558679 * r2558681;
        double r2558683 = fma(r2558671, r2558678, r2558682);
        double r2558684 = tan(r2558668);
        double r2558685 = r2558683 * r2558684;
        double r2558686 = r2558671 / r2558685;
        double r2558687 = 5.835318740221373e-87;
        bool r2558688 = r2558668 <= r2558687;
        double r2558689 = r2558671 / r2558684;
        double r2558690 = r2558668 / r2558673;
        double r2558691 = fma(r2558690, r2558690, r2558671);
        double r2558692 = cbrt(r2558672);
        double r2558693 = r2558692 * r2558692;
        double r2558694 = r2558675 / r2558693;
        double r2558695 = r2558673 / r2558694;
        double r2558696 = r2558675 / r2558692;
        double r2558697 = r2558695 / r2558696;
        double r2558698 = r2558691 * r2558697;
        double r2558699 = r2558689 / r2558698;
        double r2558700 = r2558688 ? r2558699 : r2558686;
        double r2558701 = r2558670 ? r2558686 : r2558700;
        return r2558701;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if k < -3.2137469732307354e-128 or 5.835318740221373e-87 < k

    1. Initial program 30.8

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

    2. Simplified17.6

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

    3. Taylor expanded around -inf 24.2

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

    4. Simplified14.7

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

    6. Applied times-frac4.8

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

    8. Applied associate-/l*1.8

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

    10. Applied div-inv1.8

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{\tan k}}}{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right), \left(\frac{\sin k \cdot t}{\frac{\ell}{k}} \cdot \frac{k}{\ell}\right)\right)}\]

    11. Applied associate-/l*1.9

      \[\leadsto \color{blue}{\frac{2}{\frac{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right), \left(\frac{\sin k \cdot t}{\frac{\ell}{k}} \cdot \frac{k}{\ell}\right)\right)}{\frac{1}{\tan k}}}}\]

    12. Simplified1.2

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

    if -3.2137469732307354e-128 < k < 5.835318740221373e-87

    1. Initial program 35.8

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

    2. Simplified19.4

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

    4. Applied add-cube-cbrt19.7

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

    5. Applied times-frac12.2

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

    6. Applied associate-/r*11.9

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

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -3.2137469732307354 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot \frac{t}{\frac{\ell}{t}}}{\frac{\ell}{t}}\right), \left(\frac{k}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)\right)\right) \cdot \tan k}\\ \mathbf{elif}\;k \le 5.835318740221373 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right) \cdot \frac{\frac{t}{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}}{\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \left(\frac{\sin k \cdot \frac{t}{\frac{\ell}{t}}}{\frac{\ell}{t}}\right), \left(\frac{k}{\ell} \cdot \left(t \cdot \left(\sin k \cdot \frac{k}{\ell}\right)\right)\right)\right) \cdot \tan k}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(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))))