Average Error: 32.4 → 16.8
Time: 13.7s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.5615747919334984 \cdot 10^{+82} \lor \neg \left(\ell \leq -1.1261426819451643 \cdot 10^{-89} \lor \neg \left(\ell \leq 1.0610732859761567 \cdot 10^{-149}\right) \land \ell \leq 2.604016357710232 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array}\]
\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}\;\ell \leq -1.5615747919334984 \cdot 10^{+82} \lor \neg \left(\ell \leq -1.1261426819451643 \cdot 10^{-89} \lor \neg \left(\ell \leq 1.0610732859761567 \cdot 10^{-149}\right) \land \ell \leq 2.604016357710232 \cdot 10^{+25}\right):\\
\;\;\;\;\frac{2}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\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 (<= l -1.5615747919334984e+82)
         (not
          (or (<= l -1.1261426819451643e-89)
              (and (not (<= l 1.0610732859761567e-149))
                   (<= l 2.604016357710232e+25)))))
   (/
    2.0
    (*
     (* t (* (* (/ t l) (* (/ t l) (sin k))) (tan k)))
     (+ 2.0 (pow (/ k t) 2.0))))
   (/
    2.0
    (*
     (/ (pow (sin k) 2.0) (cos k))
     (+ (/ (* t (* k k)) (* l l)) (* 2.0 (/ (pow t 3.0) (* l 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 tmp;
	if ((l <= -1.5615747919334984e+82) || !((l <= -1.1261426819451643e-89) || (!(l <= 1.0610732859761567e-149) && (l <= 2.604016357710232e+25)))) {
		tmp = 2.0 / ((t * (((t / l) * ((t / l) * sin(k))) * tan(k))) * (2.0 + pow((k / t), 2.0)));
	} else {
		tmp = 2.0 / ((pow(sin(k), 2.0) / cos(k)) * (((t * (k * k)) / (l * l)) + (2.0 * (pow(t, 3.0) / (l * 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 l < -1.5615747919334984e82 or -1.1261426819451643e-89 < l < 1.06107328597615667e-149 or 2.60401635771023186e25 < l

    1. Initial program 36.0

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

    2. Simplified36.0

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

    4. Applied unpow3_binary64_47736.0

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

    5. Applied times-frac_binary64_42027.0

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

    6. Applied associate-*l*_binary64_35725.6

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
    7. Using strategy rm

    8. Applied *-un-lft-identity_binary64_41425.6

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

    9. Applied times-frac_binary64_42016.9

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

    10. Simplified16.9

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{t} \cdot \frac{t}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
    11. Using strategy rm

    12. Applied associate-*l*_binary64_35716.6

      \[\leadsto \frac{2}{\left(\color{blue}{\left(t \cdot \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)\right)} \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\]
    13. Using strategy rm

    14. Applied associate-*l*_binary64_35717.4

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

    if -1.5615747919334984e82 < l < -1.1261426819451643e-89 or 1.06107328597615667e-149 < l < 2.60401635771023186e25

    1. Initial program 24.1

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

    2. Simplified24.1

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

    3. Taylor expanded around inf 15.9

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

    4. Simplified15.4

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.5615747919334984 \cdot 10^{+82} \lor \neg \left(\ell \leq -1.1261426819451643 \cdot 10^{-89} \lor \neg \left(\ell \leq 1.0610732859761567 \cdot 10^{-149}\right) \land \ell \leq 2.604016357710232 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{2}{\left(t \cdot \left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} + 2 \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020280 
(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))))