Toniolo and Linder, Equation (13)

Average Error: 35.0 → 19.5

Time: 2.0m

Precision: 64

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;U \le -3.04445170882942234 \cdot 10^{-304} \lor \neg \left(U \le 3.1279589487835526 \cdot 10^{-279}\right):\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot 1}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt{U}\right) \cdot \left(\sqrt{U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double temp;
	if (((U <= -3.0444517088294223e-304) || !(U <= 3.1279589487835526e-279))) {
		temp = (fabs(cbrt((2.0 * n))) * (fabs(cbrt((cbrt((2.0 * n)) * U))) * sqrt((cbrt((cbrt((2.0 * n)) * U)) * ((t - (2.0 * ((l * 1.0) / (Om / l)))) - ((n * pow((l / Om), (2.0 / 2.0))) * (pow((l / Om), (2.0 / 2.0)) * (U - U_42_))))))));
	} else {
		temp = (fabs(cbrt((2.0 * n))) * sqrt(((cbrt((2.0 * n)) * sqrt(U)) * (sqrt(U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))))));
	}
	return temp;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 2 regimes

2. if U < -3.0444517088294223e-304 or 3.1279589487835526e-279 < U

   1. Initial program 34.7

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 35.0

      \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}\right) \cdot \sqrt[3]{2 \cdot n}\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied associate-*l* 35.0

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}\right) \cdot \left(\sqrt[3]{2 \cdot n} \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Applied associate-*l* 34.2

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}\right) \cdot \left(\left(\sqrt[3]{2 \cdot n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]

   6. Applied sqrt-prod 29.4

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}} \cdot \sqrt{\left(\sqrt[3]{2 \cdot n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]

   7. Simplified 29.4

      \[\leadsto \color{blue}{\left|\sqrt[3]{2 \cdot n}\right|} \cdot \sqrt{\left(\sqrt[3]{2 \cdot n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 29.5

      \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right) \cdot \sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   10. Applied associate-*l* 29.5

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right) \cdot \left(\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]

   11. Applied sqrt-prod 23.9

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)}\]

   12. Simplified 23.9

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\color{blue}{\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right|} \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]
   13. Using strategy rm

   14. Applied *-un-lft-identity 23.9

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \color{blue}{\left(1 \cdot \ell\right)}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]

   15. Applied associate-*r* 23.9

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\color{blue}{\left(\ell \cdot 1\right) \cdot \ell}}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]

   16. Applied associate-/l* 20.2

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell \cdot 1}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\right)\]
   17. Using strategy rm

   18. Applied sqr-pow 20.2

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot 1}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)}\right)\]

   19. Applied associate-*r* 18.8

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot 1}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)}\right)\]

   20. Applied associate-*l* 18.6

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot 1}{\frac{Om}{\ell}}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)}\right)\]

   if -3.0444517088294223e-304 < U < 3.1279589487835526e-279

   1. Initial program 42.8

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 43.1

      \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}\right) \cdot \sqrt[3]{2 \cdot n}\right)} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   4. Applied associate-*l* 43.1

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}\right) \cdot \left(\sqrt[3]{2 \cdot n} \cdot U\right)\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   5. Applied associate-*l* 42.9

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}\right) \cdot \left(\left(\sqrt[3]{2 \cdot n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]

   6. Applied sqrt-prod 42.8

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{2 \cdot n} \cdot \sqrt[3]{2 \cdot n}} \cdot \sqrt{\left(\sqrt[3]{2 \cdot n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}}\]

   7. Simplified 42.8

      \[\leadsto \color{blue}{\left|\sqrt[3]{2 \cdot n}\right|} \cdot \sqrt{\left(\sqrt[3]{2 \cdot n} \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 46.2

      \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\left(\sqrt[3]{2 \cdot n} \cdot \color{blue}{\left(\sqrt{U} \cdot \sqrt{U}\right)}\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   10. Applied associate-*r* 46.2

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\color{blue}{\left(\left(\sqrt[3]{2 \cdot n} \cdot \sqrt{U}\right) \cdot \sqrt{U}\right)} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

   11. Applied associate-*l* 39.8

       \[\leadsto \left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\color{blue}{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt{U}\right) \cdot \left(\sqrt{U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 19.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -3.04445170882942234 \cdot 10^{-304} \lor \neg \left(U \le 3.1279589487835526 \cdot 10^{-279}\right):\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \left(\left|\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{2 \cdot n} \cdot U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot 1}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt[3]{2 \cdot n}\right| \cdot \sqrt{\left(\sqrt[3]{2 \cdot n} \cdot \sqrt{U}\right) \cdot \left(\sqrt{U} \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))