Average Error: 48.7 → 7.4
Time: 29.4s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -9.946564707500091 \cdot 10^{+42} \lor \neg \left(k \leq 2.3080607327321946 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{1}{\frac{k}{\cos k}} \cdot \frac{2}{\frac{\frac{k}{\ell} \cdot t_1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t_1}{\ell}}\\ \end{array} \]
\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}\\
\mathbf{if}\;k \leq -9.946564707500091 \cdot 10^{+42} \lor \neg \left(k \leq 2.3080607327321946 \cdot 10^{+185}\right):\\
\;\;\;\;\frac{1}{\frac{k}{\cos k}} \cdot \frac{2}{\frac{\frac{k}{\ell} \cdot t_1}{\ell}}\\

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


\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))))
   (if (or (<= k -9.946564707500091e+42) (not (<= k 2.3080607327321946e+185)))
     (* (/ 1.0 (/ k (cos k))) (/ 2.0 (/ (* (/ k l) t_1) l)))
     (* (* (/ (cos k) k) (/ l k)) (/ 2.0 (/ t_1 l))))))
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 tmp;
	if ((k <= -9.946564707500091e+42) || !(k <= 2.3080607327321946e+185)) {
		tmp = (1.0 / (k / cos(k))) * (2.0 / (((k / l) * t_1) / l));
	} else {
		tmp = ((cos(k) / k) * (l / k)) * (2.0 / (t_1 / l));
	}
	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 k < -9.9465647075000914e42 or 2.3080607327321946e185 < k

    1. Initial program 41.9

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

    2. Simplified35.6

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

    3. Taylor expanded in t around 0 22.1

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

    4. Applied unpow2_binary6422.1

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

    5. Applied associate-*l*_binary6418.7

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

    6. Applied times-frac_binary6415.9

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

    7. Applied *-un-lft-identity_binary6415.9

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

    8. Applied times-frac_binary6415.8

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

    9. Applied sqr-pow_binary6415.8

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

    10. Applied associate-/r*_binary6410.0

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

    11. Simplified5.5

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

    if -9.9465647075000914e42 < k < 2.3080607327321946e185

    1. Initial program 55.3

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

    2. Simplified46.3

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

    3. Taylor expanded in t around 0 24.7

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

    4. Applied unpow2_binary6424.7

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

    5. Applied associate-*l*_binary6423.8

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

    6. Applied times-frac_binary6422.1

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

    7. Applied *-un-lft-identity_binary6422.1

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

    8. Applied times-frac_binary6422.1

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

    9. Applied unpow2_binary6422.1

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

    10. Applied times-frac_binary6411.0

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

    11. Applied *-un-lft-identity_binary6411.0

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

    12. Applied times-frac_binary6410.9

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

    13. Applied associate-*r*_binary649.3

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

    14. Simplified9.2

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

  4. Final simplification7.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.946564707500091 \cdot 10^{+42} \lor \neg \left(k \leq 2.3080607327321946 \cdot 10^{+185}\right):\\ \;\;\;\;\frac{1}{\frac{k}{\cos k}} \cdot \frac{2}{\frac{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\cos k}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))