Average Error: 47.6 → 23.9
Time: 21.6s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.4547928767225641 \cdot 10^{-309}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{2} \cdot \left({\left(\frac{{-1}^{5}}{{\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{2} \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1}\right)\right) \cdot \sin k}\\ \end{array}\]
\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 -5.4547928767225641 \cdot 10^{-309}:\\
\;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{2} \cdot \left({\left(\frac{{-1}^{5}}{{\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \sin k}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{2} \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1}\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 <= -5.454792876722564e-309)) {
		VAR = ((double) (((double) (2.0 * ((double) (l * l)))) * ((double) (1.0 / ((double) (((double) (((double) pow(k, 2.0)) * ((double) (((double) pow(((double) (((double) pow(-1.0, 5.0)) / ((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / t)))))))))), 1.0)))), 1.0)) * ((double) (((double) sin(k)) / ((double) cos(k)))))))) * ((double) sin(k))))))));
	} else {
		VAR = ((double) (((double) (2.0 * ((double) (l * l)))) * ((double) (1.0 / ((double) (((double) (((double) pow(k, 2.0)) * ((double) (((double) (((double) sin(k)) / ((double) cos(k)))) * ((double) pow(((double) (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) 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 < -5.4547928767225641e-309

    1. Initial program 47.2

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

    2. Simplified38.2

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

    4. Applied div-inv38.2

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

    6. Applied div-inv38.2

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

    7. Applied unpow-prod-down46.2

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

    8. Applied associate-*l*45.4

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

    9. Taylor expanded around -inf 22.9

      \[\leadsto \left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{2} \cdot \color{blue}{\left({\left(\frac{{-1}^{5}}{{\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)}\right) \cdot \sin k}\]

    if -5.4547928767225641e-309 < t

    1. Initial program 48.1

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

    2. Simplified40.8

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

    4. Applied div-inv40.8

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

    6. Applied div-inv40.8

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

    7. Applied unpow-prod-down47.3

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

    8. Applied associate-*l*46.4

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

    9. Taylor expanded around inf 25.5

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

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.4547928767225641 \cdot 10^{-309}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{2} \cdot \left({\left(\frac{{-1}^{5}}{{\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\left({k}^{2} \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1}\right)\right) \cdot \sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020157 
(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))))