Average Error: 47.6 → 14.2
Time: 27.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
↓
\[\begin{array}{l} t_1 := t \cdot {\sin k}^{2}\\ t_2 := {\left(\sqrt[3]{k}\right)}^{2}\\ t_3 := \frac{t_2}{\ell \cdot \sqrt[3]{\ell}}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_5 := \frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}}{\cos k}\\ \mathbf{if}\;t \leq -3.917675272850077 \cdot 10^{-268}:\\ \;\;\;\;\frac{2}{t_5 \cdot \frac{t_1 \cdot t_3}{t_4}}\\ \mathbf{elif}\;t \leq 2.869646538692728 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{\frac{t_2 \cdot \left(t_2 \cdot \left(t_1 \cdot t_2\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left(t_3 \cdot \frac{t_1}{t_4}\right)}\\ \end{array} \]
\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}
t_1 := t \cdot {\sin k}^{2}\\
t_2 := {\left(\sqrt[3]{k}\right)}^{2}\\
t_3 := \frac{t_2}{\ell \cdot \sqrt[3]{\ell}}\\
t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\
t_5 := \frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}}{\cos k}\\
\mathbf{if}\;t \leq -3.917675272850077 \cdot 10^{-268}:\\
\;\;\;\;\frac{2}{t_5 \cdot \frac{t_1 \cdot t_3}{t_4}}\\

\mathbf{elif}\;t \leq 2.869646538692728 \cdot 10^{-159}:\\
\;\;\;\;\frac{2}{\frac{t_2 \cdot \left(t_2 \cdot \left(t_1 \cdot t_2\right)\right)}{\cos k \cdot {\ell}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{t_5 \cdot \left(t_3 \cdot \frac{t_1}{t_4}\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
 (let* ((t_1 (* t (pow (sin k) 2.0)))
        (t_2 (pow (cbrt k) 2.0))
        (t_3 (/ t_2 (* l (cbrt l))))
        (t_4 (pow (cbrt l) 2.0))
        (t_5 (/ (pow (* (cbrt k) (cbrt k)) 2.0) (cos k))))
   (if (<= t -3.917675272850077e-268)
     (/ 2.0 (* t_5 (/ (* t_1 t_3) t_4)))
     (if (<= t 2.869646538692728e-159)
       (/ 2.0 (/ (* t_2 (* t_2 (* t_1 t_2))) (* (cos k) (pow l 2.0))))
       (/ 2.0 (* t_5 (* t_3 (/ t_1 t_4))))))))
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 t_2 = pow(cbrt(k), 2.0);
	double t_3 = t_2 / (l * cbrt(l));
	double t_4 = pow(cbrt(l), 2.0);
	double t_5 = pow((cbrt(k) * cbrt(k)), 2.0) / cos(k);
	double tmp;
	if (t <= -3.917675272850077e-268) {
		tmp = 2.0 / (t_5 * ((t_1 * t_3) / t_4));
	} else if (t <= 2.869646538692728e-159) {
		tmp = 2.0 / ((t_2 * (t_2 * (t_1 * t_2))) / (cos(k) * pow(l, 2.0)));
	} else {
		tmp = 2.0 / (t_5 * (t_3 * (t_1 / t_4)));
	}
	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 t < -3.91767527285007704e-268

    1. Initial program 46.8

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

    2. Simplified38.8

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

    3. Taylor expanded in t around 0 22.3

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

    4. Applied add-cube-cbrt_binary6422.5

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

    5. Applied unpow-prod-down_binary6422.5

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

    6. Applied associate-*l*_binary6421.3

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

    7. Applied times-frac_binary6419.2

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

    8. Applied add-cube-cbrt_binary6419.3

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

    9. Applied unpow-prod-down_binary6419.3

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

    10. Applied associate-/r*_binary6415.4

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

    11. Simplified14.1

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

    if -3.91767527285007704e-268 < t < 2.8696465386927281e-159

    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. Taylor expanded in t around 0 28.4

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

    4. Applied add-cube-cbrt_binary6428.7

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

    5. Applied unpow-prod-down_binary6428.7

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

    6. Applied associate-*l*_binary6422.2

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

    7. Applied unpow-prod-down_binary6422.2

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

    8. Applied associate-*l*_binary6419.7

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

    if 2.8696465386927281e-159 < t

    1. Initial program 43.6

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

    2. 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 21.5

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

    4. Applied add-cube-cbrt_binary6421.7

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

    5. Applied unpow-prod-down_binary6421.7

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

    6. Applied associate-*l*_binary6421.2

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

    7. Applied times-frac_binary6418.7

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

    8. Applied add-cube-cbrt_binary6418.7

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

    9. Applied unpow-prod-down_binary6418.7

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

    10. Applied times-frac_binary6412.7

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

    11. Simplified12.6

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

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.917675272850077 \cdot 10^{-268}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}}{\cos k} \cdot \frac{\left(t \cdot {\sin k}^{2}\right) \cdot \frac{{\left(\sqrt[3]{k}\right)}^{2}}{\ell \cdot \sqrt[3]{\ell}}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{elif}\;t \leq 2.869646538692728 \cdot 10^{-159}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt[3]{k}\right)}^{2} \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \left(\left(t \cdot {\sin k}^{2}\right) \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}}{\cos k} \cdot \left(\frac{{\left(\sqrt[3]{k}\right)}^{2}}{\ell \cdot \sqrt[3]{\ell}} \cdot \frac{t \cdot {\sin k}^{2}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(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))))