Average Error: 32.2 → 16.5
Time: 17.0s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.30040421529963109 \cdot 10^{-233} \lor \neg \left(t \le 5.91502370566365492 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)\right) \cdot \sqrt[3]{\sin k}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array}\]
\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}\;t \le -6.30040421529963109 \cdot 10^{-233} \lor \neg \left(t \le 5.91502370566365492 \cdot 10^{-151}\right):\\
\;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)\right) \cdot \sqrt[3]{\sin k}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\

\end{array}
double code(double t, double l, double k) {
	return (2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)));
}
↓
double code(double t, double l, double k) {
	double VAR;
	if (((t <= -6.300404215299631e-233) || !(t <= 5.915023705663655e-151))) {
		VAR = (2.0 / (((pow(cbrt(t), 3.0) * ((pow(cbrt(t), 3.0) / l) * (((pow(cbrt(t), 3.0) / l) * (cbrt(sin(k)) * cbrt(sin(k)))) * cbrt(sin(k))))) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)));
	} else {
		VAR = (2.0 / ((2.0 * (pow((1.0 / pow(-1.0, 3.0)), 1.0) * ((pow(cbrt(-1.0), 9.0) * (pow(t, 3.0) * pow(sin(k), 2.0))) / (cos(k) * pow(l, 2.0))))) + (pow((1.0 / pow(-1.0, 3.0)), 1.0) * ((pow(cbrt(-1.0), 9.0) * (pow(sin(k), 2.0) * (pow(k, 2.0) * t))) / (cos(k) * pow(l, 2.0))))));
	}
	return VAR;
}

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 < -6.300404215299631e-233 or 5.915023705663655e-151 < t

    1. Initial program 27.9

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

    3. Applied add-cube-cbrt28.0

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

    4. Applied unpow-prod-down28.0

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

    5. Applied times-frac20.3

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

    6. Applied associate-*l*18.3

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

    8. Applied *-un-lft-identity18.3

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

    9. Applied unpow-prod-down18.3

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

    10. Applied times-frac13.4

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

    11. Simplified13.4

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

    13. Applied add-cube-cbrt13.5

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

    14. Applied associate-*r*13.5

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

    16. Applied associate-*l*13.2

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

    if -6.300404215299631e-233 < t < 5.915023705663655e-151

    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. Using strategy rm

    3. Applied add-cube-cbrt64.0

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

    4. Applied unpow-prod-down64.0

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

    5. Applied times-frac62.9

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

    6. Applied associate-*l*62.9

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

    7. Taylor expanded around -inf 40.4

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.30040421529963109 \cdot 10^{-233} \lor \neg \left(t \le 5.91502370566365492 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right)\right) \cdot \sqrt[3]{\sin k}\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(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))))