Average Error: 32.6 → 14.8
Time: 15.6s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.29821928456958346 \cdot 10^{-82} \lor \neg \left(t \le 3.814560346462089 \cdot 10^{-73}\right):\\ \;\;\;\;\left(\ell \cdot \frac{\sqrt{2}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \frac{\sqrt{2}}{\sin k \cdot \left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\tan k \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\left(\sin k\right)}^{2}}{\ell}\right) \cdot {\left(\frac{1}{{\left({\left(\frac{1}{t}\right)}^{1}\right)}^{1}}\right)}^{1}}\\ \end{array}\]
\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 -1.29821928456958346 \cdot 10^{-82} \lor \neg \left(t \le 3.814560346462089 \cdot 10^{-73}\right):\\
\;\;\;\;\left(\ell \cdot \frac{\sqrt{2}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \frac{\sqrt{2}}{\sin k \cdot \left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\tan k \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)}\\

\mathbf{else}:\\
\;\;\;\;\ell \cdot \frac{2}{\left(\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\left(\sin k\right)}^{2}}{\ell}\right) \cdot {\left(\frac{1}{{\left({\left(\frac{1}{t}\right)}^{1}\right)}^{1}}\right)}^{1}}\\

\end{array}
double code(double t, double l, double k) {
	return ((double) (2.0 / ((double) (((double) (((double) (((double) (((double) pow(t, 3.0)) / ((double) (l * l)))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))) + 1.0))))));
}
↓
double code(double t, double l, double k) {
	double VAR;
	if (((t <= -1.2982192845695835e-82) || !(t <= 3.814560346462089e-73))) {
		VAR = ((double) (((double) (l * ((double) (((double) sqrt(2.0)) / ((double) (1.0 + ((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))))))))) * ((double) (((double) sqrt(2.0)) / ((double) (((double) sin(k)) * ((double) (((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) (((double) tan(k)) * ((double) pow(((double) cbrt(t)), 3.0)))))) * ((double) (((double) pow(((double) cbrt(t)), 3.0)) / l))))))))));
	} else {
		VAR = ((double) (l * ((double) (2.0 / ((double) (((double) (((double) (k / ((double) (((double) cos(k)) / k)))) * ((double) (((double) pow(((double) sin(k)), 2.0)) / l)))) * ((double) pow(((double) (1.0 / ((double) pow(((double) pow(((double) (1.0 / t)), 1.0)), 1.0)))), 1.0))))))));
	}
	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 < -1.29821928456958346e-82 or 3.814560346462089e-73 < t

    1. Initial program 22.8

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

    2. Simplified17.8

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

    4. Applied *-un-lft-identity17.8

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

    5. Applied add-cube-cbrt18.1

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

    6. Applied unpow-prod-down18.1

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

    7. Applied times-frac16.2

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

    8. Simplified16.2

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

    10. Applied associate-*r*14.2

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

    12. Applied unpow-prod-down14.2

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

    13. Applied associate-*r*12.4

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

    15. Applied add-sqr-sqrt12.4

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

    16. Applied times-frac12.3

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

    17. Applied associate-*r*11.9

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

    if -1.29821928456958346e-82 < t < 3.814560346462089e-73

    1. Initial program 58.5

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

    2. Simplified58.1

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

    3. Taylor expanded around inf 42.8

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

    4. Simplified22.6

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.29821928456958346 \cdot 10^{-82} \lor \neg \left(t \le 3.814560346462089 \cdot 10^{-73}\right):\\ \;\;\;\;\left(\ell \cdot \frac{\sqrt{2}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \frac{\sqrt{2}}{\sin k \cdot \left(\left({\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\tan k \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\left(\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\left(\sin k\right)}^{2}}{\ell}\right) \cdot {\left(\frac{1}{{\left({\left(\frac{1}{t}\right)}^{1}\right)}^{1}}\right)}^{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))