Toniolo and Linder, Equation (10-)

Average Error: 48.5 → 3.0

Time: 36.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -3.394358466966619 \cdot 10^{-135} \lor \neg \left(k \leq 1.8331473560650358 \cdot 10^{-66}\right):\\ \;\;\;\;\left|\frac{\ell}{k}\right| \cdot \left(\left|\frac{\ell}{k}\right| \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}}\\ \end{array}\]

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 (or (<= k -3.394358466966619e-135) (not (<= k 1.8331473560650358e-66)))
   (*
    (fabs (/ l k))
    (* (fabs (/ l k)) (/ 2.0 (/ (* t (pow (sin k) 2.0)) (cos k)))))
   (* (* (/ l k) (/ l k)) (/ 2.0 (/ (* (sin k) (* t (sin k))) (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 ((k <= -3.394358466966619e-135) || !(k <= 1.8331473560650358e-66)) {
		tmp = fabs(l / k) * (fabs(l / k) * (2.0 / ((t * pow(sin(k), 2.0)) / cos(k))));
	} else {
		tmp = ((l / k) * (l / k)) * (2.0 / ((sin(k) * (t * sin(k))) / 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 k < -3.39435846696661926e-135 or 1.8331473560650358e-66 < k

   1. Initial program 47.1

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

   2. Simplified 38.6

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

   3. Taylor expanded around 0 20.3

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

   4. Simplified 20.3

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

   6. Applied times-frac_binary64_425 19.9

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

   7. Applied *-un-lft-identity_binary64_419 19.9

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

   8. Applied times-frac_binary64_425 19.9

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

   9. Simplified 19.9

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

   11. Applied add-sqr-sqrt_binary64_441 19.9

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

   12. Simplified 19.9

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

   13. Simplified 7.5

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

   15. Applied associate-*l*_binary64_360 1.6

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

   if -3.39435846696661926e-135 < k < 1.8331473560650358e-66

   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 60.6

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

   3. Taylor expanded around 0 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 times-frac_binary64_425 47.2

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

   7. Applied *-un-lft-identity_binary64_419 47.2

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

   8. Applied times-frac_binary64_425 47.4

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

   9. Simplified 47.4

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

   11. Applied add-sqr-sqrt_binary64_441 47.4

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

   12. Simplified 47.4

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

   13. Simplified 34.7

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

   15. Applied sqr-pow_binary64_391 34.7

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

   16. Applied associate-*r*_binary64_359 16.9

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

   17. Simplified 16.9

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

4. Final simplification 3.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.394358466966619 \cdot 10^{-135} \lor \neg \left(k \leq 1.8331473560650358 \cdot 10^{-66}\right):\\ \;\;\;\;\left|\frac{\ell}{k}\right| \cdot \left(\left|\frac{\ell}{k}\right| \cdot \frac{2}{\frac{t \cdot {\sin k}^{2}}{\cos k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\frac{\sin k \cdot \left(t \cdot \sin k\right)}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021097 
(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))))