Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 10.6

Time: 23.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.2439854096143383 \cdot 10^{120}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \cos k\right) \cdot \ell}{\frac{{\left(\sin k\right)}^{1}}{\frac{\ell}{\sin k}}}\\ \mathbf{elif}\;t \le -5.8071251107399111 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 1.4658926919355537 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k\right) \cdot \ell}{\frac{{\left(\sin k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \le 1.09079912784831405 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \cos k\right) \cdot \ell}{\frac{{\left(\sin k\right)}^{1}}{\frac{\ell}{\sin k}}}\\ \end{array}\]

C

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 <= -2.2439854096143383e+120)) {
		VAR = (2.0 * (((pow((1.0 / (pow(k, (2.0 / 2.0)) * (pow(k, (2.0 / 2.0)) * pow(t, 1.0)))), 1.0) * cos(k)) * l) / (pow(sin(k), 1.0) / (l / sin(k)))));
	} else {
		double VAR_1;
		if ((t <= -5.807125110739911e-37)) {
			VAR_1 = (((2.0 * l) / pow((k / t), (2.0 / 2.0))) * (l / ((pow((k / t), (2.0 / 2.0)) * (pow(t, 3.0) * tan(k))) * sin(k))));
		} else {
			double VAR_2;
			if ((t <= 1.4658926919355537e-19)) {
				VAR_2 = (2.0 * (((pow(((1.0 / pow(k, (2.0 / 2.0))) / (pow(k, (2.0 / 2.0)) * pow(t, 1.0))), 1.0) * cos(k)) * l) / (pow(sin(k), 2.0) / l)));
			} else {
				double VAR_3;
				if ((t <= 1.090799127848314e+81)) {
					VAR_3 = (((2.0 * l) / pow((k / t), (2.0 / 2.0))) * (l / ((pow((k / t), (2.0 / 2.0)) * (pow(t, 3.0) * tan(k))) * sin(k))));
				} else {
					VAR_3 = (2.0 * (((pow((1.0 / (pow(k, (2.0 / 2.0)) * (pow(k, (2.0 / 2.0)) * pow(t, 1.0)))), 1.0) * cos(k)) * l) / (pow(sin(k), 1.0) / (l / sin(k)))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -2.2439854096143383e+120 or 1.090799127848314e+81 < t

   1. Initial program 52.0

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

   2. Simplified 37.8

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

   3. Taylor expanded around inf 20.8

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

   5. Applied unpow2 20.8

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

   6. Applied associate-*r* 20.8

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

   7. Applied associate-/l* 18.7

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

   8. Applied associate-*r/ 13.9

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

   9. Simplified 13.9

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

   11. Applied sqr-pow 13.9

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

   12. Applied associate-*l* 13.9

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

   14. Applied sqr-pow 13.9

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

   15. Applied associate-/l* 10.5

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

   16. Simplified 10.5

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

   if -2.2439854096143383e+120 < t < -5.807125110739911e-37 or 1.4658926919355537e-19 < t < 1.090799127848314e+81

   1. Initial program 31.8

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

   2. Simplified 22.6

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

   4. Applied sqr-pow 22.6

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

   5. Applied associate-*l* 22.2

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

   6. Applied associate-*l* 22.1

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

   7. Applied associate-*r* 22.1

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

   8. Applied times-frac 9.3

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

   if -5.807125110739911e-37 < t < 1.4658926919355537e-19

   1. Initial program 53.1

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

   2. Simplified 52.3

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

   3. Taylor expanded around inf 25.2

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

   5. Applied unpow2 25.2

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

   6. Applied associate-*r* 25.2

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

   7. Applied associate-/l* 24.0

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

   8. Applied associate-*r/ 19.9

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

   9. Simplified 19.9

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

   11. Applied sqr-pow 19.9

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

   12. Applied associate-*l* 11.5

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

   14. Applied associate-/r* 11.3

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.2439854096143383 \cdot 10^{120}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \cos k\right) \cdot \ell}{\frac{{\left(\sin k\right)}^{1}}{\frac{\ell}{\sin k}}}\\ \mathbf{elif}\;t \le -5.8071251107399111 \cdot 10^{-37}:\\ \;\;\;\;\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 1.4658926919355537 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k\right) \cdot \ell}{\frac{{\left(\sin k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;t \le 1.09079912784831405 \cdot 10^{81}:\\ \;\;\;\;\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\ell}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \cos k\right) \cdot \ell}{\frac{{\left(\sin k\right)}^{1}}{\frac{\ell}{\sin k}}}\\ \end{array}\]

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))))