Average Error: 48.2 → 9.4
Time: 25.6s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 3.26618261886 \cdot 10^{-313}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{k}}} \cdot \cos k}{t_1}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 1.2583695792669349 \cdot 10^{+60}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(k \cdot t_1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t_1}}}\\ \end{array} \]
\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}
t_1 := t \cdot {\sin k}^{2}\\
\mathbf{if}\;\ell \cdot \ell \leq 3.26618261886 \cdot 10^{-313}:\\
\;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{k}}} \cdot \cos k}{t_1}}}\\

\mathbf{elif}\;\ell \cdot \ell \leq 1.2583695792669349 \cdot 10^{+60}:\\
\;\;\;\;\frac{2}{\frac{1}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(k \cdot t_1\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{1}{\frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t_1}}}\\


\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
 (let* ((t_1 (* t (pow (sin k) 2.0))))
   (if (<= (* l l) 3.26618261886e-313)
     (/ 2.0 (/ 1.0 (/ (* (/ l (/ k (/ l k))) (cos k)) t_1)))
     (if (<= (* l l) 1.2583695792669349e+60)
       (/ 2.0 (/ 1.0 (/ (* (* l l) (cos k)) (* k (* k t_1)))))
       (/ 2.0 (/ 1.0 (/ (* (cos k) (* (/ l k) (/ l k))) t_1)))))))
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 t_1 = t * pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 3.26618261886e-313) {
		tmp = 2.0 / (1.0 / (((l / (k / (l / k))) * cos(k)) / t_1));
	} else if ((l * l) <= 1.2583695792669349e+60) {
		tmp = 2.0 / (1.0 / (((l * l) * cos(k)) / (k * (k * t_1))));
	} else {
		tmp = 2.0 / (1.0 / ((cos(k) * ((l / k) * (l / k))) / t_1));
	}
	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 3 regimes

  2. if (*.f64 l l) < 3.26618261886e-313

    1. Initial program 46.6

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

    2. Simplified37.9

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

    3. Taylor expanded in t around 0 21.2

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

    4. Simplified21.2

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

    6. Applied clear-num_binary6421.2

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

    7. Simplified21.2

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

    9. Applied associate-/r*_binary6420.9

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

    10. Simplified20.9

      \[\leadsto \frac{2}{\frac{1}{\frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}}{t \cdot {\sin k}^{2}}}} \]
    11. Using strategy rm

    12. Applied associate-/l*_binary6414.8

      \[\leadsto \frac{2}{\frac{1}{\frac{\color{blue}{\frac{\ell}{\frac{k \cdot k}{\ell}}} \cdot \cos k}{t \cdot {\sin k}^{2}}}} \]

    13. Simplified14.8

      \[\leadsto \frac{2}{\frac{1}{\frac{\frac{\ell}{\color{blue}{\frac{k}{\frac{\ell}{k}}}} \cdot \cos k}{t \cdot {\sin k}^{2}}}} \]

    if 3.26618261886e-313 < (*.f64 l l) < 1.2583695792669349e60

    1. Initial program 43.3

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

    2. Simplified33.4

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

    3. Taylor expanded in t around 0 7.8

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

    4. Simplified7.8

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

    6. Applied clear-num_binary647.8

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

    7. Simplified7.8

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

    9. Applied associate-*l*_binary645.5

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

    if 1.2583695792669349e60 < (*.f64 l l)

    1. Initial program 55.4

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

    2. Simplified51.7

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

    3. Taylor expanded in t around 0 41.1

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

    4. Simplified41.1

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

    6. Applied clear-num_binary6441.1

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

    7. Simplified41.1

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

    9. Applied associate-/r*_binary6439.0

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

    10. Simplified39.0

      \[\leadsto \frac{2}{\frac{1}{\frac{\color{blue}{\frac{\ell \cdot \ell}{k \cdot k} \cdot \cos k}}{t \cdot {\sin k}^{2}}}} \]
    11. Using strategy rm

    12. Applied add-sqr-sqrt_binary6439.1

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

    13. Simplified39.0

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

    14. Simplified8.1

      \[\leadsto \frac{2}{\frac{1}{\frac{\left(\left|\frac{\ell}{k}\right| \cdot \color{blue}{\left|\frac{\ell}{k}\right|}\right) \cdot \cos k}{t \cdot {\sin k}^{2}}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 3.26618261886 \cdot 10^{-313}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{k}}} \cdot \cos k}{t \cdot {\sin k}^{2}}}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 1.2583695792669349 \cdot 10^{+60}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{1}{\frac{\cos k \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}{t \cdot {\sin k}^{2}}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021212 
(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))))