Average Error: 34.8 → 27.9
Time: 34.8s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.8141662603144044 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \left(\frac{2}{Om} + U \cdot \frac{n}{Om \cdot Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 2.772988692385023 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(n \cdot 2\right)\right) + U \cdot \left(\left(\left(n \cdot 2\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \frac{\ell \cdot n}{Om} - \ell \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{\frac{n \cdot n}{\frac{\frac{Om}{U}}{U*}}}{Om} - \frac{n \cdot n}{\frac{Om}{U} \cdot \frac{Om}{U}}\right) - 4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\ \end{array} \]
\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 \leq -1.8141662603144044 \cdot 10^{+225}:\\
\;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \left(\frac{2}{Om} + U \cdot \frac{n}{Om \cdot Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\

\mathbf{elif}\;\ell \leq 2.772988692385023 \cdot 10^{+135}:\\
\;\;\;\;\sqrt{U \cdot \left(t \cdot \left(n \cdot 2\right)\right) + U \cdot \left(\left(\left(n \cdot 2\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \frac{\ell \cdot n}{Om} - \ell \cdot 2\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{\frac{n \cdot n}{\frac{\frac{Om}{U}}{U*}}}{Om} - \frac{n \cdot n}{\frac{Om}{U} \cdot \frac{Om}{U}}\right) - 4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\


\end{array}

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
↓
(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= l -1.8141662603144044e+225)
   (*
    (sqrt
     (*
      (* n U)
      (- (* (/ n Om) (/ U* Om)) (+ (/ 2.0 Om) (* U (/ n (* Om Om)))))))
    (* (sqrt 2.0) (- l)))
   (if (<= l 2.772988692385023e+135)
     (sqrt
      (+
       (* U (* t (* n 2.0)))
       (*
        U
        (* (* (* n 2.0) (/ l Om)) (- (* (- U* U) (/ (* l n) Om)) (* l 2.0))))))
     (*
      l
      (sqrt
       (-
        (*
         2.0
         (-
          (/ (/ (* n n) (/ (/ Om U) U*)) Om)
          (/ (* n n) (* (/ Om U) (/ Om U)))))
        (* 4.0 (* U (/ n Om)))))))))
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 tmp;
	if (l <= -1.8141662603144044e+225) {
		tmp = sqrt((n * U) * (((n / Om) * (U_42_ / Om)) - ((2.0 / Om) + (U * (n / (Om * Om)))))) * (sqrt(2.0) * -l);
	} else if (l <= 2.772988692385023e+135) {
		tmp = sqrt((U * (t * (n * 2.0))) + (U * (((n * 2.0) * (l / Om)) * (((U_42_ - U) * ((l * n) / Om)) - (l * 2.0)))));
	} else {
		tmp = l * sqrt((2.0 * ((((n * n) / ((Om / U) / U_42_)) / Om) - ((n * n) / ((Om / U) * (Om / U))))) - (4.0 * (U * (n / Om))));
	}
	return tmp;
}

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 l < -1.8141662603144044e225

    1. Initial program 64.0

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

    2. Simplified55.7

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

    3. Taylor expanded around -inf 32.9

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

    4. Simplified29.6

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

    if -1.8141662603144044e225 < l < 2.77298869238502309e135

    1. Initial program 29.9

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

    2. Simplified27.2

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

    4. Applied sub-neg_binary64_41227.2

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

    5. Applied distribute-rgt-in_binary64_36927.2

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

    6. Applied distribute-rgt-in_binary64_36927.2

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

    7. Simplified26.6

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

    if 2.77298869238502309e135 < l

    1. Initial program 60.5

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

    2. Simplified45.6

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

    4. Applied sub-neg_binary64_41245.6

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

    5. Applied distribute-rgt-in_binary64_36945.6

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

    6. Applied distribute-rgt-in_binary64_36945.6

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

    7. Simplified39.4

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

    8. Taylor expanded around inf 43.8

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

    9. Simplified36.9

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

  4. Final simplification27.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.8141662603144044 \cdot 10^{+225}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot \left(\frac{n}{Om} \cdot \frac{U*}{Om} - \left(\frac{2}{Om} + U \cdot \frac{n}{Om \cdot Om}\right)\right)} \cdot \left(\sqrt{2} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 2.772988692385023 \cdot 10^{+135}:\\ \;\;\;\;\sqrt{U \cdot \left(t \cdot \left(n \cdot 2\right)\right) + U \cdot \left(\left(\left(n \cdot 2\right) \cdot \frac{\ell}{Om}\right) \cdot \left(\left(U* - U\right) \cdot \frac{\ell \cdot n}{Om} - \ell \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \sqrt{2 \cdot \left(\frac{\frac{n \cdot n}{\frac{\frac{Om}{U}}{U*}}}{Om} - \frac{n \cdot n}{\frac{Om}{U} \cdot \frac{Om}{U}}\right) - 4 \cdot \left(U \cdot \frac{n}{Om}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021210 
(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*))))))