Average Error: 34.5 → 28.3
Time: 24.4s
Precision: binary64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]
↓
\[\begin{array}{l} t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{if}\;\ell \leq -1.140102103818506 \cdot 10^{+118}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\ell \leq 6.715360645371229 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
\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}
t_1 := \sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\
\mathbf{if}\;\ell \leq -1.140102103818506 \cdot 10^{+118}:\\
\;\;\;\;-t_1\\

\mathbf{elif}\;\ell \leq 6.715360645371229 \cdot 10^{+231}:\\
\;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\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
 (let* ((t_1
         (*
          (sqrt
           (*
            n
            (*
             U
             (- (/ (* n U*) (* Om Om)) (+ (/ 2.0 Om) (/ (* n U) (* Om Om)))))))
          (* l (sqrt 2.0)))))
   (if (<= l -1.140102103818506e+118)
     (- t_1)
     (if (<= l 6.715360645371229e+231)
       (sqrt
        (*
         (* n 2.0)
         (* U (+ t (* (/ l Om) (fma l -2.0 (* (- U* U) (* n (/ l Om)))))))))
       t_1))))
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 t_1 = sqrt(n * (U * (((n * U_42_) / (Om * Om)) - ((2.0 / Om) + ((n * U) / (Om * Om)))))) * (l * sqrt(2.0));
	double tmp;
	if (l <= -1.140102103818506e+118) {
		tmp = -t_1;
	} else if (l <= 6.715360645371229e+231) {
		tmp = sqrt((n * 2.0) * (U * (t + ((l / Om) * fma(l, -2.0, ((U_42_ - U) * (n * (l / Om))))))));
	} else {
		tmp = t_1;
	}
	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*

Derivation

  1. Split input into 3 regimes

  2. if l < -1.14010210381850598e118

    1. Initial program 57.2

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

    2. Simplified45.5

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

    3. Taylor expanded in l around -inf 35.6

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

    4. Simplified35.6

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

    if -1.14010210381850598e118 < l < 6.7153606453712287e231

    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. Simplified27.3

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

    3. Applied associate-*l*_binary6427.1

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

    4. Applied *-un-lft-identity_binary6427.1

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

    if 6.7153606453712287e231 < l

    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. Simplified58.3

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

    3. Taylor expanded in l around inf 31.5

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

    4. Simplified31.5

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

  4. Final simplification28.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.140102103818506 \cdot 10^{+118}:\\ \;\;\;\;-\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \mathbf{elif}\;\ell \leq 6.715360645371229 \cdot 10^{+231}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(U \cdot \left(t + \frac{\ell}{Om} \cdot \mathsf{fma}\left(\ell, -2, \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(U \cdot \left(\frac{n \cdot U*}{Om \cdot Om} - \left(\frac{2}{Om} + \frac{n \cdot U}{Om \cdot Om}\right)\right)\right)} \cdot \left(\ell \cdot \sqrt{2}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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*))))))