Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 27.8

Time: 45.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.75270015826472054 \cdot 10^{215}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{{-1}^{3}}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}\right)}^{1} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;t \le -2.4058546412849183 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k}}{\sin k}\\ \mathbf{elif}\;t \le 1.9359051995686767 \cdot 10^{-58}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k\right)\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 1.12539180684310793 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k\right)\right)\right) \cdot \sin k}\\ \end{array}\]

C

double code(double t, double l, double k) {
	return ((double) (2.0 / ((double) (((double) (((double) (((double) (((double) pow(t, 3.0)) / ((double) (l * l)))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))) - 1.0))))));
}

↓

double code(double t, double l, double k) {
	double VAR;
	if ((t <= -5.7527001582647205e+215)) {
		VAR = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) pow(((double) (((double) pow(-1.0, 3.0)) / ((double) (((double) pow(((double) exp(((double) (2.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / k)))))))))), 1.0)) * ((double) pow(((double) exp(((double) (1.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / t)))))))))), 1.0)))))), 1.0)) * ((double) (((double) pow(((double) sin(k)), 2.0)) / ((double) cos(k))))))));
	} else {
		double VAR_1;
		if ((t <= -2.4058546412849183e-144)) {
			VAR_1 = ((double) (((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), 3.0)))))) * ((double) (l / ((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) tan(k)))))))) / ((double) sin(k))));
		} else {
			double VAR_2;
			if ((t <= 1.9359051995686767e-58)) {
				VAR_2 = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) tan(k)))))))) * ((double) sin(k))))));
			} else {
				double VAR_3;
				if ((t <= 1.1253918068431079e+111)) {
					VAR_3 = ((double) (((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), 3.0)))))) * ((double) (l / ((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) tan(k)))))))) / ((double) sin(k))));
				} else {
					VAR_3 = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), ((double) (3.0 / 2.0)))))) * ((double) (((double) pow(((double) cbrt(t)), 3.0)) * ((double) tan(k)))))))) * ((double) 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 < -5.75270015826472054e215

   1. Initial program 57.8

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

   2. Simplified 42.6

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

   3. Taylor expanded around -inf 42.8

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

   if -5.75270015826472054e215 < t < -2.4058546412849183e-144 or 1.9359051995686767e-58 < t < 1.12539180684310793e111

   1. Initial program 37.0

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

   2. Simplified 28.8

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

   4. Applied sqr-pow 28.8

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

   5. Applied associate-*l* 26.5

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

   7. Applied add-cube-cbrt 26.8

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

   8. Applied unpow-prod-down 26.8

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

   9. Applied associate-*l* 26.5

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

   11. Applied associate-*r* 22.8

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

   13. Applied associate-/r* 22.0

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

   14. Simplified 19.1

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

   if -2.4058546412849183e-144 < t < 1.9359051995686767e-58 or 1.12539180684310793e111 < t

   1. Initial program 57.3

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

   2. Simplified 51.0

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

   4. Applied sqr-pow 51.0

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

   5. Applied associate-*l* 47.2

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

   7. Applied add-cube-cbrt 47.2

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

   8. Applied unpow-prod-down 47.2

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

   9. Applied associate-*l* 47.2

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

   11. Applied associate-*r* 42.6

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

   13. Applied sqr-pow 42.6

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

   14. Applied associate-*r* 33.1

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

4. Final simplification 27.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.75270015826472054 \cdot 10^{215}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{{\left(\frac{{-1}^{3}}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}\right)}^{1} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;t \le -2.4058546412849183 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k}}{\sin k}\\ \mathbf{elif}\;t \le 1.9359051995686767 \cdot 10^{-58}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k\right)\right)\right) \cdot \sin k}\\ \mathbf{elif}\;t \le 1.12539180684310793 \cdot 10^{111}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{t}\right)}^{3} \cdot \tan k\right)\right)\right) \cdot \sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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))))