Average Error: 43.0 → 8.5
Time: 1.6min
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -8.421989759483264 \cdot 10^{+63}:\\ \;\;\;\;\frac{t}{\sqrt{2 \cdot \left(\frac{1}{x + -1} + \frac{x}{x + -1}\right)} \cdot \frac{-t}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -8.753971576230512 \cdot 10^{-164}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{x} \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -9.937225512484842 \cdot 10^{-284}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)} \cdot \left(-t\right) - \sqrt{\frac{1}{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right)}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.094632978090295 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{2 \cdot \frac{t}{x \cdot {\left(\sqrt{2}\right)}^{2}} + \left(t + \frac{{\ell}^{2}}{t \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{2}\right)}\right)}\\ \mathbf{elif}\;t \leq 3.1287133522825756 \cdot 10^{+147}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{x} \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{t}{x} + \left(t + \frac{\frac{\ell}{\frac{t}{\ell}}}{2 \cdot x}\right)}\\ \end{array}\]
\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \leq -8.421989759483264 \cdot 10^{+63}:\\
\;\;\;\;\frac{t}{\sqrt{2 \cdot \left(\frac{1}{x + -1} + \frac{x}{x + -1}\right)} \cdot \frac{-t}{\sqrt{2}}}\\

\mathbf{elif}\;t \leq -8.753971576230512 \cdot 10^{-164}:\\
\;\;\;\;\frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{x} \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}}{\sqrt{2}}}\\

\mathbf{elif}\;t \leq -9.937225512484842 \cdot 10^{-284}:\\
\;\;\;\;\frac{t}{\frac{\sqrt{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)} \cdot \left(-t\right) - \sqrt{\frac{1}{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right)}{\sqrt{2}}}\\

\mathbf{elif}\;t \leq 1.094632978090295 \cdot 10^{-158}:\\
\;\;\;\;\frac{t}{2 \cdot \frac{t}{x \cdot {\left(\sqrt{2}\right)}^{2}} + \left(t + \frac{{\ell}^{2}}{t \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{2}\right)}\right)}\\

\mathbf{elif}\;t \leq 3.1287133522825756 \cdot 10^{+147}:\\
\;\;\;\;\frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{x} \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}}{\sqrt{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{t}{\frac{t}{x} + \left(t + \frac{\frac{\ell}{\frac{t}{\ell}}}{2 \cdot x}\right)}\\

\end{array}
(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))
↓
(FPCore (x l t)
 :precision binary64
 (if (<= t -8.421989759483264e+63)
   (/
    t
    (*
     (sqrt (* 2.0 (+ (/ 1.0 (+ x -1.0)) (/ x (+ x -1.0)))))
     (/ (- t) (sqrt 2.0))))
   (if (<= t -8.753971576230512e-164)
     (/
      t
      (/
       (sqrt
        (+
         (/ 2.0 (/ 1.0 (/ l (/ x l))))
         (*
          (sqrt
           (+
            (* 4.0 (+ (/ t (/ x t)) (* (/ t x) (/ t x))))
            (* 2.0 (+ (* t t) (/ l (/ (* x x) l))))))
          (sqrt
           (+
            (* 4.0 (* (/ t x) (+ t (/ t x))))
            (* 2.0 (+ (* t t) (* (/ l x) (/ l x)))))))))
       (sqrt 2.0)))
     (if (<= t -9.937225512484842e-284)
       (/
        t
        (/
         (-
          (* (sqrt (+ (/ (/ 4.0 x) x) (+ 2.0 (/ 4.0 x)))) (- t))
          (*
           (sqrt (/ 1.0 (+ (/ (/ 4.0 x) x) (+ 2.0 (/ 4.0 x)))))
           (+ (* (/ l x) (/ l t)) (/ (* (/ l x) (/ l x)) t))))
         (sqrt 2.0)))
       (if (<= t 1.094632978090295e-158)
         (/
          t
          (+
           (* 2.0 (/ t (* x (pow (sqrt 2.0) 2.0))))
           (+ t (/ (pow l 2.0) (* t (* x (pow (sqrt 2.0) 2.0)))))))
         (if (<= t 3.1287133522825756e+147)
           (/
            t
            (/
             (sqrt
              (+
               (/ 2.0 (/ 1.0 (/ l (/ x l))))
               (*
                (sqrt
                 (+
                  (* 4.0 (+ (/ t (/ x t)) (* (/ t x) (/ t x))))
                  (* 2.0 (+ (* t t) (/ l (/ (* x x) l))))))
                (sqrt
                 (+
                  (* 4.0 (* (/ t x) (+ t (/ t x))))
                  (* 2.0 (+ (* t t) (* (/ l x) (/ l x)))))))))
             (sqrt 2.0)))
           (/ t (+ (/ t x) (+ t (/ (/ l (/ t l)) (* 2.0 x)))))))))))
double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l));
}
↓
double code(double x, double l, double t) {
	double tmp;
	if (t <= -8.421989759483264e+63) {
		tmp = t / (sqrt(2.0 * ((1.0 / (x + -1.0)) + (x / (x + -1.0)))) * (-t / sqrt(2.0)));
	} else if (t <= -8.753971576230512e-164) {
		tmp = t / (sqrt((2.0 / (1.0 / (l / (x / l)))) + (sqrt((4.0 * ((t / (x / t)) + ((t / x) * (t / x)))) + (2.0 * ((t * t) + (l / ((x * x) / l))))) * sqrt((4.0 * ((t / x) * (t + (t / x)))) + (2.0 * ((t * t) + ((l / x) * (l / x))))))) / sqrt(2.0));
	} else if (t <= -9.937225512484842e-284) {
		tmp = t / (((sqrt(((4.0 / x) / x) + (2.0 + (4.0 / x))) * -t) - (sqrt(1.0 / (((4.0 / x) / x) + (2.0 + (4.0 / x)))) * (((l / x) * (l / t)) + (((l / x) * (l / x)) / t)))) / sqrt(2.0));
	} else if (t <= 1.094632978090295e-158) {
		tmp = t / ((2.0 * (t / (x * pow(sqrt(2.0), 2.0)))) + (t + (pow(l, 2.0) / (t * (x * pow(sqrt(2.0), 2.0))))));
	} else if (t <= 3.1287133522825756e+147) {
		tmp = t / (sqrt((2.0 / (1.0 / (l / (x / l)))) + (sqrt((4.0 * ((t / (x / t)) + ((t / x) * (t / x)))) + (2.0 * ((t * t) + (l / ((x * x) / l))))) * sqrt((4.0 * ((t / x) * (t + (t / x)))) + (2.0 * ((t * t) + ((l / x) * (l / x))))))) / sqrt(2.0));
	} else {
		tmp = t / ((t / x) + (t + ((l / (t / l)) / (2.0 * x))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if t < -8.421989759483264e63

    1. Initial program 46.5

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Simplified46.5

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2}}}}\]

    3. Taylor expanded around -inf 3.6

      \[\leadsto \frac{t}{\color{blue}{-1 \cdot \left(\frac{t}{\sqrt{2}} \cdot \sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}}\right)}}\]

    4. Simplified3.6

      \[\leadsto \frac{t}{\color{blue}{-\sqrt{2 \cdot \left(\frac{1}{x - 1} + \frac{x}{x - 1}\right)} \cdot \frac{t}{\sqrt{2}}}}\]

    if -8.421989759483264e63 < t < -8.7539715762305119e-164 or 1.094632978090295e-158 < t < 3.12871335228257563e147

    1. Initial program 26.1

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Simplified26.1

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2}}}}\]

    3. Taylor expanded around inf 10.1

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{{x}^{2}}\right)\right)\right)}}}{\sqrt{2}}}\]

    4. Simplified9.7

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{\frac{2}{\frac{x}{\ell \cdot \ell}} + \left(4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)\right)}}}{\sqrt{2}}}\]
    5. Using strategy rm

    6. Applied clear-num_binary64_779.7

      \[\leadsto \frac{t}{\frac{\sqrt{\frac{2}{\color{blue}{\frac{1}{\frac{\ell \cdot \ell}{x}}}} + \left(4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)\right)}}{\sqrt{2}}}\]

    7. Simplified5.0

      \[\leadsto \frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\color{blue}{\frac{\ell}{\frac{x}{\ell}}}}} + \left(4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)\right)}}{\sqrt{2}}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt_binary64_1005.0

      \[\leadsto \frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \color{blue}{\sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)}}}}{\sqrt{2}}}\]

    10. Simplified5.0

      \[\leadsto \frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \color{blue}{\sqrt{4 \cdot \left(\frac{t}{x} \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}}}{\sqrt{2}}}\]

    if -8.7539715762305119e-164 < t < -9.93722551248484173e-284

    1. Initial program 62.4

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Simplified62.4

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2}}}}\]

    3. Taylor expanded around inf 40.6

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(2 \cdot {t}^{2} + 2 \cdot \frac{{\ell}^{2}}{{x}^{2}}\right)\right)\right)}}}{\sqrt{2}}}\]

    4. Simplified36.1

      \[\leadsto \frac{t}{\frac{\sqrt{\color{blue}{\frac{2}{\frac{x}{\ell \cdot \ell}} + \left(4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)\right)}}}{\sqrt{2}}}\]

    5. Taylor expanded around -inf 29.5

      \[\leadsto \frac{t}{\frac{\color{blue}{-\left(\sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 2\right)} \cdot t + \left(\sqrt{\frac{1}{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 2\right)}} \cdot \frac{{\ell}^{2}}{x \cdot t} + \sqrt{\frac{1}{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 2\right)}} \cdot \frac{{\ell}^{2}}{{x}^{2} \cdot t}\right)\right)}}{\sqrt{2}}}\]

    6. Simplified24.6

      \[\leadsto \frac{t}{\frac{\color{blue}{\sqrt{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)} \cdot \left(-t\right) - \sqrt{\frac{1}{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right)}}{\sqrt{2}}}\]

    if -9.93722551248484173e-284 < t < 1.094632978090295e-158

    1. Initial program 62.5

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Simplified62.5

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2}}}}\]

    3. Taylor expanded around inf 28.1

      \[\leadsto \frac{t}{\color{blue}{2 \cdot \frac{t}{x \cdot {\left(\sqrt{2}\right)}^{2}} + \left(t + \frac{{\ell}^{2}}{t \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot x\right)}\right)}}\]

    if 3.12871335228257563e147 < t

    1. Initial program 60.7

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Simplified60.7

      \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{\sqrt{2}}}}\]

    3. Taylor expanded around inf 14.6

      \[\leadsto \frac{t}{\color{blue}{2 \cdot \frac{t}{x \cdot {\left(\sqrt{2}\right)}^{2}} + \left(t + \frac{{\ell}^{2}}{t \cdot \left({\left(\sqrt{2}\right)}^{2} \cdot x\right)}\right)}}\]

    4. Simplified4.3

      \[\leadsto \frac{t}{\color{blue}{1 \cdot \frac{t}{x} + \left(t + \frac{\frac{\ell}{\frac{t}{\ell}}}{2 \cdot x}\right)}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.421989759483264 \cdot 10^{+63}:\\ \;\;\;\;\frac{t}{\sqrt{2 \cdot \left(\frac{1}{x + -1} + \frac{x}{x + -1}\right)} \cdot \frac{-t}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -8.753971576230512 \cdot 10^{-164}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{x} \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq -9.937225512484842 \cdot 10^{-284}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)} \cdot \left(-t\right) - \sqrt{\frac{1}{\frac{\frac{4}{x}}{x} + \left(2 + \frac{4}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t} + \frac{\frac{\ell}{x} \cdot \frac{\ell}{x}}{t}\right)}{\sqrt{2}}}\\ \mathbf{elif}\;t \leq 1.094632978090295 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{2 \cdot \frac{t}{x \cdot {\left(\sqrt{2}\right)}^{2}} + \left(t + \frac{{\ell}^{2}}{t \cdot \left(x \cdot {\left(\sqrt{2}\right)}^{2}\right)}\right)}\\ \mathbf{elif}\;t \leq 3.1287133522825756 \cdot 10^{+147}:\\ \;\;\;\;\frac{t}{\frac{\sqrt{\frac{2}{\frac{1}{\frac{\ell}{\frac{x}{\ell}}}} + \sqrt{4 \cdot \left(\frac{t}{\frac{x}{t}} + \frac{t}{x} \cdot \frac{t}{x}\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x \cdot x}{\ell}}\right)} \cdot \sqrt{4 \cdot \left(\frac{t}{x} \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(t \cdot t + \frac{\ell}{x} \cdot \frac{\ell}{x}\right)}}}{\sqrt{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{t}{x} + \left(t + \frac{\frac{\ell}{\frac{t}{\ell}}}{2 \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))