Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 27.2

Time: 28.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.7756472975929318 \cdot 10^{237}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\ \end{array}\]

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 ((((double) (l * l)) <= 9.775647297592932e+237)) {
		VAR = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) (((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) pow(((double) cbrt(t)), 3.0)))) * ((double) pow(((double) cbrt(t)), 3.0)))) * ((double) pow(((double) cbrt(t)), 3.0)))) * ((double) tan(k)))))) * ((double) sin(k))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) pow(t, 3.0)))))) * ((double) (l / ((double) tan(k)))))) / ((double) sin(k))));
	}
	return VAR;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if (* l l) < 9.7756472975929318e237

   1. Initial program 44.9

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

   2. Simplified 35.5

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

   4. Applied sqr-pow 35.5

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

   5. Applied associate-*l* 31.1

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

   7. Applied associate-*r* 31.0

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

   9. Applied add-cube-cbrt 31.2

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

   10. Applied unpow-prod-down 31.2

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

   11. Applied associate-*r* 26.6

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

   13. Applied unpow-prod-down 26.6

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

   14. Applied associate-*r* 21.2

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

   if 9.7756472975929318e237 < (* l l)

   1. Initial program 60.6

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

   2. Simplified 59.3

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

   4. Applied sqr-pow 59.3

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

   5. Applied associate-*l* 58.6

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

   7. Applied associate-*r* 58.6

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

   9. Applied associate-/r* 58.6

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

   10. Simplified 50.8

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

4. Final simplification 27.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.7756472975929318 \cdot 10^{237}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right) \cdot \tan k\right)\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot {t}^{3}} \cdot \frac{\ell}{\tan k}}{\sin k}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(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))))