Average Error: 48.3 → 30.6
Time: 42.8s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.804694207801877 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{elif}\;t \le -5.9337126311956078 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 5.832499001294496 \cdot 10^{-124}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \end{array}\]
\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 -1.804694207801877 \cdot 10^{-108}:\\
\;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\

\mathbf{elif}\;t \le -5.9337126311956078 \cdot 10^{-309}:\\
\;\;\;\;2 \cdot \left({\left(\frac{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\

\mathbf{elif}\;t \le 5.832499001294496 \cdot 10^{-124}:\\
\;\;\;\;\frac{2}{{\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\

\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 <= -1.804694207801877e-108)) {
		VAR = ((double) (((double) (1.0 / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))) * ((double) (l * ((double) (l / ((double) sin(k))))))))));
	} else {
		double VAR_1;
		if ((t <= -5.93371263119561e-309)) {
			VAR_1 = ((double) (2.0 * ((double) (((double) pow(((double) (((double) (((double) pow(((double) exp(((double) (2.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / k)))))))))), 1.0)) * ((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / t)))))))))), 1.0)))) / ((double) pow(-1.0, 3.0)))), 1.0)) * ((double) (((double) (((double) cos(k)) * ((double) pow(l, 2.0)))) / ((double) pow(((double) sin(k)), 2.0))))))));
		} else {
			double VAR_2;
			if ((t <= 5.832499001294496e-124)) {
				VAR_2 = ((double) (((double) (2.0 / ((double) (((double) pow(((double) (1.0 / ((double) (((double) pow(((double) exp(((double) (2.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (1.0 / k)))))))))), 1.0)) * ((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(((double) (1.0 / t)))) + ((double) log(1.0)))))))), 1.0)))))), 1.0)) * ((double) (((double) sin(k)) / ((double) cos(k)))))))) * ((double) (l * ((double) (l / ((double) sin(k))))))));
			} else {
				VAR_2 = ((double) (((double) (1.0 / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))) * ((double) (l * ((double) (l / ((double) sin(k))))))))));
			}
			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 < -1.804694207801877e-108 or 5.832499001294496e-124 < t

    1. Initial program 42.7

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

    2. Simplified32.5

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

    4. Applied sqr-pow32.5

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

    5. Applied associate-*l*27.4

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

    7. Applied times-frac27.0

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

    9. Applied *-un-lft-identity27.0

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

    10. Applied times-frac26.8

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

    11. Simplified26.8

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

    13. Applied *-un-lft-identity26.8

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

    14. Applied times-frac26.9

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

    15. Applied associate-*l*25.6

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

    if -1.804694207801877e-108 < t < -5.9337126311956078e-309

    1. Initial program 63.9

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

    2. Simplified63.9

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

    3. Taylor expanded around -inf 44.5

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

    if -5.9337126311956078e-309 < t < 5.832499001294496e-124

    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

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

    4. Applied sqr-pow64.0

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

    5. Applied associate-*l*64.0

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

    7. Applied times-frac64.0

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

    9. Applied *-un-lft-identity64.0

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

    10. Applied times-frac64.0

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

    11. Simplified64.0

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

    12. Taylor expanded around inf 45.1

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

  4. Final simplification30.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.804694207801877 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{elif}\;t \le -5.9337126311956078 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 5.832499001294496 \cdot 10^{-124}:\\ \;\;\;\;\frac{2}{{\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))))