Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 8.1

Time: 23.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq 2.9517580616168344 \cdot 10^{+103}:\\ \;\;\;\;\frac{\ell}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{\sqrt[3]{k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array}\]

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 2.9517580616168344e+103)
   (*
    (/ l (* (cbrt k) (cbrt k)))
    (* (/ 2.0 (/ (* k (* t (pow (sin k) 2.0))) (cos k))) (/ l (cbrt k))))
   (* (* (/ l k) (/ l k)) (/ 2.0 (/ (* t (pow (sin k) 2.0)) (cos k))))))

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 tmp;
	if (l <= 2.9517580616168344e+103) {
		tmp = (l / (cbrt(k) * cbrt(k))) * ((2.0 / ((k * (t * pow(sin(k), 2.0))) / cos(k))) * (l / cbrt(k)));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 / ((t * pow(sin(k), 2.0)) / cos(k)));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if l < 2.9517580616168344e103

   1. Initial program 47.0

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

   2. Simplified 39.0

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

   3. Taylor expanded around inf 19.2

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

   4. Simplified 19.2

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

   6. Applied associate-*l*_binary64_360 17.4

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

   8. Applied times-frac_binary64_425 15.3

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

   9. Applied *-un-lft-identity_binary64_419 15.3

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

   10. Applied times-frac_binary64_425 15.3

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

   11. Simplified 15.2

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

   13. Applied add-cube-cbrt_binary64_454 15.4

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

   14. Applied times-frac_binary64_425 12.2

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

   15. Applied associate-*l*_binary64_360 8.1

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

   16. Simplified 8.1

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

   if 2.9517580616168344e103 < l

   1. Initial program 60.3

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

   2. Simplified 58.5

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

   3. Taylor expanded around inf 53.2

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

   4. Simplified 53.2

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

   6. Applied associate-*l*_binary64_360 51.1

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

   8. Applied times-frac_binary64_425 47.4

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

   9. Applied *-un-lft-identity_binary64_419 47.4

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

   10. Applied times-frac_binary64_425 47.4

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

   11. Simplified 47.4

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

   13. Applied *-un-lft-identity_binary64_419 47.4

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

   14. Applied times-frac_binary64_425 47.4

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

   15. Applied *-un-lft-identity_binary64_419 47.4

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

   16. Applied times-frac_binary64_425 47.2

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

   17. Applied associate-*r*_binary64_359 46.3

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

   18. Simplified 7.8

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

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.9517580616168344 \cdot 10^{+103}:\\ \;\;\;\;\frac{\ell}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{\sqrt[3]{k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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))))