Average Error: 32.2 → 9.9
Time: 14.6s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 3.85487638616180756 \cdot 10^{48}:\\ \;\;\;\;\frac{2}{\frac{-\left(2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{t}^{3} \cdot \sin k}{\ell}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot k}{\frac{\frac{\ell}{\sin k}}{k}}\right)}{\frac{\ell}{\sin k} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \end{array}\]
\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 3.85487638616180756 \cdot 10^{48}:\\
\;\;\;\;\frac{2}{\frac{-\left(2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{t}^{3} \cdot \sin k}{\ell}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot k}{\frac{\frac{\ell}{\sin k}}{k}}\right)}{\frac{\ell}{\sin k} \cdot \cos k}}\\

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

\end{array}
double code(double t, double l, double k) {
	return (2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0)));
}
↓
double code(double t, double l, double k) {
	double VAR;
	if ((t <= 3.8548763861618076e+48)) {
		VAR = (2.0 / (-((2.0 * (pow((1.0 / pow(-1.0, 3.0)), 1.0) * ((pow(t, 3.0) * sin(k)) / l))) + (pow((1.0 / pow(-1.0, 3.0)), 1.0) * ((t * k) / ((l / sin(k)) / k)))) / ((l / sin(k)) * cos(k))));
	} else {
		VAR = (2.0 / ((pow(t, (3.0 / 2.0)) / l) * ((((pow(t, (3.0 / 2.0)) / l) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0))));
	}
	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 < 3.8548763861618076e+48

    1. Initial program 35.7

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

    3. Applied associate-/r*33.0

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

    4. Applied associate-*l/31.5

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

    6. Applied tan-quot31.6

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

    7. Applied associate-/l*31.6

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

    8. Applied frac-times31.3

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

    9. Applied associate-*l/30.5

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

    10. Taylor expanded around -inf 18.1

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

    12. Applied unpow218.1

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

    13. Applied associate-*l*18.1

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

    14. Applied associate-*r*14.4

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

    15. Applied associate-/l*10.0

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

    16. Simplified10.0

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

    if 3.8548763861618076e+48 < t

    1. Initial program 22.2

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

    3. Applied sqr-pow22.3

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

    4. Applied times-frac11.8

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

    5. Applied associate-*l*9.6

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

    6. Applied associate-*l*9.6

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

    7. Applied associate-*l*9.4

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.85487638616180756 \cdot 10^{48}:\\ \;\;\;\;\frac{2}{\frac{-\left(2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{t}^{3} \cdot \sin k}{\ell}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot k}{\frac{\frac{\ell}{\sin k}}{k}}\right)}{\frac{\ell}{\sin k} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))