Average Error: 48.0 → 8.6
Time: 26.0s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ t_2 := \frac{\cos k}{k}\\ \mathbf{if}\;\ell \cdot \ell \leq 31752309304847.016 \lor \neg \left(\ell \cdot \ell \leq 2.28298209324412 \cdot 10^{+305}\right):\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\ell}{\frac{k}{\frac{\ell}{t_1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\frac{{\ell}^{2}}{k}}{t_1}\right)\\ \end{array} \]
\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 := t \cdot {\sin k}^{2}\\
t_2 := \frac{\cos k}{k}\\
\mathbf{if}\;\ell \cdot \ell \leq 31752309304847.016 \lor \neg \left(\ell \cdot \ell \leq 2.28298209324412 \cdot 10^{+305}\right):\\
\;\;\;\;2 \cdot \left(t_2 \cdot \frac{\ell}{\frac{k}{\frac{\ell}{t_1}}}\right)\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \left(t_2 \cdot \frac{\frac{{\ell}^{2}}{k}}{t_1}\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 (* t (pow (sin k) 2.0))) (t_2 (/ (cos k) k)))
   (if (or (<= (* l l) 31752309304847.016)
           (not (<= (* l l) 2.28298209324412e+305)))
     (* 2.0 (* t_2 (/ l (/ k (/ l t_1)))))
     (* 2.0 (* t_2 (/ (/ (pow l 2.0) k) t_1))))))
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 = t * pow(sin(k), 2.0);
	double t_2 = cos(k) / k;
	double tmp;
	if (((l * l) <= 31752309304847.016) || !((l * l) <= 2.28298209324412e+305)) {
		tmp = 2.0 * (t_2 * (l / (k / (l / t_1))));
	} else {
		tmp = 2.0 * (t_2 * ((pow(l, 2.0) / k) / t_1));
	}
	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) < 31752309304847.016 or 2.28298209324411989e305 < (*.f64 l l)

    1. Initial program 48.1

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

    2. Simplified40.7

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

    3. Taylor expanded in t around 0 24.0

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

    4. Applied unpow2_binary6424.0

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

    5. Applied associate-*l*_binary6423.1

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

    6. Applied times-frac_binary6422.3

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

    7. Applied unpow2_binary6422.3

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

    8. Applied associate-/l*_binary6414.4

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

    9. Simplified9.9

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

    if 31752309304847.016 < (*.f64 l l) < 2.28298209324411989e305

    1. Initial program 47.6

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

    2. Simplified39.0

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

    3. Taylor expanded in t around 0 18.6

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

    4. Applied unpow2_binary6418.6

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

    5. Applied associate-*l*_binary6412.2

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

    6. Applied times-frac_binary645.6

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

    7. Applied associate-/r*_binary643.7

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 31752309304847.016 \lor \neg \left(\ell \cdot \ell \leq 2.28298209324412 \cdot 10^{+305}\right):\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\ell}{\frac{k}{\frac{\ell}{t \cdot {\sin k}^{2}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k} \cdot \frac{\frac{{\ell}^{2}}{k}}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))