Average Error: 32.2 → 22.4
Time: 12.3s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.36233798607962222 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(-\frac{2}{\sin k}\right) \cdot \ell}{\frac{\left(\left(-\sin k\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot {t}^{3}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(-\tan k\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot {t}^{\left(\frac{3}{2}\right)}} \cdot \frac{\left(-\frac{1}{\sin k}\right) \cdot \ell}{{t}^{\left(\frac{3}{2}\right)}}\\ \end{array}\]
\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}\;t \le -1.36233798607962222 \cdot 10^{-108}:\\
\;\;\;\;\frac{\left(-\frac{2}{\sin k}\right) \cdot \ell}{\frac{\left(\left(-\sin k\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot {t}^{3}}{\cos k}}\\

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

\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 <= -1.3623379860796222e-108)) {
		VAR = ((-(2.0 / sin(k)) * l) / (((-sin(k) * (fma(2.0, 1.0, pow((k / t), 2.0)) / l)) * pow(t, 3.0)) / cos(k)));
	} else {
		VAR = ((2.0 / ((-tan(k) * (fma(2.0, 1.0, pow((k / t), 2.0)) / l)) * pow(t, (3.0 / 2.0)))) * ((-(1.0 / sin(k)) * l) / pow(t, (3.0 / 2.0))));
	}
	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 2 regimes

  2. if t < -1.3623379860796222e-108

    1. Initial program 23.2

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

    2. Simplified23.2

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

    4. Applied associate-/l*22.2

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

    5. Applied frac-2neg22.2

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

    6. Applied frac-times17.3

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

    8. Applied *-commutative17.3

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

    9. Applied associate-/r*17.3

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

    10. Applied distribute-neg-frac17.3

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

    11. Applied associate-*l/15.9

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

    12. Applied associate-/l/15.7

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

    14. Applied tan-quot15.8

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

    15. Applied distribute-neg-frac15.8

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

    16. Applied associate-*l/15.8

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

    17. Applied associate-*l/15.8

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

    if -1.3623379860796222e-108 < t

    1. Initial program 38.0

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

    2. Simplified38.6

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

    4. Applied associate-/l*37.9

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

    5. Applied frac-2neg37.9

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

    6. Applied frac-times34.5

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

    8. Applied *-commutative34.5

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

    9. Applied associate-/r*34.5

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

    10. Applied distribute-neg-frac34.5

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

    11. Applied associate-*l/33.6

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

    12. Applied associate-/l/33.5

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

    14. Applied sqr-pow33.5

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

    15. Applied associate-*r*28.7

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

    16. Applied div-inv28.7

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

    17. Applied distribute-rgt-neg-in28.7

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

    18. Applied associate-*l*28.7

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

    19. Applied times-frac26.8

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

  4. Final simplification22.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.36233798607962222 \cdot 10^{-108}:\\ \;\;\;\;\frac{\left(-\frac{2}{\sin k}\right) \cdot \ell}{\frac{\left(\left(-\sin k\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot {t}^{3}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(-\tan k\right) \cdot \frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}\right) \cdot {t}^{\left(\frac{3}{2}\right)}} \cdot \frac{\left(-\frac{1}{\sin k}\right) \cdot \ell}{{t}^{\left(\frac{3}{2}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))))