Average Error: 43.3 → 11.0
Time: 1.3min
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -7.544252562526922 \cdot 10^{-07}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -6.594436387197525 \cdot 10^{-150}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \sqrt[3]{\frac{{t}^{6}}{{x}^{6}}}\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -3.167662387401811 \cdot 10^{-308}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot {x}^{2}} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot {x}^{3}} + t \cdot \sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + \left(2 + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.903811773111177 \cdot 10^{-183}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 2.151441207155239 \cdot 10^{+61}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]
\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 -7.544252562526922 \cdot 10^{-07}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\

\mathbf{elif}\;t \leq -6.594436387197525 \cdot 10^{-150}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \sqrt[3]{\frac{{t}^{6}}{{x}^{6}}}\right)\right)\right)\right)}}\\

\mathbf{elif}\;t \leq -3.167662387401811 \cdot 10^{-308}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{-\left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot {x}^{2}} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot {x}^{3}} + t \cdot \sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + \left(2 + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\right)\right)}\\

\mathbf{elif}\;t \leq 1.903811773111177 \cdot 10^{-183}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\

\mathbf{elif}\;t \leq 2.151441207155239 \cdot 10^{+61}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\

\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 -7.544252562526922e-07)
   (/
    (* t (sqrt 2.0))
    (- (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))
   (if (<= t -6.594436387197525e-150)
     (/
      (* t (sqrt 2.0))
      (sqrt
       (+
        (* 2.0 (/ (* l l) x))
        (+
         (* 4.0 (/ (* t t) x))
         (+
          (* 2.0 (/ (* l l) (pow x 3.0)))
          (+
           (* 2.0 (+ (* t t) (/ (* l l) (* x x))))
           (*
            4.0
            (+
             (/ (* t t) (pow x 3.0))
             (cbrt (/ (pow t 6.0) (pow x 6.0)))))))))))
     (if (<= t -3.167662387401811e-308)
       (/
        (* t (sqrt 2.0))
        (-
         (+
          (*
           (sqrt
            (/
             1.0
             (+
              2.0
              (+
               (* 4.0 (/ 1.0 (pow x 2.0)))
               (+ (* 4.0 (/ 1.0 x)) (* 4.0 (/ 1.0 (pow x 3.0))))))))
           (/ (pow l 2.0) (* t (pow x 2.0))))
          (+
           (*
            (sqrt
             (/
              1.0
              (+
               2.0
               (+
                (* 4.0 (/ 1.0 (pow x 2.0)))
                (+ (* 4.0 (/ 1.0 x)) (* 4.0 (/ 1.0 (pow x 3.0))))))))
            (/ (pow l 2.0) (* t x)))
           (+
            (*
             (sqrt
              (/
               1.0
               (+
                2.0
                (+
                 (* 4.0 (/ 1.0 (pow x 2.0)))
                 (+ (* 4.0 (/ 1.0 x)) (* 4.0 (/ 1.0 (pow x 3.0))))))))
             (/ (pow l 2.0) (* t (pow x 3.0))))
            (*
             t
             (sqrt
              (+
               (* 4.0 (/ 1.0 (pow x 2.0)))
               (+
                (* 4.0 (/ 1.0 x))
                (+ 2.0 (* 4.0 (/ 1.0 (pow x 3.0)))))))))))))
       (if (<= t 1.903811773111177e-183)
         (/
          (* t (sqrt 2.0))
          (+
           (* t (sqrt 2.0))
           (+
            (* 2.0 (/ t (* (sqrt 2.0) x)))
            (/ (* l l) (* t (* (sqrt 2.0) x))))))
         (if (<= t 2.151441207155239e+61)
           (/
            (* t (sqrt 2.0))
            (sqrt
             (+
              (* 2.0 (/ (* l l) x))
              (+ (* 4.0 (/ (* t t) x)) (* 2.0 (* t t))))))
           (/
            (* t (sqrt 2.0))
            (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0)))))))))))))
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 <= -7.544252562526922e-07) {
		tmp = (t * sqrt(2.0)) / -(t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	} else if (t <= -6.594436387197525e-150) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * ((l * l) / x)) + ((4.0 * ((t * t) / x)) + ((2.0 * ((l * l) / pow(x, 3.0))) + ((2.0 * ((t * t) + ((l * l) / (x * x)))) + (4.0 * (((t * t) / pow(x, 3.0)) + cbrt(pow(t, 6.0) / pow(x, 6.0))))))));
	} else if (t <= -3.167662387401811e-308) {
		tmp = (t * sqrt(2.0)) / -((sqrt(1.0 / (2.0 + ((4.0 * (1.0 / pow(x, 2.0))) + ((4.0 * (1.0 / x)) + (4.0 * (1.0 / pow(x, 3.0))))))) * (pow(l, 2.0) / (t * pow(x, 2.0)))) + ((sqrt(1.0 / (2.0 + ((4.0 * (1.0 / pow(x, 2.0))) + ((4.0 * (1.0 / x)) + (4.0 * (1.0 / pow(x, 3.0))))))) * (pow(l, 2.0) / (t * x))) + ((sqrt(1.0 / (2.0 + ((4.0 * (1.0 / pow(x, 2.0))) + ((4.0 * (1.0 / x)) + (4.0 * (1.0 / pow(x, 3.0))))))) * (pow(l, 2.0) / (t * pow(x, 3.0)))) + (t * sqrt((4.0 * (1.0 / pow(x, 2.0))) + ((4.0 * (1.0 / x)) + (2.0 + (4.0 * (1.0 / pow(x, 3.0))))))))));
	} else if (t <= 1.903811773111177e-183) {
		tmp = (t * sqrt(2.0)) / ((t * sqrt(2.0)) + ((2.0 * (t / (sqrt(2.0) * x))) + ((l * l) / (t * (sqrt(2.0) * x)))));
	} else if (t <= 2.151441207155239e+61) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * ((l * l) / x)) + ((4.0 * ((t * t) / x)) + (2.0 * (t * t))));
	} else {
		tmp = (t * sqrt(2.0)) / (t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	}
	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 6 regimes

  2. if t < -7.5442525625269223e-7

    1. Initial program 41.9

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

    2. Taylor expanded around -inf 4.8

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

    3. Simplified4.8

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

    if -7.5442525625269223e-7 < t < -6.59443638719752451e-150

    1. Initial program 32.5

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

    2. Taylor expanded around inf 10.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{{x}^{3}} + 4 \cdot \frac{{t}^{2}}{{x}^{2}}\right)\right)\right)\right)\right)}}}\]

    3. Simplified10.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \frac{t \cdot t}{x \cdot x}\right)\right)\right)\right)}}}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube_binary64_11410.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \color{blue}{\sqrt[3]{\left(\frac{t \cdot t}{x \cdot x} \cdot \frac{t \cdot t}{x \cdot x}\right) \cdot \frac{t \cdot t}{x \cdot x}}}\right)\right)\right)\right)}}\]

    6. Simplified10.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \sqrt[3]{\color{blue}{\frac{{t}^{6}}{{x}^{6}}}}\right)\right)\right)\right)}}\]

    if -6.59443638719752451e-150 < t < -3.1676623874018108e-308

    1. Initial program 61.4

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

    2. Taylor expanded around inf 37.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot {t}^{2} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{{x}^{3}} + 4 \cdot \frac{{t}^{2}}{{x}^{2}}\right)\right)\right)\right)\right)}}}\]

    3. Simplified37.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \frac{t \cdot t}{x \cdot x}\right)\right)\right)\right)}}}\]

    4. Taylor expanded around -inf 32.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{{x}^{2} \cdot t} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{x \cdot t} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{{x}^{3} \cdot t} + \sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + \left(4 \cdot \frac{1}{{x}^{3}} + 2\right)\right)} \cdot t\right)\right)\right)}}\]

    if -3.1676623874018108e-308 < t < 1.9038117731111768e-183

    1. Initial program 62.8

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

    2. Taylor expanded around inf 26.3

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

    3. Simplified26.3

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

    if 1.9038117731111768e-183 < t < 2.1514412071552391e61

    1. Initial program 31.0

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

    2. Taylor expanded around inf 12.8

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

    3. Simplified12.8

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

    if 2.1514412071552391e61 < t

    1. Initial program 45.8

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

    2. Taylor expanded around inf 3.4

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

    3. Simplified3.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]
  3. Recombined 6 regimes into one program.

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.544252562526922 \cdot 10^{-07}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -6.594436387197525 \cdot 10^{-150}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + \left(2 \cdot \frac{\ell \cdot \ell}{{x}^{3}} + \left(2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x \cdot x}\right) + 4 \cdot \left(\frac{t \cdot t}{{x}^{3}} + \sqrt[3]{\frac{{t}^{6}}{{x}^{6}}}\right)\right)\right)\right)}}\\ \mathbf{elif}\;t \leq -3.167662387401811 \cdot 10^{-308}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot {x}^{2}} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \left(\sqrt{\frac{1}{2 + \left(4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}} \cdot \frac{{\ell}^{2}}{t \cdot {x}^{3}} + t \cdot \sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(4 \cdot \frac{1}{x} + \left(2 + 4 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.903811773111177 \cdot 10^{-183}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 2.151441207155239 \cdot 10^{+61}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{x} + \left(4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021079 
(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)))))