Average Error: 47.9 → 16.5
Time: 26.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := k \cdot \left(\sin k \cdot \sqrt{t}\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 9.250010914405445 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 1.0427932792743221 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{t_1}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
\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}
t_1 := k \cdot \left(\sin k \cdot \sqrt{t}\right)\\
\mathbf{if}\;\ell \cdot \ell \leq 9.250010914405445 \cdot 10^{+203}:\\
\;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}{\ell}}\\

\mathbf{elif}\;\ell \cdot \ell \leq 1.0427932792743221 \cdot 10^{+305}:\\
\;\;\;\;\frac{2}{t_1 \cdot \frac{t_1}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\left(\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\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
 (let* ((t_1 (* k (* (sin k) (sqrt t)))))
   (if (<= (* l l) 9.250010914405445e+203)
     (/ 2.0 (* (* (* k k) t) (/ (/ (pow (sin k) 2.0) (* l (cos k))) l)))
     (if (<= (* l l) 1.0427932792743221e+305)
       (/ 2.0 (* t_1 (/ t_1 (* (* l l) (cos k)))))
       (/
        2.0
        (*
         (* (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0)) (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 t_1 = k * (sin(k) * sqrt(t));
	double tmp;
	if ((l * l) <= 9.250010914405445e+203) {
		tmp = 2.0 / (((k * k) * t) * ((pow(sin(k), 2.0) / (l * cos(k))) / l));
	} else if ((l * l) <= 1.0427932792743221e+305) {
		tmp = 2.0 / (t_1 * (t_1 / ((l * l) * cos(k))));
	} else {
		tmp = 2.0 / (((sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)) * 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 3 regimes

  2. if (*.f64 l l) < 9.2500109144054447e203

    1. Initial program 44.8

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

    2. Simplified35.3

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

    3. Taylor expanded in t around 0 14.4

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

    4. Applied egg-rr13.1

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

    5. Applied egg-rr10.6

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

    if 9.2500109144054447e203 < (*.f64 l l) < 1.0427932792743221e305

    1. Initial program 47.7

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

    2. Simplified41.9

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

    3. Taylor expanded in t around 0 28.0

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

    4. Applied egg-rr33.6

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

    if 1.0427932792743221e305 < (*.f64 l l)

    1. Initial program 63.8

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

    2. Simplified63.8

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

    3. Applied egg-rr39.0

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 9.250010914405445 \cdot 10^{+203}:\\ \;\;\;\;\frac{2}{\left(\left(k \cdot k\right) \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 1.0427932792743221 \cdot 10^{+305}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(\sin k \cdot \sqrt{t}\right)\right) \cdot \frac{k \cdot \left(\sin k \cdot \sqrt{t}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))))