Average Error: 48.5 → 27.1
Time: 43.4s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.9313676215627785 \cdot 10^{120}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le -7.7911602116379966 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\ \mathbf{elif}\;t \le 2.0255652128622138 \cdot 10^{-101}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 4.95901555545514021 \cdot 10^{102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \end{array}\]
\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}\;t \le -2.9313676215627785 \cdot 10^{120}:\\
\;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\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(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\

\mathbf{elif}\;t \le -7.7911602116379966 \cdot 10^{-98}:\\
\;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\

\mathbf{elif}\;t \le 2.0255652128622138 \cdot 10^{-101}:\\
\;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\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(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\

\mathbf{elif}\;t \le 4.95901555545514021 \cdot 10^{102}:\\
\;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\

\mathbf{else}:\\
\;\;\;\;\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 {t}^{\left(\frac{3}{2}\right)}\right) \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot \tan k\right)\right) \cdot \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 ((t <= -2.9313676215627785e+120)) {
		VAR = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((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) pow(((double) cbrt(t)), 3.0)))) * ((double) tan(k)))))) * ((double) sin(k))))));
	} else {
		double VAR_1;
		if ((t <= -7.791160211637997e-98)) {
			VAR_1 = ((double) (((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) pow(t, 3.0)))))) * ((double) (l / ((double) tan(k)))))) / ((double) sin(k))));
		} else {
			double VAR_2;
			if ((t <= 2.0255652128622138e-101)) {
				VAR_2 = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((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) pow(((double) cbrt(t)), 3.0)))) * ((double) tan(k)))))) * ((double) sin(k))))));
			} else {
				double VAR_3;
				if ((t <= 4.95901555545514e+102)) {
					VAR_3 = ((double) (((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) pow(t, 3.0)))))) * ((double) (l / ((double) tan(k)))))) / ((double) sin(k))));
				} else {
					VAR_3 = ((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(t, ((double) (3.0 / 2.0)))))) * ((double) pow(t, ((double) (3.0 / 2.0)))))) * ((double) tan(k)))))) * ((double) sin(k))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	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 3 regimes

  2. if t < -2.9313676215627785e120 or -7.7911602116379966e-98 < t < 2.0255652128622138e-101

    1. Initial program 59.5

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

    2. Simplified53.9

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

    4. Applied sqr-pow53.9

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

    5. Applied associate-*l*50.1

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

    7. Applied associate-*r*50.0

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

    9. Applied add-cube-cbrt50.0

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

    10. Applied unpow-prod-down50.0

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

    11. Applied associate-*r*43.0

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

    13. Applied unpow-prod-down43.0

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

    14. Applied associate-*r*33.6

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

    if -2.9313676215627785e120 < t < -7.7911602116379966e-98 or 2.0255652128622138e-101 < t < 4.95901555545514021e102

    1. Initial program 31.8

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

    2. Simplified24.3

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

    4. Applied sqr-pow24.3

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

    5. Applied associate-*l*21.9

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

    7. Applied associate-*r*21.8

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

    9. Applied associate-/r*21.4

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

    10. Simplified18.3

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

    if 4.95901555545514021e102 < t

    1. Initial program 52.6

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

    2. Simplified38.0

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

    4. Applied sqr-pow38.0

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

    5. Applied associate-*l*30.7

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

    7. Applied associate-*r*30.7

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

    9. Applied sqr-pow30.7

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

    10. Applied associate-*r*27.8

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.9313676215627785 \cdot 10^{120}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le -7.7911602116379966 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\ \mathbf{elif}\;t \le 2.0255652128622138 \cdot 10^{-101}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 4.95901555545514021 \cdot 10^{102}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(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))))