Average Error: 48.5 → 15.3
Time: 27.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4.9928687176460827 \cdot 10^{+297}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]
\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}\;\ell \cdot \ell \leq 4.9928687176460827 \cdot 10^{+297}:\\
\;\;\;\;\frac{2}{\frac{k}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)}}}\\

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

\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
 (if (<= (* l l) 4.9928687176460827e+297)
   (/ 2.0 (/ k (/ (* (* l l) (cos k)) (* k (* (sin k) (* t (sin k)))))))
   (/
    2.0
    (* (* (* (sin k) (* (/ t l) (* t (/ t l)))) (tan k)) (pow (/ k t) 2.0)))))
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 tmp;
	if ((l * l) <= 4.9928687176460827e+297) {
		tmp = 2.0 / (k / (((l * l) * cos(k)) / (k * (sin(k) * (t * sin(k))))));
	} else {
		tmp = 2.0 / (((sin(k) * ((t / l) * (t * (t / l)))) * tan(k)) * pow((k / t), 2.0));
	}
	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 2 regimes

  2. if (*.f64 l l) < 4.99287e297

    1. Initial program 45.8

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

    2. Simplified36.5

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

    3. Taylor expanded around inf 16.1

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

    4. Simplified16.1

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

    6. Applied associate-*l*_binary6413.5

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

    8. Applied associate-/l*_binary6410.9

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

    10. Applied unpow2_binary6410.9

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

    11. Applied associate-*r*_binary6410.7

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

    if 4.99287e297 < (*.f64 l l)

    1. Initial program 63.5

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

    2. Simplified63.3

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

    4. Applied cube-mult_binary6463.3

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

    5. Applied times-frac_binary6450.0

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

    6. Simplified40.5

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

  4. Final simplification15.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4.9928687176460827 \cdot 10^{+297}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))