Average Error: 32.8 → 16.2
Time: 12.9s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.8655696603753476 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{\tan k}{\ell}\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;t \le 1.3710261062242033 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot \ell}\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \end{array}\]
\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}\;t \le -3.8655696603753476 \cdot 10^{-5}:\\
\;\;\;\;\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{\tan k}{\ell}\right)\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;t \le 1.3710261062242033 \cdot 10^{-160}:\\
\;\;\;\;\frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot \ell}\right) \cdot \frac{1}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\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 <= -3.8655696603753476e-05)) {
		VAR = (2.0 / (((pow(t, 3.0) * sin(k)) * (((1.0 + pow((k / t), 2.0)) + 1.0) * (tan(k) / l))) * (1.0 / l)));
	} else {
		double VAR_1;
		if ((t <= 1.3710261062242033e-160)) {
			VAR_1 = (2.0 / (((2.0 * ((pow(t, 3.0) * pow(sin(k), 2.0)) / (cos(k) * l))) - (pow((1.0 / pow(-1.0, 3.0)), 1.0) * ((t * (pow(k, 2.0) * pow(sin(k), 2.0))) / (cos(k) * l)))) * (1.0 / l)));
		} else {
			VAR_1 = (2.0 / ((pow(t, (3.0 / 2.0)) / l) * ((((pow(t, (3.0 / 2.0)) / l) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0))));
		}
		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 < -3.8655696603753476e-05

    1. Initial program 22.8

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

    3. Applied associate-/r*19.0

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

    4. Applied associate-*l/17.8

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

    6. Applied div-inv17.8

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

    7. Applied associate-*l*17.6

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

    8. Applied associate-*l*17.1

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

    9. Simplified17.1

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

    11. Applied div-inv17.1

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

    12. Applied associate-*r*17.1

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

    13. Applied associate-*r*17.7

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

    14. Simplified16.3

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

    if -3.8655696603753476e-05 < t < 1.3710261062242033e-160

    1. Initial program 53.5

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

    3. Applied associate-/r*52.2

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

    4. Applied associate-*l/51.2

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

    6. Applied div-inv51.2

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

    7. Applied associate-*l*50.8

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

    8. Applied associate-*l*50.7

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

    9. Simplified50.7

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

    11. Applied div-inv50.7

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

    12. Applied associate-*r*50.7

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

    13. Applied associate-*r*51.9

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

    14. Simplified52.3

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

    15. Taylor expanded around -inf 22.6

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

    if 1.3710261062242033e-160 < t

    1. Initial program 26.7

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

    3. Applied sqr-pow26.7

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

    4. Applied times-frac16.2

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

    5. Applied associate-*l*13.1

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

    6. Applied associate-*l*13.6

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

    7. Applied associate-*l*12.1

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.8655696603753476 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{\tan k}{\ell}\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;t \le 1.3710261062242033 \cdot 10^{-160}:\\ \;\;\;\;\frac{2}{\left(2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot \ell} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot \ell}\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 
(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))))