Average Error: 34.4 → 31.5
Time: 21.1s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;Om \leq -2.502409614250465 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.8007965386908381 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \left({\left(\frac{1}{{\left({\left(\frac{1}{\ell}\right)}^{2}\right)}^{1}}\right)}^{1} \cdot \left(\frac{U}{Om} \cdot \frac{n \cdot \left(n \cdot U*\right)}{Om}\right) - 2 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 6.559726813680222 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}\\ \end{array}\]
\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}\;Om \leq -2.502409614250465 \cdot 10^{-190}:\\
\;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\

\mathbf{elif}\;Om \leq 1.8007965386908381 \cdot 10^{-276}:\\
\;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \left({\left(\frac{1}{{\left({\left(\frac{1}{\ell}\right)}^{2}\right)}^{1}}\right)}^{1} \cdot \left(\frac{U}{Om} \cdot \frac{n \cdot \left(n \cdot U*\right)}{Om}\right) - 2 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)}\\

\mathbf{elif}\;Om \leq 6.559726813680222 \cdot 10^{-73}:\\
\;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}\\

\end{array}
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) (l * l)) / Om))))) - ((double) (((double) (n * ((double) pow((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 ((Om <= -2.502409614250465e-190)) {
		VAR = ((double) sqrt(((double) (2.0 * ((double) (((double) (n * U)) * ((double) (t + ((double) (((double) (((double) (n * ((double) pow((l / Om), 2.0)))) * ((double) (U_42_ - U)))) - ((double) (2.0 * ((double) (l * (l / Om)))))))))))))));
	} else {
		double VAR_1;
		if ((Om <= 1.8007965386908381e-276)) {
			VAR_1 = ((double) sqrt(((double) (2.0 * ((double) (((double) (n * ((double) (U * t)))) + ((double) (((double) (((double) pow((1.0 / ((double) pow(((double) pow((1.0 / l), 2.0)), 1.0))), 1.0)) * ((double) ((U / Om) * (((double) (n * ((double) (n * U_42_)))) / Om))))) - ((double) (2.0 * ((double) ((U / Om) * ((double) (n * ((double) (l * l))))))))))))))));
		} else {
			double VAR_2;
			if ((Om <= 6.559726813680222e-73)) {
				VAR_2 = ((double) sqrt(((double) (2.0 * ((double) (((double) (n * U)) * ((double) (t + ((double) (((double) (((double) (n * ((double) pow((l / Om), 2.0)))) * ((double) (U_42_ - U)))) - ((double) (2.0 * ((double) (l * (l / Om)))))))))))))));
			} else {
				VAR_2 = ((double) sqrt(((double) (2.0 * ((double) (n * ((double) (U * ((double) (t + ((double) (((double) (((double) (n * ((double) pow((l / Om), (2.0 / 2.0))))) * ((double) (((double) (U_42_ - U)) * ((double) pow((l / Om), (2.0 / 2.0))))))) - ((double) (2.0 * ((double) (l * (l / Om)))))))))))))))));
			}
			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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if Om < -2.50240961425046487e-190 or 1.8007965386908381e-276 < Om < 6.55972681368022225e-73

    1. Initial program 34.5

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

    2. Simplified32.3

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

    4. Applied associate-*r*31.8

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

    6. Applied associate-*r*31.8

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

    if -2.50240961425046487e-190 < Om < 1.8007965386908381e-276

    1. Initial program 46.0

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

    2. Simplified47.5

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

    4. Applied sqr-pow47.5

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

    5. Applied associate-*l*45.8

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

    6. Simplified45.8

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

    7. Taylor expanded around inf 64.0

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

    8. Simplified46.6

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

    if 6.55972681368022225e-73 < Om

    1. Initial program 32.0

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

    2. Simplified28.9

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

    4. Applied sqr-pow28.9

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

    5. Applied associate-*l*28.4

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

    6. Simplified28.4

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

    8. Applied associate-*r*28.0

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

  4. Final simplification31.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;Om \leq -2.502409614250465 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 1.8007965386908381 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \left({\left(\frac{1}{{\left({\left(\frac{1}{\ell}\right)}^{2}\right)}^{1}}\right)}^{1} \cdot \left(\frac{U}{Om} \cdot \frac{n \cdot \left(n \cdot U*\right)}{Om}\right) - 2 \cdot \left(\frac{U}{Om} \cdot \left(n \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;Om \leq 6.559726813680222 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(n \cdot U\right) \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U* - U\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot \left(t + \left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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*))))))