Average Error: 32.8 → 17.2
Time: 9.9s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -3.443483628374146 \cdot 10^{-199} \lor \neg \left(t \leq 8.168012831157414 \cdot 10^{-142}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sin k \cdot \frac{\sqrt[3]{t}}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot {t}^{3}}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \end{array}\]
\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 -3.443483628374146 \cdot 10^{-199} \lor \neg \left(t \leq 8.168012831157414 \cdot 10^{-142}\right):\\
\;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sin k \cdot \frac{\sqrt[3]{t}}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot {t}^{3}}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\

\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 -3.443483628374146e-199) (not (<= t 8.168012831157414e-142)))
   (/
    2.0
    (*
     (*
      (* (* t (/ t l)) (* (* (cbrt t) (cbrt t)) (* (sin k) (/ (cbrt t) l))))
      (tan k))
     (+ 2.0 (pow (/ k t) 2.0))))
   (/
    2.0
    (+
     (/ (* (* k k) (* t (pow (sin k) 2.0))) (* (* l l) (cos k)))
     (* 2.0 (/ (* (pow (sin k) 2.0) (pow t 3.0)) (* (* l l) (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 <= -3.443483628374146e-199) || !(t <= 8.168012831157414e-142)) {
		tmp = 2.0 / ((((t * (t / l)) * ((cbrt(t) * cbrt(t)) * (sin(k) * (cbrt(t) / l)))) * tan(k)) * (2.0 + pow((k / t), 2.0)));
	} else {
		tmp = 2.0 / ((((k * k) * (t * pow(sin(k), 2.0))) / ((l * l) * cos(k))) + (2.0 * ((pow(sin(k), 2.0) * pow(t, 3.0)) / ((l * l) * 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 < -3.4434836283741459e-199 or 8.16801283115741363e-142 < t

    1. Initial program 27.3

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

    2. Simplified27.3

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

    4. Applied unpow3_binary6427.3

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

    5. Applied times-frac_binary6419.1

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

    6. Applied associate-*l*_binary6416.9

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

    8. Applied *-un-lft-identity_binary6416.9

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

    9. Applied times-frac_binary6412.0

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

    10. Simplified12.0

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

    12. Applied *-un-lft-identity_binary6412.0

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

    13. Applied add-cube-cbrt_binary6412.2

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

    14. Applied times-frac_binary6412.2

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

    15. Applied associate-*l*_binary6413.1

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

    16. Simplified13.1

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

    if -3.4434836283741459e-199 < t < 8.16801283115741363e-142

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around inf 41.0

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

    4. Simplified41.0

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

  4. Final simplification17.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.443483628374146 \cdot 10^{-199} \lor \neg \left(t \leq 8.168012831157414 \cdot 10^{-142}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sin k \cdot \frac{\sqrt[3]{t}}{\ell}\right)\right)\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\left(\ell \cdot \ell\right) \cdot \cos k} + 2 \cdot \frac{{\sin k}^{2} \cdot {t}^{3}}{\left(\ell \cdot \ell\right) \cdot \cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020220 
(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))))