Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 31.9

Time: 27.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.28823798784846508 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;t \le -6.1574780140413241 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 2.9462302763418328 \cdot 10^{-117}:\\ \;\;\;\;2 \cdot \left({\left({\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt{t}\right)}^{3} \cdot \left({\left(\sqrt{t}\right)}^{3} \cdot \tan k\right)\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

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 <= -1.288237987848465e-117)) {
		VAR = ((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))) * ((double) (l * ((double) (l / ((double) sin(k))))))));
	} else {
		double VAR_1;
		if ((t <= -6.157478014041324e-309)) {
			VAR_1 = ((double) (2.0 * ((double) (((double) pow(((double) (((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)))) / ((double) pow(-1.0, 3.0)))), 1.0)) * ((double) (((double) (((double) cos(k)) * ((double) pow(l, 2.0)))) / ((double) pow(((double) sin(k)), 2.0))))))));
		} else {
			double VAR_2;
			if ((t <= 2.9462302763418328e-117)) {
				VAR_2 = ((double) (2.0 * ((double) (((double) pow(((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(((double) (1.0 / t)))) + ((double) log(1.0)))))))), 1.0)))), 1.0)) * ((double) (((double) (((double) cos(k)) * ((double) pow(l, 2.0)))) / ((double) pow(((double) sin(k)), 2.0))))))));
			} else {
				VAR_2 = ((double) (((double) (2.0 / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) sqrt(t)), 3.0)) * ((double) (((double) pow(((double) sqrt(t)), 3.0)) * ((double) tan(k)))))))))))) * ((double) (l * ((double) (l / ((double) sin(k))))))));
			}
			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 4 regimes

2. if t < -1.28823798784846508e-117

   1. Initial program 43.5

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

   2. Simplified 33.5

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

   4. Applied sqr-pow 33.5

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

   5. Applied associate-*l* 29.2

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

   7. Applied times-frac 28.9

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

   9. Applied *-un-lft-identity 28.9

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

   10. Applied times-frac 28.6

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

   11. Simplified 28.6

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

   if -1.28823798784846508e-117 < t < -6.1574780140413241e-309

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded around -inf 44.8

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

   if -6.1574780140413241e-309 < t < 2.9462302763418328e-117

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded around inf 44.0

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

   if 2.9462302763418328e-117 < t

   1. Initial program 42.6

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

   2. Simplified 32.7

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

   4. Applied sqr-pow 32.7

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

   5. Applied associate-*l* 27.4

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

   7. Applied times-frac 27.0

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

   9. Applied *-un-lft-identity 27.0

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

   10. Applied times-frac 26.7

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

   11. Simplified 26.7

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

   13. Applied add-sqr-sqrt 26.8

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

   14. Applied unpow-prod-down 26.8

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

   15. Applied associate-*l* 26.5

       \[\leadsto \frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left({\left(\sqrt{t}\right)}^{3} \cdot \left({\left(\sqrt{t}\right)}^{3} \cdot \tan k\right)\right)}\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 31.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.28823798784846508 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;t \le -6.1574780140413241 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 2.9462302763418328 \cdot 10^{-117}:\\ \;\;\;\;2 \cdot \left({\left({\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt{t}\right)}^{3} \cdot \left({\left(\sqrt{t}\right)}^{3} \cdot \tan k\right)\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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))))