Toniolo and Linder, Equation (13)

Average Error: 34.6 → 31.1

Time: 37.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.13321154877586516 \cdot 10^{167}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - 0\right)\right)}\\ \mathbf{elif}\;t \le -3.62528278840627114 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 9.7277242626137345 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 3.4992493273958607 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{elif}\;t \le 1.26333709025898057 \cdot 10^{191}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

C

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

↓

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double VAR;
	if ((t <= -1.1332115487758652e+167)) {
		VAR = ((double) sqrt(((double) (((double) (2.0 * n)) * ((double) (U * ((double) (((double) (t - ((double) (2.0 * ((double) (((double) (l * l)) / Om)))))) - 0.0))))))));
	} else {
		double VAR_1;
		if ((t <= -3.625282788406271e-207)) {
			VAR_1 = ((double) sqrt(((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * ((double) (l / ((double) (Om / l)))))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), 2.0)))) * ((double) (U - U_42_))))))))));
		} else {
			double VAR_2;
			if ((t <= 9.727724262613734e-120)) {
				VAR_2 = ((double) sqrt(((double) (((double) (2.0 * n)) * ((double) (U * ((double) (((double) (t - ((double) (2.0 * ((double) (((double) (l * l)) / Om)))))) - ((double) (((double) (((double) (n * ((double) pow(((double) (l / Om)), ((double) (2.0 / 2.0)))))) * ((double) pow(((double) (l / Om)), ((double) (2.0 / 2.0)))))) * ((double) (U - U_42_))))))))))));
			} else {
				double VAR_3;
				if ((t <= 3.4992493273958607e-56)) {
					VAR_3 = ((double) (((double) sqrt(((double) (((double) (2.0 * n)) * U)))) * ((double) sqrt(((double) (((double) (t - ((double) (2.0 * ((double) (l / ((double) (Om / l)))))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), 2.0)))) * ((double) (U - U_42_))))))))));
				} else {
					double VAR_4;
					if ((t <= 1.2633370902589806e+191)) {
						VAR_4 = ((double) sqrt(((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * ((double) (l / ((double) (Om / l)))))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), 2.0)))) * ((double) (U - U_42_))))))))));
					} else {
						VAR_4 = ((double) (((double) sqrt(((double) (((double) (2.0 * n)) * U)))) * ((double) sqrt(((double) (((double) (t - ((double) (2.0 * ((double) (l / ((double) (Om / l)))))))) - ((double) (((double) (n * ((double) pow(((double) (l / Om)), 2.0)))) * ((double) (U - U_42_))))))))));
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 4 regimes

2. if t < -1.13321154877586516e167

   1. Initial program 38.5

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

   3. Applied associate-*l* 38.7

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

   4. Taylor expanded around 0 38.4

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \color{blue}{0}\right)\right)}\]

   if -1.13321154877586516e167 < t < -3.62528278840627114e-207 or 3.4992493273958607e-56 < t < 1.26333709025898057e191

   1. Initial program 31.2

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

   3. Applied associate-/l* 28.3

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

   if -3.62528278840627114e-207 < t < 9.7277242626137345e-120

   1. Initial program 38.4

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

   3. Applied associate-*l* 39.2

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

   5. Applied sqr-pow 39.2

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

   6. Applied associate-*r* 37.5

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

   if 9.7277242626137345e-120 < t < 3.4992493273958607e-56 or 1.26333709025898057e191 < t

   1. Initial program 36.9

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

   3. Applied associate-/l* 34.5

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

   5. Applied sqrt-prod 24.8

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 31.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.13321154877586516 \cdot 10^{167}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - 0\right)\right)}\\ \mathbf{elif}\;t \le -3.62528278840627114 \cdot 10^{-207}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 9.7277242626137345 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;t \le 3.4992493273958607 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{elif}\;t \le 1.26333709025898057 \cdot 10^{191}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))