Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 17.2

Time: 15.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.9384138131524779 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\frac{1}{\sin k}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}{\tan k} \cdot \frac{\sqrt[3]{\ell}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 1.001927250660164 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot {\left(\sin k\right)}^{2}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1}}{{t}^{\left(\frac{3}{2}\right)}} \cdot \frac{\frac{\sqrt[3]{\ell}}{\sin k}}{{t}^{\left(\frac{3}{2}\right)}}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

C

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

↓

double code(double t, double l, double k) {
	double VAR;
	if ((t <= -1.938413813152478e-87)) {
		VAR = ((((2.0 * ((l / pow((cbrt(t) * cbrt(t)), 3.0)) * ((1.0 / sin(k)) / pow(cbrt(t), 3.0)))) * (cbrt(l) * cbrt(l))) / tan(k)) * (cbrt(l) / fma(2.0, 1.0, pow((k / t), 2.0))));
	} else {
		double VAR_1;
		if ((t <= 1.0019272506601638e-82)) {
			VAR_1 = (2.0 * (((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * pow(sin(k), 2.0))) * pow((1.0 / pow(t, 1.0)), 1.0)));
		} else {
			VAR_1 = (((2.0 * ((((cbrt(l) * cbrt(l)) / 1.0) / pow(t, (3.0 / 2.0))) * ((cbrt(l) / sin(k)) / pow(t, (3.0 / 2.0))))) / tan(k)) * (l / fma(2.0, 1.0, pow((k / t), 2.0))));
		}
		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 < -1.938413813152478e-87

   1. Initial program 22.9

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

   2. Simplified 22.6

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

   4. Applied *-un-lft-identity 22.6

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

   5. Applied times-frac 21.5

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

   6. Applied associate-*r* 18.1

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

   7. Simplified 16.8

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

   9. Applied div-inv 16.8

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

   10. Applied associate-*l* 16.8

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

   11. Simplified 16.5

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

   13. Applied add-cube-cbrt 16.8

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

   14. Applied unpow-prod-down 16.8

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

   15. Applied div-inv 16.8

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

   16. Applied times-frac 14.5

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

   18. Applied *-un-lft-identity 14.5

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

   19. Applied add-cube-cbrt 14.6

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

   20. Applied times-frac 14.6

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

   21. Applied associate-*r* 14.9

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

   22. Simplified 14.3

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

   if -1.938413813152478e-87 < t < 1.0019272506601638e-82

   1. Initial program 60.2

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

   2. Simplified 61.7

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

   4. Applied *-un-lft-identity 61.7

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

   5. Applied times-frac 61.6

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

   6. Applied associate-*r* 61.9

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

   7. Simplified 61.7

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

   9. Applied div-inv 61.7

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

   10. Applied associate-*l* 61.7

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

   11. Simplified 60.2

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

   12. Taylor expanded around inf 30.0

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

   if 1.0019272506601638e-82 < t

   1. Initial program 22.4

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

   2. Simplified 22.7

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

   4. Applied *-un-lft-identity 22.7

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

   5. Applied times-frac 21.6

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

   6. Applied associate-*r* 18.0

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

   7. Simplified 16.5

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

   9. Applied div-inv 16.5

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

   10. Applied associate-*l* 16.5

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

   11. Simplified 16.4

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

   13. Applied sqr-pow 16.4

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

   14. Applied *-un-lft-identity 16.4

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

   15. Applied add-cube-cbrt 16.6

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

   16. Applied times-frac 16.6

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

   17. Applied times-frac 11.5

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

4. Final simplification 17.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.9384138131524779 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}} \cdot \frac{\frac{1}{\sin k}}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}{\tan k} \cdot \frac{\sqrt[3]{\ell}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \le 1.001927250660164 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot {\left(\sin k\right)}^{2}} \cdot {\left(\frac{1}{{t}^{1}}\right)}^{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1}}{{t}^{\left(\frac{3}{2}\right)}} \cdot \frac{\frac{\sqrt[3]{\ell}}{\sin k}}{{t}^{\left(\frac{3}{2}\right)}}\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))