Average Error: 48.3 → 16.8
Time: 28.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := \frac{{t}^{1.5}}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 3.262102074554716 \cdot 10^{+303}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t_1 \cdot \left(\sin k \cdot t_1\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]
\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}
t_1 := \frac{{t}^{1.5}}{\ell}\\
\mathbf{if}\;\ell \cdot \ell \leq 3.262102074554716 \cdot 10^{+303}:\\
\;\;\;\;\frac{2}{\frac{k}{\frac{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\left(\left(t_1 \cdot \left(\sin k \cdot t_1\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
 (let* ((t_1 (/ (pow t 1.5) l)))
   (if (<= (* l l) 3.262102074554716e+303)
     (/ 2.0 (/ k (/ (* (* l l) (/ (cos k) k)) (* t (pow (sin k) 2.0)))))
     (/ 2.0 (* (* (* t_1 (* (sin k) t_1)) (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 = pow(t, 1.5) / l;
	double tmp;
	if ((l * l) <= 3.262102074554716e+303) {
		tmp = 2.0 / (k / (((l * l) * (cos(k) / k)) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = 2.0 / (((t_1 * (sin(k) * t_1)) * 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) < 3.26210207455471629e303

    1. Initial program 45.5

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

    2. Simplified36.3

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

    3. Taylor expanded in t around 0 15.5

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

    4. Applied unpow2_binary6415.5

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

    5. Applied associate-*l*_binary6413.1

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

    6. Applied associate-/l*_binary6410.6

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

    7. Applied associate-/r*_binary6410.1

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

    8. Simplified10.1

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

    if 3.26210207455471629e303 < (*.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 sqr-pow_binary6463.9

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

    4. Applied times-frac_binary6453.8

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

    5. Applied associate-*l*_binary6453.8

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

    6. Simplified53.8

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)}\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.262102074554716 \cdot 10^{+303}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{k}}{t \cdot {\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{1.5}}{\ell} \cdot \left(\sin k \cdot \frac{{t}^{1.5}}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))