Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 28.4

Time: 25.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.71194921138226557 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{2}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\sqrt{2} \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;t \le -7.306022719573431 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 1.60231479643995 \cdot 10^{-108}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\sqrt{2} \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

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 <= -2.7119492113822656e-108)) {
		VAR = ((double) (((double) (((double) sqrt(2.0)) / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (((double) (((double) sqrt(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 <= -7.30602271957343e-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 <= 1.60231479643995e-108)) {
				VAR_2 = ((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 {
				VAR_2 = ((double) (((double) (((double) sqrt(2.0)) / ((double) pow(((double) (k / t)), ((double) (2.0 / 2.0)))))) * ((double) (((double) (((double) (((double) sqrt(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 < -2.71194921138226557e-108 or 1.60231479643995e-108 < t

   1. Initial program 42.5

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

   2. Simplified 31.8

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

   4. Applied sqr-pow 31.8

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

   5. Applied associate-*l* 26.7

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

   7. Applied times-frac 26.4

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

   9. Applied add-sqr-sqrt 26.5

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

   10. Applied times-frac 26.6

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

   11. Applied associate-*l* 25.9

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

   13. Applied *-un-lft-identity 25.9

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

   14. Applied times-frac 25.3

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

   15. Applied associate-*r* 21.4

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

   16. Simplified 21.4

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

   if -2.71194921138226557e-108 < t < -7.306022719573431e-309

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded around -inf 45.7

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

   if -7.306022719573431e-309 < t < 1.60231479643995e-108

   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. Using strategy rm

   4. Applied sqr-pow 64.0

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

   5. Applied associate-*l* 64.0

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

   7. Applied times-frac 64.0

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

   9. Applied add-log-exp 64.0

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

   10. Simplified 50.3

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

4. Final simplification 28.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.71194921138226557 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{2}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\sqrt{2} \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{elif}\;t \le -7.306022719573431 \cdot 10^{-309}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left(e^{1 \cdot \left(\log 1 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 1.60231479643995 \cdot 10^{-108}:\\ \;\;\;\;\log \left({\left(e^{\frac{2 \cdot 1}{{\left(\frac{k}{t}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)}}\right)}^{\left(\frac{\ell \cdot \ell}{\sin k}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\sqrt{2} \cdot \ell}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{3} \cdot \tan k\right)} \cdot \frac{\ell}{\sin k}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(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))))