Average Error: 48.7 → 7.5
Time: 31.0s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \leq -1.2700187514190925 \cdot 10^{-51} \lor \neg \left(k \leq 7.410408846849746 \cdot 10^{-103}\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left({\left({\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1} \cdot \left(\sqrt[3]{\ell} \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{{\left({t}^{1}\right)}^{1} \cdot {\left({k}^{2}\right)}^{1}}\right)}^{1} \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) - 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\ \end{array}\]
\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}\;k \leq -1.2700187514190925 \cdot 10^{-51} \lor \neg \left(k \leq 7.410408846849746 \cdot 10^{-103}\right):\\
\;\;\;\;\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left({\left({\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1} \cdot \left(\sqrt[3]{\ell} \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;{\left(\frac{1}{{\left({t}^{1}\right)}^{1} \cdot {\left({k}^{2}\right)}^{1}}\right)}^{1} \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) - 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\

\end{array}
double code(double t, double l, double k) {
	return (2.0 / ((double) (((double) (((double) ((((double) pow(t, 3.0)) / ((double) (l * l))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow((k / t), 2.0)))) - 1.0)))));
}
↓
double code(double t, double l, double k) {
	double VAR;
	if (((k <= -1.2700187514190925e-51) || !(k <= 7.410408846849746e-103))) {
		VAR = ((double) (l * ((double) (2.0 * ((double) (((double) pow(((double) pow(((double) pow(k, (((double) -(2.0)) / 2.0))), 1.0)), 1.0)) * ((double) (((double) (((double) cbrt(l)) * ((double) cbrt(l)))) * ((double) (((double) pow(((double) pow(((double) pow(t, ((double) -(1.0)))), 1.0)), 1.0)) * ((double) (((double) cbrt(l)) * ((double) (((double) pow(((double) pow(((double) pow(k, (((double) -(2.0)) / 2.0))), 1.0)), 1.0)) * (((double) cos(k)) / ((double) pow(((double) sin(k)), 2.0)))))))))))))))));
	} else {
		VAR = ((double) (((double) pow((1.0 / ((double) (((double) pow(((double) pow(t, 1.0)), 1.0)) * ((double) pow(((double) pow(k, 2.0)), 1.0))))), 1.0)) * ((double) (((double) (2.0 * ((double) (l * (l / ((double) (k * k))))))) - ((double) (0.3333333333333333 * ((double) (l * l))))))));
	}
	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 k < -1.27001875141909251e-51 or 7.4104088468497459e-103 < k

    1. Initial program Error: 46.7 bits

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

    2. SimplifiedError: 36.3 bits

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

    3. Taylor expanded around inf Error: 51.7 bits

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

    4. SimplifiedError: 13.8 bits

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

    6. Applied unpow-prod-downError: 13.8 bits

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

    7. Applied associate-*l*Error: 11.7 bits

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

    8. SimplifiedError: 12.0 bits

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

    10. Applied sqr-powError: 12.1 bits

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

    11. Applied unpow-prod-downError: 12.1 bits

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

    12. Applied unpow-prod-downError: 12.1 bits

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

    13. Applied associate-*l*Error: 8.2 bits

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

    14. SimplifiedError: 5.8 bits

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

    16. Applied add-cube-cbrtError: 6.1 bits

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

    17. Applied associate-*l*Error: 6.1 bits

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

    18. SimplifiedError: 4.6 bits

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

    if -1.27001875141909251e-51 < k < 7.4104088468497459e-103

    1. Initial program Error: 63.7 bits

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

    2. SimplifiedError: 62.5 bits

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

    3. Taylor expanded around 0 Error: 61.8 bits

      \[\leadsto \color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot {\left(\frac{1}{{\left(e^{1 \cdot \left(\log t + \log 1\right)}\right)}^{1} \cdot {\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1}\right) - 0.3333333333333333 \cdot \left({\left(\frac{1}{{\left(e^{1 \cdot \left(\log t + \log 1\right)}\right)}^{1} \cdot {\left(e^{2 \cdot \left(\log 1 + \log k\right)}\right)}^{1}}\right)}^{1} \cdot {\ell}^{2}\right)}\]

    4. SimplifiedError: 29.8 bits

      \[\leadsto \color{blue}{{\left(\frac{1}{{\left(1 \cdot {t}^{1}\right)}^{1} \cdot {\left(1 \cdot {k}^{2}\right)}^{1}}\right)}^{1} \cdot \left(2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \ell\right) - 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 7.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.2700187514190925 \cdot 10^{-51} \lor \neg \left(k \leq 7.410408846849746 \cdot 10^{-103}\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \left({\left({\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1} \cdot \left(\sqrt[3]{\ell} \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{{\left({t}^{1}\right)}^{1} \cdot {\left({k}^{2}\right)}^{1}}\right)}^{1} \cdot \left(2 \cdot \left(\ell \cdot \frac{\ell}{k \cdot k}\right) - 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))