Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 8.8

Time: 23.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -6.210609360145837 \cdot 10^{-162}:\\ \;\;\;\;\frac{\ell}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{\sqrt[3]{k}}\right)\\ \mathbf{elif}\;k \leq 4.5629687678940777 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sqrt{k}} \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{\sqrt{k}}\right)\\ \end{array}\]

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 (<= k -6.210609360145837e-162)
   (*
    (/ l (* (cbrt k) (cbrt k)))
    (* (/ 2.0 (/ (* k (* t (pow (sin k) 2.0))) (cos k))) (/ l (cbrt k))))
   (if (<= k 4.5629687678940777e-128)
     (/
      2.0
      (* (* (* (/ (* t t) l) (* (sin k) (/ t l))) (tan k)) (pow (/ k t) 2.0)))
     (*
      (/ l (sqrt k))
      (* (/ 2.0 (/ (* k (* t (pow (sin k) 2.0))) (cos k))) (/ l (sqrt 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 (k <= -6.210609360145837e-162) {
		tmp = (l / (cbrt(k) * cbrt(k))) * ((2.0 / ((k * (t * pow(sin(k), 2.0))) / cos(k))) * (l / cbrt(k)));
	} else if (k <= 4.5629687678940777e-128) {
		tmp = 2.0 / (((((t * t) / l) * (sin(k) * (t / l))) * tan(k)) * pow((k / t), 2.0));
	} else {
		tmp = (l / sqrt(k)) * ((2.0 / ((k * (t * pow(sin(k), 2.0))) / cos(k))) * (l / sqrt(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 k < -6.210609360145837e-162

   1. Initial program 48.2

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

   2. Simplified 40.2

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

   3. Taylor expanded around inf 22.0

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

   4. Simplified 22.0

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

   6. Applied associate-*l*_binary64 19.7

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

   8. Applied times-frac_binary64 17.5

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

   9. Applied *-un-lft-identity_binary64 17.5

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

   10. Applied times-frac_binary64 17.5

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

   11. Simplified 17.3

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

   13. Applied add-cube-cbrt_binary64 17.5

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

   14. Applied times-frac_binary64 12.0

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

   15. Applied associate-*l*_binary64 7.2

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

   16. Simplified 7.2

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

   if -6.210609360145837e-162 < k < 4.5629687678940777e-128

   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 62.9

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

   4. Applied unpow3_binary64 62.9

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

   5. Applied times-frac_binary64 60.2

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

   6. Applied associate-*l*_binary64 57.0

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

   7. Simplified 57.0

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

   if 4.5629687678940777e-128 < k

   1. Initial program 47.3

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

   2. Simplified 38.9

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

   3. Taylor expanded around inf 20.9

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

   4. Simplified 20.9

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

   6. Applied associate-*l*_binary64 18.5

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

   8. Applied times-frac_binary64 16.1

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

   9. Applied *-un-lft-identity_binary64 16.1

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

   10. Applied times-frac_binary64 16.2

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

   11. Simplified 16.0

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

   13. Applied add-sqr-sqrt_binary64 16.1

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

   14. Applied times-frac_binary64 10.7

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

   15. Applied associate-*l*_binary64 6.1

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

   16. Simplified 6.1

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.210609360145837 \cdot 10^{-162}:\\ \;\;\;\;\frac{\ell}{\sqrt[3]{k} \cdot \sqrt[3]{k}} \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{\sqrt[3]{k}}\right)\\ \mathbf{elif}\;k \leq 4.5629687678940777 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sqrt{k}} \cdot \left(\frac{2}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}} \cdot \frac{\ell}{\sqrt{k}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020231 
(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))))