Average Error: 48.2 → 16.1
Time: 25.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.780421029131753 \cdot 10^{+161} \lor \neg \left(\ell \leq 1.348002792593927 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({k}^{\left(\frac{2}{2}\right)}\right)}^{1} \cdot \left(\frac{\sin k}{\cos k} \cdot \left({\left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}^{1} \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array}\]
\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}\;\ell \leq -4.780421029131753 \cdot 10^{+161} \lor \neg \left(\ell \leq 1.348002792593927 \cdot 10^{+154}\right):\\
\;\;\;\;\frac{2}{\left(\left(\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(\frac{k}{t}\right)}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{{\left({k}^{\left(\frac{2}{2}\right)}\right)}^{1} \cdot \left(\frac{\sin k}{\cos k} \cdot \left({\left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}^{1} \cdot \frac{\sin k}{\ell \cdot \ell}\right)\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 (<= l -4.780421029131753e+161) (not (<= l 1.348002792593927e+154)))
   (/
    2.0
    (*
     (*
      (*
       (* (/ (pow (* (cbrt t) (cbrt t)) 3.0) l) (/ (pow (cbrt t) 3.0) l))
       (sin k))
      (tan k))
     (pow (/ k t) 2.0)))
   (/
    2.0
    (*
     (pow (pow k (/ 2.0 2.0)) 1.0)
     (*
      (/ (sin k) (cos k))
      (*
       (pow (* (pow k (/ 2.0 2.0)) (pow t 1.0)) 1.0)
       (/ (sin k) (* l l))))))))
double code(double t, double l, double k) {
	return (2.0 / ((double) (((double) (((double) ((((double) pow(t, 3.0)) / ((double) (l * l))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow((k / t), 2.0)))) - 1.0)))));
}
↓
double code(double t, double l, double k) {
	double tmp;
	if (((l <= -4.780421029131753e+161) || !(l <= 1.348002792593927e+154))) {
		tmp = (2.0 / ((double) (((double) (((double) (((double) ((((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), 3.0)) / l) * (((double) pow(((double) cbrt(t)), 3.0)) / l))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) pow((k / t), 2.0)))));
	} else {
		tmp = (2.0 / ((double) (((double) pow(((double) pow(k, (2.0 / 2.0))), 1.0)) * ((double) ((((double) sin(k)) / ((double) cos(k))) * ((double) (((double) pow(((double) (((double) pow(k, (2.0 / 2.0))) * ((double) pow(t, 1.0)))), 1.0)) * (((double) sin(k)) / ((double) (l * l))))))))));
	}
	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 l < -4.7804210291317526e161 or 1.348002792593927e154 < l

    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

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

    4. Applied add-cube-cbrt_binary6464.0

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

    5. Applied unpow-prod-down_binary6464.0

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

    6. Applied times-frac_binary6449.5

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

    if -4.7804210291317526e161 < l < 1.348002792593927e154

    1. Initial program 45.5

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

    2. Simplified35.8

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

    3. Taylor expanded around inf 14.9

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

    4. Simplified14.9

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

    6. Applied sqr-pow_binary6414.9

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

    7. Applied associate-*l*_binary6412.4

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

    8. Simplified12.4

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

    10. Applied unpow-prod-down_binary6412.4

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

    11. Applied associate-*l*_binary6410.6

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

    12. Simplified10.6

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

    14. Applied unpow2_binary6410.6

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

    15. Applied times-frac_binary6410.3

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

    16. Applied associate-*l*_binary6410.4

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

    17. Simplified10.4

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.780421029131753 \cdot 10^{+161} \lor \neg \left(\ell \leq 1.348002792593927 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({k}^{\left(\frac{2}{2}\right)}\right)}^{1} \cdot \left(\frac{\sin k}{\cos k} \cdot \left({\left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}^{1} \cdot \frac{\sin k}{\ell \cdot \ell}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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))))