Average Error: 48.0 → 26.3
Time: 23.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 5.1946241434167455 \cdot 10^{227}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k\right)\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k}}{\sin k}\\ \end{array}\]
\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}\;\ell \cdot \ell \le 5.1946241434167455 \cdot 10^{227}:\\
\;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k\right)\right)\right) \cdot \sin k}\\

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

\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 ((((double) (l * l)) <= 5.1946241434167455e+227)) {
		VAR = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) pow(((double) cbrt(t)), 3.0)))) * ((double) pow(((double) cbrt(t)), 3.0)))) * ((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) tan(k)))))))) * ((double) sin(k))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), 3.0)))))) * ((double) (l / ((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) tan(k)))))))) / ((double) sin(k))));
	}
	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 (* l l) < 5.1946241434167455e227

    1. Initial program 44.9

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

    2. Simplified35.7

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

    4. Applied sqr-pow35.7

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

    5. Applied associate-*l*31.3

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

    7. Applied add-cube-cbrt31.4

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

    8. Applied unpow-prod-down31.4

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

    9. Applied associate-*l*31.2

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

    11. Applied associate-*r*26.9

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

    13. Applied unpow-prod-down26.9

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

    14. Applied associate-*r*21.1

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

    if 5.1946241434167455e227 < (* l l)

    1. Initial program 60.6

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

    2. Simplified59.1

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

    4. Applied sqr-pow59.1

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

    5. Applied associate-*l*58.3

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

    7. Applied add-cube-cbrt58.4

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

    8. Applied unpow-prod-down58.4

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

    9. Applied associate-*l*58.4

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

    11. Applied associate-*r*57.5

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

    13. Applied associate-/r*57.5

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

    14. Simplified47.7

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 5.1946241434167455 \cdot 10^{227}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k\right)\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k}}{\sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020152 
(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))))