Average Error: 48.3 → 10.7
Time: 1.3m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 7.034952289559081821119819809298347258955 \cdot 10^{-94} \lor \neg \left(t \le 9.443522340558764843901876198839124615428 \cdot 10^{67}\right):\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\frac{\sin k}{\ell} \cdot {t}^{3}}} \cdot \sqrt[3]{\frac{2}{\frac{\sin k}{\ell} \cdot {t}^{3}}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\frac{\ell}{\tan k} \cdot \sqrt[3]{\frac{2}{\frac{\sin k \cdot {t}^{3}}{\ell}}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]
\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}\;t \le 7.034952289559081821119819809298347258955 \cdot 10^{-94} \lor \neg \left(t \le 9.443522340558764843901876198839124615428 \cdot 10^{67}\right):\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right)\right)\\

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

\end{array}
double f(double t, double l, double k) {
        double r117568 = 2.0;
        double r117569 = t;
        double r117570 = 3.0;
        double r117571 = pow(r117569, r117570);
        double r117572 = l;
        double r117573 = r117572 * r117572;
        double r117574 = r117571 / r117573;
        double r117575 = k;
        double r117576 = sin(r117575);
        double r117577 = r117574 * r117576;
        double r117578 = tan(r117575);
        double r117579 = r117577 * r117578;
        double r117580 = 1.0;
        double r117581 = r117575 / r117569;
        double r117582 = pow(r117581, r117568);
        double r117583 = r117580 + r117582;
        double r117584 = r117583 - r117580;
        double r117585 = r117579 * r117584;
        double r117586 = r117568 / r117585;
        return r117586;
}
↓
double f(double t, double l, double k) {
        double r117587 = t;
        double r117588 = 7.034952289559082e-94;
        bool r117589 = r117587 <= r117588;
        double r117590 = 9.443522340558765e+67;
        bool r117591 = r117587 <= r117590;
        double r117592 = !r117591;
        bool r117593 = r117589 || r117592;
        double r117594 = l;
        double r117595 = k;
        double r117596 = tan(r117595);
        double r117597 = r117594 / r117596;
        double r117598 = sin(r117595);
        double r117599 = r117594 / r117598;
        double r117600 = 1.0;
        double r117601 = 2.0;
        double r117602 = 2.0;
        double r117603 = r117601 / r117602;
        double r117604 = pow(r117595, r117603);
        double r117605 = r117600 / r117604;
        double r117606 = 1.0;
        double r117607 = pow(r117587, r117606);
        double r117608 = r117605 / r117607;
        double r117609 = r117608 / r117604;
        double r117610 = pow(r117609, r117606);
        double r117611 = r117610 * r117601;
        double r117612 = r117599 * r117611;
        double r117613 = r117597 * r117612;
        double r117614 = r117598 / r117594;
        double r117615 = 3.0;
        double r117616 = pow(r117587, r117615);
        double r117617 = r117614 * r117616;
        double r117618 = r117601 / r117617;
        double r117619 = cbrt(r117618);
        double r117620 = r117619 * r117619;
        double r117621 = r117595 / r117587;
        double r117622 = pow(r117621, r117603);
        double r117623 = r117620 / r117622;
        double r117624 = r117598 * r117616;
        double r117625 = r117624 / r117594;
        double r117626 = r117601 / r117625;
        double r117627 = cbrt(r117626);
        double r117628 = r117597 * r117627;
        double r117629 = r117628 / r117622;
        double r117630 = r117623 * r117629;
        double r117631 = r117593 ? r117613 : r117630;
        return r117631;
}

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 t < 7.034952289559082e-94 or 9.443522340558765e+67 < t

    1. Initial program 50.9

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

    2. Simplified40.4

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

    3. Taylor expanded around inf 16.5

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

    4. Simplified16.3

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

    6. Applied sqr-pow16.3

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

    7. Applied associate-/r*11.0

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

    8. Simplified11.0

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

    10. Applied associate-/r*11.0

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

    if 7.034952289559082e-94 < t < 9.443522340558765e+67

    1. Initial program 29.9

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

    2. Simplified14.8

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

    4. Applied sqr-pow14.8

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

    5. Applied add-cube-cbrt15.1

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

    6. Applied times-frac8.9

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

    7. Applied associate-*l*6.1

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

    8. Simplified8.9

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 7.034952289559081821119819809298347258955 \cdot 10^{-94} \lor \neg \left(t \le 9.443522340558764843901876198839124615428 \cdot 10^{67}\right):\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\frac{\sin k}{\ell} \cdot {t}^{3}}} \cdot \sqrt[3]{\frac{2}{\frac{\sin k}{\ell} \cdot {t}^{3}}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\frac{\ell}{\tan k} \cdot \sqrt[3]{\frac{2}{\frac{\sin k \cdot {t}^{3}}{\ell}}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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))))