Average Error: 32.5 → 13.5
Time: 16.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -4.995686352806614 \cdot 10^{-145} \lor \neg \left(t \leq 3.400435998125358 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{-\frac{\ell}{t}} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(-1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]
\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 -4.995686352806614 \cdot 10^{-145} \lor \neg \left(t \leq 3.400435998125358 \cdot 10^{-56}\right):\\
\;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{-\frac{\ell}{t}} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(-1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\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 -4.995686352806614e-145) (not (<= t 3.400435998125358e-56)))
   (/
    2.0
    (*
     (* (tan k) (* (/ t (- (/ l t))) (* (sin k) (/ t l))))
     (- -1.0 (+ 1.0 (pow (/ k t) 2.0)))))
   (* 2.0 (/ (* (cos k) (* l l)) (* (* k k) (* t (pow (sin k) 2.0)))))))
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 <= -4.995686352806614e-145) || !(t <= 3.400435998125358e-56)) {
		tmp = 2.0 / ((tan(k) * ((t / -(l / t)) * (sin(k) * (t / l)))) * (-1.0 - (1.0 + pow((k / t), 2.0))));
	} else {
		tmp = 2.0 * ((cos(k) * (l * l)) / ((k * k) * (t * pow(sin(k), 2.0))));
	}
	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 < -4.9956863528066138e-145 or 3.4004359981253578e-56 < t

    1. Initial program 24.4

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

    3. Applied add-cube-cbrt_binary6424.5

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

    4. Applied unpow-prod-down_binary6424.5

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

    5. Applied times-frac_binary6417.3

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

    6. Simplified17.2

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

    7. Simplified17.0

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

    9. Applied add-cube-cbrt_binary6417.2

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

    10. Applied associate-/r*_binary6417.2

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

    11. Simplified13.9

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

    13. Applied frac-2neg_binary6413.9

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

    14. Applied distribute-frac-neg_binary6413.9

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

    15. Simplified12.7

      \[\leadsto \frac{2}{\left(\left(\left(\left(-\color{blue}{\frac{t}{-\frac{\ell}{t}}}\right) \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
    16. Using strategy rm

    17. Applied associate-*l*_binary649.7

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

    18. Simplified9.7

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

    if -4.9956863528066138e-145 < t < 3.4004359981253578e-56

    1. Initial program 59.1

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

    3. Applied add-cube-cbrt_binary6459.1

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

    4. Applied unpow-prod-down_binary6459.1

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

    5. Applied times-frac_binary6452.3

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

    6. Simplified52.2

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

    7. Simplified52.2

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

    9. Applied add-cube-cbrt_binary6452.2

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

    10. Applied associate-/r*_binary6452.2

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

    11. Simplified45.0

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

    13. Applied frac-2neg_binary6445.0

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

    14. Applied distribute-frac-neg_binary6445.0

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

    15. Simplified43.3

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

    16. Taylor expanded around 0 25.7

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

    17. Simplified25.7

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

  4. Final simplification13.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.995686352806614 \cdot 10^{-145} \lor \neg \left(t \leq 3.400435998125358 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\frac{t}{-\frac{\ell}{t}} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right) \cdot \left(-1 - \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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))))