Average Error: 32.1 → 10.0
Time: 16.2s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -5.032329797783243 \cdot 10^{-164} \lor \neg \left(t \leq 1.4087779612255968 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \end{array} \]
\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 -5.032329797783243 \cdot 10^{-164} \lor \neg \left(t \leq 1.4087779612255968 \cdot 10^{-70}\right):\\
\;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\


\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 -5.032329797783243e-164) (not (<= t 1.4087779612255968e-70)))
   (/
    2.0
    (/
     (* (* (* t (/ (* t (sin k)) l)) (tan k)) (+ 2.0 (pow (/ k t) 2.0)))
     (/ l t)))
   (/ 2.0 (/ (/ (* (pow k 2.0) (pow (sin k) 2.0)) (* l (cos k))) (/ l t)))))
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 <= -5.032329797783243e-164) || !(t <= 1.4087779612255968e-70)) {
		tmp = 2.0 / ((((t * ((t * sin(k)) / l)) * tan(k)) * (2.0 + pow((k / t), 2.0))) / (l / t));
	} else {
		tmp = 2.0 / (((pow(k, 2.0) * pow(sin(k), 2.0)) / (l * cos(k))) / (l / t));
	}
	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 < -5.0323297977832429e-164 or 1.4087779612255968e-70 < t

    1. Initial program 24.5

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

    2. Simplified24.5

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

    3. Applied unpow3_binary6424.5

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

    4. Applied times-frac_binary6417.1

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

    5. Applied associate-*l*_binary6415.1

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

    6. Applied associate-/l*_binary6410.5

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

    7. Applied associate-*l/_binary649.5

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

    8. Applied associate-*l/_binary647.4

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

    9. Applied associate-*l/_binary646.4

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

    10. Applied *-un-lft-identity_binary646.4

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

    11. Applied *-un-lft-identity_binary646.4

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

    12. Applied times-frac_binary646.4

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

    13. Applied associate-*l*_binary646.4

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

    14. Simplified6.3

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

    if -5.0323297977832429e-164 < t < 1.4087779612255968e-70

    1. Initial program 60.3

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

    2. Simplified60.3

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

    3. Applied unpow3_binary6460.3

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

    4. Applied times-frac_binary6454.3

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

    5. Applied associate-*l*_binary6454.1

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

    6. Applied associate-/l*_binary6445.0

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

    7. Applied associate-*l/_binary6445.0

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

    8. Applied associate-*l/_binary6445.7

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

    9. Applied associate-*l/_binary6442.1

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

    10. Taylor expanded in t around 0 23.6

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.032329797783243 \cdot 10^{-164} \lor \neg \left(t \leq 1.4087779612255968 \cdot 10^{-70}\right):\\ \;\;\;\;\frac{2}{\frac{\left(\left(t \cdot \frac{t \cdot \sin k}{\ell}\right) \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{2} \cdot {\sin k}^{2}}{\ell \cdot \cos k}}{\frac{\ell}{t}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022039 
(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))))