Average Error: 34.8 → 31.1
Time: 23.5s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -9.27291794463707 \cdot 10^{-307} \lor \neg \left(t \leq 1.3152323392947841 \cdot 10^{-194}\right) \land t \leq 7.158335102890174 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + n \cdot \left(\sqrt[3]{U - U*} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right)\right)\right)}\\ \end{array}\]
\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}\;t \leq -9.27291794463707 \cdot 10^{-307} \lor \neg \left(t \leq 1.3152323392947841 \cdot 10^{-194}\right) \land t \leq 7.158335102890174 \cdot 10^{+73}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \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)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + n \cdot \left(\sqrt[3]{U - U*} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right)\right)\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 (or (<= t -9.27291794463707e-307)
         (and (not (<= t 1.3152323392947841e-194))
              (<= t 7.158335102890174e+73)))
   (sqrt
    (*
     (* 2.0 n)
     (*
      U
      (+ t (- (* n (* (pow (/ l Om) 2.0) (- U* U))) (* 2.0 (* l (/ l Om))))))))
   (*
    (sqrt (* 2.0 (* n U)))
    (sqrt
     (-
      t
      (+
       (* 2.0 (* l (/ l Om)))
       (*
        n
        (*
         (cbrt (- U U*))
         (* (pow (/ l Om) 2.0) (* (cbrt (- U U*)) (cbrt (- U U*))))))))))))
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 tmp;
	if (((t <= -9.27291794463707e-307) || (!(t <= 1.3152323392947841e-194) && (t <= 7.158335102890174e+73)))) {
		tmp = ((double) sqrt(((double) (((double) (2.0 * n)) * ((double) (U * ((double) (t + ((double) (((double) (n * ((double) (((double) pow((l / Om), 2.0)) * ((double) (U_42_ - U)))))) - ((double) (2.0 * ((double) (l * (l / Om)))))))))))))));
	} else {
		tmp = ((double) (((double) sqrt(((double) (2.0 * ((double) (n * U)))))) * ((double) sqrt(((double) (t - ((double) (((double) (2.0 * ((double) (l * (l / Om))))) + ((double) (n * ((double) (((double) cbrt(((double) (U - U_42_)))) * ((double) (((double) pow((l / Om), 2.0)) * ((double) (((double) cbrt(((double) (U - U_42_)))) * ((double) cbrt(((double) (U - U_42_))))))))))))))))))));
	}
	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 2 regimes

  2. if t < -9.27291794463706929e-307 or 1.315232339294784e-194 < t < 7.15833510289017356e73

    1. Initial program Error: 34.2 bits

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

    3. Applied associate-*l*Error: 34.1 bits

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

    4. SimplifiedError: 32.0 bits

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

    if -9.27291794463706929e-307 < t < 1.315232339294784e-194 or 7.15833510289017356e73 < t

    1. Initial program Error: 36.3 bits

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

    3. Applied add-cube-cbrtError: 36.3 bits

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

    4. Applied associate-*r*Error: 36.3 bits

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

    5. SimplifiedError: 36.7 bits

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

    7. Applied sqrt-prodError: 30.9 bits

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

    8. SimplifiedError: 30.9 bits

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

    9. SimplifiedError: 28.9 bits

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

  4. Final simplificationError: 31.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.27291794463707 \cdot 10^{-307} \lor \neg \left(t \leq 1.3152323392947841 \cdot 10^{-194}\right) \land t \leq 7.158335102890174 \cdot 10^{+73}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t + \left(n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right)\right) - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot U\right)} \cdot \sqrt{t - \left(2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right) + n \cdot \left(\sqrt[3]{U - U*} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\sqrt[3]{U - U*} \cdot \sqrt[3]{U - U*}\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(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*))))))