Average Error: 32.8 → 13.7
Time: 30.8s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -7.571105654180937 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \sin k}} \cdot \tan k\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 3.704175901510959 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \frac{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}{\frac{\ell}{t}}\right)}\\ \end{array} \]
\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}
t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;t \leq -7.571105654180937 \cdot 10^{-19}:\\
\;\;\;\;\frac{2}{\left(\frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \sin k}} \cdot \tan k\right) \cdot t_1}\\

\mathbf{elif}\;t \leq 3.704175901510959 \cdot 10^{-156}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \frac{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}{\frac{\ell}{t}}\right)}\\


\end{array}

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
↓
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -7.571105654180937e-19)
     (/ 2.0 (* (* (/ t (* (/ l t) (/ l (* t (sin k))))) (tan k)) t_1))
     (if (<= t 3.704175901510959e-156)
       (*
        2.0
        (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
       (/ 2.0 (* t_1 (* (tan k) (/ (* t (* (sin k) (/ t l))) (/ l t)))))))))
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 t_1 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -7.571105654180937e-19) {
		tmp = 2.0 / (((t / ((l / t) * (l / (t * sin(k))))) * tan(k)) * t_1);
	} else if (t <= 3.704175901510959e-156) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / (t_1 * (tan(k) * ((t * (sin(k) * (t / l))) / (l / t))));
	}
	return tmp;
}

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 < -7.571105654180937e-19

    1. Initial program 23.3

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

    2. Simplified23.3

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

    3. Applied add-cube-cbrt_binary6423.5

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

    4. Applied unpow-prod-down_binary6423.5

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

    5. Applied times-frac_binary6416.5

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

    6. Simplified16.3

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

    7. Simplified16.2

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

    8. Applied associate-/l*_binary6411.2

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

    9. Applied associate-*l*_binary648.4

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

    10. Applied associate-*l/_binary648.4

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

    11. Applied frac-times_binary6413.8

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

    12. Applied associate-/l*_binary6411.0

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

    13. Simplified7.8

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

    if -7.571105654180937e-19 < t < 3.70417590151095916e-156

    1. Initial program 55.9

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

    2. Simplified55.9

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

    3. Applied add-cube-cbrt_binary6456.0

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

    4. Applied unpow-prod-down_binary6456.0

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

    5. Applied times-frac_binary6451.0

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

    6. Simplified50.9

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

    7. Simplified50.8

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

    8. Applied associate-/l*_binary6442.0

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

    9. Taylor expanded in t around 0 27.5

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

    if 3.70417590151095916e-156 < t

    1. Initial program 26.1

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

    2. Simplified26.1

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

    3. Applied add-cube-cbrt_binary6426.3

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

    4. Applied unpow-prod-down_binary6426.3

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

    5. Applied times-frac_binary6418.2

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

    6. Simplified18.0

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

    7. Simplified17.9

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

    8. Applied associate-/l*_binary6414.1

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

    9. Applied associate-*l*_binary6411.1

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

    10. Applied associate-*l/_binary6410.1

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

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.571105654180937 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot \sin k}} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 3.704175901510959 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \frac{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(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))))