Average Error: 35.4 → 33.1
Time: 37.4s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.1662789012153849 \cdot 10^{139}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \le -6.1705074707341553 \cdot 10^{96}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right) + 2 \cdot \left(\frac{U \cdot \left({n}^{2} \cdot U*\right)}{{Om}^{2}} \cdot {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{\ell}\right)\right)}\right)}^{1}}\right)}^{1}\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}\\ \mathbf{elif}\;\ell \le -4.59960513501680617 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \le 8.396131385583266 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - 0\right)}\\ \end{array}\]
\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}\;\ell \le -1.1662789012153849 \cdot 10^{139}:\\
\;\;\;\;\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}\;\ell \le -6.1705074707341553 \cdot 10^{96}:\\
\;\;\;\;\sqrt{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right) + 2 \cdot \left(\frac{U \cdot \left({n}^{2} \cdot U*\right)}{{Om}^{2}} \cdot {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{\ell}\right)\right)}\right)}^{1}}\right)}^{1}\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}\\

\mathbf{elif}\;\ell \le -4.59960513501680617 \cdot 10^{-150}:\\
\;\;\;\;\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}\;\ell \le 8.396131385583266 \cdot 10^{135}:\\
\;\;\;\;\sqrt{\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)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - 0\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) (((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 ((l <= -1.166278901215385e+139)) {
		VAR = ((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_1;
		if ((l <= -6.170507470734155e+96)) {
			VAR_1 = ((double) sqrt(((double) (((double) (((double) (2.0 * ((double) (t * ((double) (U * n)))))) + ((double) (2.0 * ((double) (((double) (((double) (U * ((double) (((double) pow(n, 2.0)) * U_42_)))) / ((double) pow(Om, 2.0)))) * ((double) pow(((double) (1.0 / ((double) pow(((double) exp(((double) (2.0 * ((double) (((double) log(1.0)) + ((double) log(((double) (-1.0 / l)))))))))), 1.0)))), 1.0)))))))) - ((double) (4.0 * ((double) (((double) (U * ((double) (n * ((double) pow(l, 2.0)))))) / Om))))))));
		} else {
			double VAR_2;
			if ((l <= -4.599605135016806e-150)) {
				VAR_2 = ((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_3;
				if ((l <= 8.396131385583266e+135)) {
					VAR_3 = ((double) sqrt(((double) (((double) (2.0 * n)) * ((double) (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_))))))))))));
				} else {
					VAR_3 = ((double) sqrt(((double) (((double) (((double) (2.0 * n)) * U)) * ((double) (((double) (t - ((double) (2.0 * ((double) (l / ((double) (Om / l)))))))) - 0.0))))));
				}
				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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if l < -1.1662789012153849e139 or -6.1705074707341553e96 < l < -4.59960513501680617e-150

    1. Initial program 40.6

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

    3. Applied associate-/l*36.8

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

    if -1.1662789012153849e139 < l < -6.1705074707341553e96

    1. Initial program 33.1

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

    2. Taylor expanded around -inf 32.1

      \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right) + 2 \cdot \left(\frac{U \cdot \left({n}^{2} \cdot U*\right)}{{Om}^{2}} \cdot {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{\ell}\right)\right)}\right)}^{1}}\right)}^{1}\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\]

    if -4.59960513501680617e-150 < l < 8.396131385583266e135

    1. Initial program 27.8

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

    3. Applied associate-*l*28.4

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

    if 8.396131385583266e135 < l

    1. Initial program 60.6

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

    3. Applied associate-/l*46.8

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

    4. Taylor expanded around 0 47.6

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{0}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification33.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.1662789012153849 \cdot 10^{139}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \le -6.1705074707341553 \cdot 10^{96}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right) + 2 \cdot \left(\frac{U \cdot \left({n}^{2} \cdot U*\right)}{{Om}^{2}} \cdot {\left(\frac{1}{{\left(e^{2 \cdot \left(\log 1 + \log \left(\frac{-1}{\ell}\right)\right)}\right)}^{1}}\right)}^{1}\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}\\ \mathbf{elif}\;\ell \le -4.59960513501680617 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;\ell \le 8.396131385583266 \cdot 10^{135}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - 0\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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*))))))