Average Error: 48.2 → 29.5
Time: 42.6s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -8.77596861926105034 \cdot 10^{-80} \lor \neg \left(t \le 1.02748589139423835 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \end{array}\]
\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 -8.77596861926105034 \cdot 10^{-80} \lor \neg \left(t \le 1.02748589139423835 \cdot 10^{-95}\right):\\
\;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \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 <= -8.77596861926105e-80) || !(t <= 1.0274858913942383e-95))) {
		VAR = ((double) (((double) (1.0 / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))) * ((double) (l / ((double) sin(k))))))));
	} else {
		VAR = ((double) log(((double) pow(((double) exp(((double) (((double) (2.0 * 1.0)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))), ((double) (((double) (l * l)) / ((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 t < -8.77596861926105034e-80 or 1.02748589139423835e-95 < t

    1. Initial program 43.0

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

    2. Simplified31.9

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

    4. Applied sqr-pow31.9

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

    5. Applied associate-*l*27.5

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

    7. Applied times-frac27.2

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

    9. Applied *-un-lft-identity27.2

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

    10. Applied times-frac27.3

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

    11. Applied associate-*l*26.3

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

    13. Applied *-un-lft-identity26.3

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

    14. Applied times-frac25.5

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

    15. Applied associate-*r*21.8

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

    16. Simplified21.7

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

    if -8.77596861926105034e-80 < t < 1.02748589139423835e-95

    1. Initial program 61.0

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

    2. Simplified61.0

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

    4. Applied sqr-pow61.0

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

    5. Applied associate-*l*60.0

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

    7. Applied times-frac60.0

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

    9. Applied add-log-exp61.7

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

    10. Simplified48.8

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

  4. Final simplification29.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -8.77596861926105034 \cdot 10^{-80} \lor \neg \left(t \le 1.02748589139423835 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020164 
(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))))