Toniolo and Linder, Equation (10-)

Average Error: 47.9 → 28.3

Time: 42.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.79240876530840681 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;t \le 1.9643182186219191 \cdot 10^{-192}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{elif}\;t \le 1.55542222486645039 \cdot 10^{-97}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

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 <= -2.792408765308407e-107)) {
		VAR = ((double) (((double) (1.0 / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))) * ((double) (l / ((double) sin(k))))))));
	} else {
		double VAR_1;
		if ((t <= 1.964318218621919e-192)) {
			VAR_1 = ((double) log(((double) pow(((double) exp(((double) (((double) (2.0 * 1.0)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 * ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))))), ((double) (((double) (l * l)) / ((double) sin(k))))))));
		} else {
			double VAR_2;
			if ((t <= 1.5554222248664504e-97)) {
				VAR_2 = ((double) (((double) (2.0 * ((double) (l * l)))) / ((double) (((double) (((double) pow(((double) (1.0 / ((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(((double) (1.0 / t)))) + ((double) log(1.0)))))))), 1.0)))))), 1.0)) * ((double) (((double) sin(k)) / ((double) cos(k)))))) * ((double) sin(k))))));
			} else {
				VAR_2 = ((double) (((double) (1.0 / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (((double) (2.0 * l)) / ((double) (((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(t, 3.0)) * ((double) tan(k)))))))) * ((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 3 regimes

2. if t < -2.79240876530840681e-107 or 1.55542222486645039e-97 < t

   1. Initial program 42.0

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

   2. Simplified 31.1

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

   4. Applied sqr-pow 31.1

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

   5. Applied associate-*l* 26.2

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

   7. Applied times-frac 25.8

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

   9. Applied *-un-lft-identity 25.8

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

   10. Applied times-frac 26.0

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

   11. Applied associate-*l* 25.2

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

   13. Applied *-un-lft-identity 25.2

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

   14. Applied times-frac 24.5

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

   15. Applied associate-*r* 20.6

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

   16. Simplified 20.5

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

   if -2.79240876530840681e-107 < t < 1.9643182186219191e-192

   1. Initial program 63.8

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

   2. Simplified 63.8

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

   4. Applied sqr-pow 63.8

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

   5. Applied associate-*l* 63.8

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

   7. Applied times-frac 63.8

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

   9. Applied add-log-exp 63.8

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

   10. Simplified 49.4

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

   if 1.9643182186219191e-192 < t < 1.55542222486645039e-97

   1. Initial program 61.0

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

   2. Simplified 61.0

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

   3. Taylor expanded around inf 45.5

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

4. Final simplification 28.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.79240876530840681 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;t \le 1.9643182186219191 \cdot 10^{-192}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{elif}\;t \le 1.55542222486645039 \cdot 10^{-97}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{\left({\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}^{1}}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{2 \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020169 
(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))))