Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 18.0

Time: 23.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5.324378565018582 \cdot 10^{+304}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}\right)}^{1} \cdot \left(\left(\left({\left({t}^{1}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot {\left({\left(\sqrt[3]{k}\right)}^{2}\right)}^{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 5.324378565018582e+304)
   (/
    2.0
    (*
     (pow (pow (* (cbrt k) (cbrt k)) 2.0) 1.0)
     (*
      (* (* (pow (pow t 1.0) 1.0) (/ (sin k) (cos k))) (/ (sin k) (* l l)))
      (pow (pow (cbrt k) 2.0) 1.0))))
   (/
    2.0
    (*
     (*
      (*
       (/ (pow (* (cbrt t) (cbrt t)) 3.0) l)
       (* (sin k) (/ (pow (cbrt t) 3.0) l)))
      (tan k))
     (pow (/ k t) 2.0)))))

C

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

↓

double code(double t, double l, double k) {
	double tmp;
	if ((((double) (l * l)) <= 5.324378565018582e+304)) {
		tmp = (2.0 / ((double) (((double) pow(((double) pow(((double) (((double) cbrt(k)) * ((double) cbrt(k)))), 2.0)), 1.0)) * ((double) (((double) (((double) (((double) pow(((double) pow(t, 1.0)), 1.0)) * (((double) sin(k)) / ((double) cos(k))))) * (((double) sin(k)) / ((double) (l * l))))) * ((double) pow(((double) pow(((double) cbrt(k)), 2.0)), 1.0)))))));
	} else {
		tmp = (2.0 / ((double) (((double) (((double) ((((double) pow(((double) (((double) cbrt(t)) * ((double) cbrt(t)))), 3.0)) / l) * ((double) (((double) sin(k)) * (((double) pow(((double) cbrt(t)), 3.0)) / l))))) * ((double) tan(k)))) * ((double) pow((k / t), 2.0)))));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 5.3243785650185817e304

   1. Initial program 44.8

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

   2. Simplified 35.8

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

   3. Taylor expanded around inf 14.0

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

   4. Simplified 14.0

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

   6. Applied unpow-prod-down_binary64 14.0

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

   7. Applied associate-*l*_binary64 14.2

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

   8. Simplified 14.2

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

   10. Applied add-cube-cbrt_binary64 14.5

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

   11. Applied unpow-prod-down_binary64 14.5

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

   12. Applied unpow-prod-down_binary64 14.5

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

   13. Applied associate-*l*_binary64 12.6

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

   14. Simplified 12.6

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

   16. Applied unpow2_binary64 12.6

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

   17. Applied times-frac_binary64 12.3

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

   18. Applied associate-*r*_binary64 12.2

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

   if 5.3243785650185817e304 < (*.f64 l l)

   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 63.7

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

   4. Applied add-cube-cbrt_binary64 63.7

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

   5. Applied unpow-prod-down_binary64 63.7

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

   6. Applied times-frac_binary64 48.7

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

   7. Applied associate-*l*_binary64 48.7

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

   8. Simplified 48.7

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5.324378565018582 \cdot 10^{+304}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2}\right)}^{1} \cdot \left(\left(\left({\left({t}^{1}\right)}^{1} \cdot \frac{\sin k}{\cos k}\right) \cdot \frac{\sin k}{\ell \cdot \ell}\right) \cdot {\left({\left(\sqrt[3]{k}\right)}^{2}\right)}^{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\sin k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))))