Average Error: 32.6 → 10.0
Time: 18.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -198860.82714289907 \lor \neg \left(t \leq 2.3156856470715744 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{1}{t \cdot \frac{t \cdot \sin k}{\ell}} \cdot \left(\frac{2}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \end{array} \]
\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}
\mathbf{if}\;t \leq -198860.82714289907 \lor \neg \left(t \leq 2.3156856470715744 \cdot 10^{-81}\right):\\
\;\;\;\;\frac{1}{t \cdot \frac{t \cdot \sin k}{\ell}} \cdot \left(\frac{2}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\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
 (if (or (<= t -198860.82714289907) (not (<= t 2.3156856470715744e-81)))
   (*
    (/ 1.0 (* t (/ (* t (sin k)) l)))
    (* (/ 2.0 (+ 1.0 (+ 1.0 (pow (/ k t) 2.0)))) (/ (/ l t) (tan k))))
   (/
    2.0
    (*
     (/ (pow (sin k) 2.0) (* l l))
     (+ (/ (* t (* k k)) (cos k)) (* 2.0 (/ (pow t 3.0) (cos k))))))))
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 ((t <= -198860.82714289907) || !(t <= 2.3156856470715744e-81)) {
		tmp = (1.0 / (t * ((t * sin(k)) / l))) * ((2.0 / (1.0 + (1.0 + pow((k / t), 2.0)))) * ((l / t) / tan(k)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / (l * l)) * (((t * (k * k)) / cos(k)) + (2.0 * (pow(t, 3.0) / cos(k)))));
	}
	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 t < -198860.82714289907 or 2.31568564707157445e-81 < t

    1. Initial program 22.9

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

    2. Applied cube-mult_binary6422.9

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

    3. Applied times-frac_binary6416.6

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

    4. Applied associate-*l*_binary6414.4

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

    5. Applied *-un-lft-identity_binary6414.4

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

    6. Applied times-frac_binary648.9

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

    7. Applied associate-*l*_binary647.4

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

    8. Applied *-un-lft-identity_binary647.4

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

    9. Applied times-frac_binary647.4

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

    10. Simplified4.3

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

    11. Simplified4.3

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

    12. Applied *-un-lft-identity_binary644.3

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

    13. Applied times-frac_binary643.6

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

    14. Applied associate-*l*_binary642.8

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

    15. Simplified2.8

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

    if -198860.82714289907 < t < 2.31568564707157445e-81

    1. Initial program 53.4

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

    2. Applied cube-mult_binary6453.4

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

    3. Applied times-frac_binary6445.0

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

    4. Applied associate-*l*_binary6444.1

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

    5. Taylor expanded in t around 0 36.1

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

    6. Simplified25.5

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -198860.82714289907 \lor \neg \left(t \leq 2.3156856470715744 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{1}{t \cdot \frac{t \cdot \sin k}{\ell}} \cdot \left(\frac{2}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{\frac{\ell}{t}}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\ell \cdot \ell} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\cos k} + 2 \cdot \frac{{t}^{3}}{\cos k}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(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))))