Average Error: 47.9 → 16.5
Time: 59.4s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -2.42796361625678879 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\tan k \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 1.3396743901283733 \cdot 10^{154}:\\ \;\;\;\;\left(\frac{\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\sin k}}{\sin k} \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k}}{\tan k \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]
\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 -2.42796361625678879 \cdot 10^{162}:\\
\;\;\;\;\frac{\frac{2}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\tan k \cdot {\left(\frac{k}{t}\right)}^{2}}\\

\mathbf{elif}\;\ell \le 1.3396743901283733 \cdot 10^{154}:\\
\;\;\;\;\left(\frac{\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\sin k}}{\sin k} \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\

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

\end{array}
double f(double t, double l, double k) {
        double r85505 = 2.0;
        double r85506 = t;
        double r85507 = 3.0;
        double r85508 = pow(r85506, r85507);
        double r85509 = l;
        double r85510 = r85509 * r85509;
        double r85511 = r85508 / r85510;
        double r85512 = k;
        double r85513 = sin(r85512);
        double r85514 = r85511 * r85513;
        double r85515 = tan(r85512);
        double r85516 = r85514 * r85515;
        double r85517 = 1.0;
        double r85518 = r85512 / r85506;
        double r85519 = pow(r85518, r85505);
        double r85520 = r85517 + r85519;
        double r85521 = r85520 - r85517;
        double r85522 = r85516 * r85521;
        double r85523 = r85505 / r85522;
        return r85523;
}
↓
double f(double t, double l, double k) {
        double r85524 = l;
        double r85525 = -2.4279636162567888e+162;
        bool r85526 = r85524 <= r85525;
        double r85527 = 2.0;
        double r85528 = t;
        double r85529 = cbrt(r85528);
        double r85530 = r85529 * r85529;
        double r85531 = 3.0;
        double r85532 = pow(r85530, r85531);
        double r85533 = r85532 / r85524;
        double r85534 = pow(r85529, r85531);
        double r85535 = r85534 / r85524;
        double r85536 = k;
        double r85537 = sin(r85536);
        double r85538 = r85535 * r85537;
        double r85539 = r85533 * r85538;
        double r85540 = r85527 / r85539;
        double r85541 = tan(r85536);
        double r85542 = r85536 / r85528;
        double r85543 = pow(r85542, r85527);
        double r85544 = r85541 * r85543;
        double r85545 = r85540 / r85544;
        double r85546 = 1.3396743901283733e+154;
        bool r85547 = r85524 <= r85546;
        double r85548 = 1.0;
        double r85549 = 2.0;
        double r85550 = r85527 / r85549;
        double r85551 = pow(r85536, r85550);
        double r85552 = r85548 / r85551;
        double r85553 = 1.0;
        double r85554 = pow(r85552, r85553);
        double r85555 = cos(r85536);
        double r85556 = pow(r85524, r85549);
        double r85557 = r85555 * r85556;
        double r85558 = r85554 * r85557;
        double r85559 = r85558 / r85537;
        double r85560 = r85559 / r85537;
        double r85561 = pow(r85528, r85553);
        double r85562 = r85561 * r85551;
        double r85563 = r85548 / r85562;
        double r85564 = pow(r85563, r85553);
        double r85565 = r85560 * r85564;
        double r85566 = r85565 * r85527;
        double r85567 = r85533 * r85535;
        double r85568 = r85567 * r85537;
        double r85569 = r85527 / r85568;
        double r85570 = r85569 / r85544;
        double r85571 = r85547 ? r85566 : r85570;
        double r85572 = r85526 ? r85545 : r85571;
        return r85572;
}

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 3 regimes

  2. if l < -2.4279636162567888e+162

    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

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

    4. Applied add-cube-cbrt64.0

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

    5. Applied unpow-prod-down64.0

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

    6. Applied times-frac51.6

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

    7. Applied associate-*l*51.6

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

    if -2.4279636162567888e+162 < l < 1.3396743901283733e+154

    1. Initial program 45.1

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

    2. Simplified36.6

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

    3. Taylor expanded around inf 15.0

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

    5. Applied sqr-pow15.0

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

    6. Applied associate-*r*12.5

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

    8. Applied add-cube-cbrt12.5

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

    9. Applied times-frac12.3

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

    10. Applied unpow-prod-down12.3

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

    11. Applied associate-*l*10.9

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

    12. Simplified10.9

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

    14. Applied unpow210.9

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

    15. Applied associate-/r*10.7

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

    if 1.3396743901283733e+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

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

    4. Applied add-cube-cbrt64.0

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

    5. Applied unpow-prod-down64.0

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

    6. Applied times-frac48.2

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.42796361625678879 \cdot 10^{162}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\tan k \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 1.3396743901283733 \cdot 10^{154}:\\ \;\;\;\;\left(\frac{\frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\cos k \cdot {\ell}^{2}\right)}{\sin k}}{\sin k} \cdot {\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k}}{\tan k \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))))