Average Error: 32.7 → 10.1
Time: 27.1s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -4.738216409662838 \cdot 10^{-164} \lor \neg \left(t \leq 1.0926168737272691 \cdot 10^{-193}\right):\\ \;\;\;\;\frac{1}{t \cdot \frac{t \cdot \sin k}{\ell}} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]
\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 -4.738216409662838 \cdot 10^{-164} \lor \neg \left(t \leq 1.0926168737272691 \cdot 10^{-193}\right):\\
\;\;\;\;\frac{1}{t \cdot \frac{t \cdot \sin k}{\ell}} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\

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


\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 -4.738216409662838e-164) (not (<= t 1.0926168737272691e-193)))
   (*
    (/ 1.0 (* t (/ (* t (sin k)) l)))
    (* (/ (/ l t) (tan k)) (/ 2.0 (+ 2.0 (pow (/ k t) 2.0)))))
   (/
    2.0
    (/ (* (pow k 2.0) (* t (pow (sin k) 2.0))) (* (cos k) (pow l 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 <= -4.738216409662838e-164) || !(t <= 1.0926168737272691e-193)) {
		tmp = (1.0 / (t * ((t * sin(k)) / l))) * (((l / t) / tan(k)) * (2.0 / (2.0 + pow((k / t), 2.0))));
	} else {
		tmp = 2.0 / ((pow(k, 2.0) * (t * pow(sin(k), 2.0))) / (cos(k) * pow(l, 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 < -4.738216409662838e-164 or 1.0926168737272691e-193 < t

    1. Initial program 27.7

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

    2. Simplified27.7

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

    3. Applied cube-mult_binary6427.7

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

    4. Applied times-frac_binary6419.4

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

    5. Applied associate-*l*_binary6417.3

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

    6. Applied *-un-lft-identity_binary6417.3

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

    7. Applied times-frac_binary6412.4

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

    8. Applied associate-*l*_binary6411.5

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

    9. Applied *-un-lft-identity_binary6411.5

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

    10. Applied times-frac_binary6411.6

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

    11. Simplified9.9

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

    12. Applied *-un-lft-identity_binary649.9

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

    13. Applied times-frac_binary648.9

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

    14. Applied associate-*l*_binary647.4

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

    if -4.738216409662838e-164 < t < 1.0926168737272691e-193

    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 in t around 0 26.9

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.738216409662838 \cdot 10^{-164} \lor \neg \left(t \leq 1.0926168737272691 \cdot 10^{-193}\right):\\ \;\;\;\;\frac{1}{t \cdot \frac{t \cdot \sin k}{\ell}} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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))))