Average Error: 32.8 → 16.8
Time: 11.9s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.0511429488171726 \cdot 10^{-61}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 5.0235376559012423 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2} \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan k \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}{\frac{2}{\sin k}}}\\ \end{array}\]
\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 \le -1.0511429488171726 \cdot 10^{-61}:\\
\;\;\;\;\frac{2 \cdot \ell}{\left(\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot \sin k}\\

\mathbf{elif}\;t \le 5.0235376559012423 \cdot 10^{-124}:\\
\;\;\;\;2 \cdot \left({\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2} \cdot {k}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\tan k \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}{\frac{2}{\sin k}}}\\

\end{array}
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 VAR;
	if ((t <= -1.0511429488171726e-61)) {
		VAR = ((2.0 * l) / (((tan(k) * pow(t, 3.0)) * (fma(2.0, 1.0, pow((k / t), 2.0)) / l)) * sin(k)));
	} else {
		double VAR_1;
		if ((t <= 5.023537655901242e-124)) {
			VAR_1 = (2.0 * (pow((1.0 / (pow(-1.0, 2.0) * pow(t, 1.0))), 1.0) * ((cos(k) * pow(l, 2.0)) / (pow(sin(k), 2.0) * pow(k, 2.0)))));
		} else {
			VAR_1 = (1.0 / (((tan(k) * pow(t, (3.0 / 2.0))) / l) * ((pow(t, (3.0 / 2.0)) * (fma(2.0, 1.0, pow((k / t), 2.0)) / l)) / (2.0 / sin(k)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -1.0511429488171726e-61

    1. Initial program 23.1

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

    2. Simplified22.9

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
    3. Using strategy rm

    4. Applied associate-/l*21.6

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}}\]

    5. Applied *-commutative21.6

      \[\leadsto \frac{\frac{2}{\color{blue}{\sin k \cdot {t}^{3}}}}{\tan k} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    6. Applied associate-/r*21.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sin k}}{{t}^{3}}}}{\tan k} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    7. Applied associate-/l/21.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\tan k \cdot {t}^{3}}} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    8. Applied frac-times15.9

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k} \cdot \ell}{\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}}\]
    9. Using strategy rm

    10. Applied associate-*l/15.9

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    11. Applied associate-/l/16.3

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot \sin k}}\]

    if -1.0511429488171726e-61 < t < 5.023537655901242e-124

    1. Initial program 59.7

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

    2. Simplified60.6

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
    3. Using strategy rm

    4. Applied associate-/l*60.5

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}}\]

    5. Applied *-commutative60.5

      \[\leadsto \frac{\frac{2}{\color{blue}{\sin k \cdot {t}^{3}}}}{\tan k} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    6. Applied associate-/r*60.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sin k}}{{t}^{3}}}}{\tan k} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    7. Applied associate-/l/60.5

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\tan k \cdot {t}^{3}}} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    8. Applied frac-times59.8

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k} \cdot \ell}{\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}}\]
    9. Using strategy rm

    10. Applied associate-*l/59.8

      \[\leadsto \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    11. Applied associate-/l/59.8

      \[\leadsto \color{blue}{\frac{2 \cdot \ell}{\left(\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot \sin k}}\]

    12. Taylor expanded around -inf 27.9

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

    if 5.023537655901242e-124 < 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. Simplified24.6

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
    3. Using strategy rm

    4. Applied associate-/l*23.4

      \[\leadsto \frac{\frac{2}{{t}^{3} \cdot \sin k}}{\tan k} \cdot \color{blue}{\frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}}\]

    5. Applied *-commutative23.4

      \[\leadsto \frac{\frac{2}{\color{blue}{\sin k \cdot {t}^{3}}}}{\tan k} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    6. Applied associate-/r*23.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sin k}}{{t}^{3}}}}{\tan k} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    7. Applied associate-/l/23.4

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\tan k \cdot {t}^{3}}} \cdot \frac{\ell}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\]

    8. Applied frac-times18.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\sin k} \cdot \ell}{\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}}\]
    9. Using strategy rm

    10. Applied clear-num18.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}{\frac{2}{\sin k} \cdot \ell}}}\]
    11. Using strategy rm

    12. Applied *-commutative18.4

      \[\leadsto \frac{1}{\frac{\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}{\color{blue}{\ell \cdot \frac{2}{\sin k}}}}\]

    13. Applied sqr-pow18.5

      \[\leadsto \frac{1}{\frac{\left(\tan k \cdot \color{blue}{\left({t}^{\left(\frac{3}{2}\right)} \cdot {t}^{\left(\frac{3}{2}\right)}\right)}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}{\ell \cdot \frac{2}{\sin k}}}\]

    14. Applied associate-*r*16.8

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(\tan k \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot {t}^{\left(\frac{3}{2}\right)}\right)} \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}{\ell \cdot \frac{2}{\sin k}}}\]

    15. Applied associate-*l*13.4

      \[\leadsto \frac{1}{\frac{\color{blue}{\left(\tan k \cdot {t}^{\left(\frac{3}{2}\right)}\right) \cdot \left({t}^{\left(\frac{3}{2}\right)} \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right)}}{\ell \cdot \frac{2}{\sin k}}}\]

    16. Applied times-frac10.3

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0511429488171726 \cdot 10^{-61}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(\tan k \cdot {t}^{3}\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 5.0235376559012423 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2} \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\tan k \cdot {t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)} \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}{\frac{2}{\sin k}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))