Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 9.3

Time: 28.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5.571154850637776 \cdot 10^{-290}:\\ \;\;\;\;\frac{2}{\frac{\frac{1}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}{\cos k}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2.99273882252485 \cdot 10^{+306}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell \cdot \ell} \cdot \left(k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{\sqrt[3]{k}}{\ell}\right)}{\cos k}}\\ \end{array}\]

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 l) 5.571154850637776e-290)
   (/ 2.0 (/ (* (/ 1.0 l) (* (/ k l) (* k (* t (pow (sin k) 2.0))))) (cos k)))
   (if (<= (* l l) 2.99273882252485e+306)
     (/ 2.0 (/ (* (/ k (* l l)) (* k (* (sin k) (* t (sin k))))) (cos k)))
     (/
      2.0
      (/
       (*
        (/ (* (cbrt k) (cbrt k)) l)
        (* (* k (* t (pow (sin k) 2.0))) (/ (cbrt k) l)))
       (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 * l) <= 5.571154850637776e-290) {
		tmp = 2.0 / (((1.0 / l) * ((k / l) * (k * (t * pow(sin(k), 2.0))))) / cos(k));
	} else if ((l * l) <= 2.99273882252485e+306) {
		tmp = 2.0 / (((k / (l * l)) * (k * (sin(k) * (t * sin(k))))) / cos(k));
	} else {
		tmp = 2.0 / ((((cbrt(k) * cbrt(k)) / l) * ((k * (t * pow(sin(k), 2.0))) * (cbrt(k) / l))) / cos(k));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.5711548506377763e-290

   1. Initial program 46.5

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

   2. Simplified 37.5

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

   3. Taylor expanded around 0 20.0

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

   4. Simplified 20.0

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

   6. Applied associate-*l*_binary64_360 20.0

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

   8. Applied associate-/r*_binary64_363 20.0

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

   9. Simplified 19.8

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

   11. Applied *-un-lft-identity_binary64_419 19.8

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

   12. Applied times-frac_binary64_425 18.1

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

   13. Applied associate-*l*_binary64_360 12.3

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

   if 5.5711548506377763e-290 < (*.f64 l l) < 2.9927388225248498e306

   1. Initial program 44.7

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

   2. Simplified 36.0

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

   3. Taylor expanded around 0 13.2

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

   4. Simplified 13.2

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

   6. Applied associate-*l*_binary64_360 8.8

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

   8. Applied associate-/r*_binary64_363 8.8

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

   9. Simplified 4.3

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

   11. Applied unpow2_binary64_484 4.3

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

   12. Applied associate-*r*_binary64_359 4.0

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

   if 2.9927388225248498e306 < (*.f64 l l)

   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

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

   3. Taylor expanded around 0 63.8

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

   4. Simplified 63.8

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

   6. Applied associate-*l*_binary64_360 63.8

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

   8. Applied associate-/r*_binary64_363 63.8

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

   9. Simplified 63.6

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

   11. Applied add-cube-cbrt_binary64_454 63.6

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

   12. Applied times-frac_binary64_425 33.7

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

   13. Applied associate-*l*_binary64_360 19.0

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5.571154850637776 \cdot 10^{-290}:\\ \;\;\;\;\frac{2}{\frac{\frac{1}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)\right)}{\cos k}}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2.99273882252485 \cdot 10^{+306}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell \cdot \ell} \cdot \left(k \cdot \left(\sin k \cdot \left(t \cdot \sin k\right)\right)\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\sqrt[3]{k} \cdot \sqrt[3]{k}}{\ell} \cdot \left(\left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right) \cdot \frac{\sqrt[3]{k}}{\ell}\right)}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021058 
(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))))