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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ t_3 := \frac{t_1}{t_2 \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(\sqrt{2}, t, \frac{\sqrt{2}}{\frac{\frac{t}{\ell}}{\ell}} \cdot \frac{0.5}{x}\right)}\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_2}\\ \end{array} \]

(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
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (sqrt (/ (+ x 1.0) (+ x -1.0))))
        (t_3 (/ t_1 (* t_2 (* t (- (sqrt 2.0)))))))
   (if (<= t -1.3682754575575214e+144)
     t_3
     (if (<= t -1.2e-164)
       (/ t_1 (sqrt (* 2.0 (+ (/ l (/ x l)) (* t t)))))
       (if (<= t -3.2e-201)
         t_3
         (if (<= t 1.55e-252)
           (/ t_1 (* (sqrt 2.0) (hypot t (/ l (sqrt x)))))
           (if (<= t 9.5e-178)
             (/
              t_1
              (fma (sqrt 2.0) t (* (/ (sqrt 2.0) (/ (/ t l) l)) (/ 0.5 x))))
             (if (<= t 1.301854491159175e+143)
               (/ t_1 (sqrt (* 2.0 (+ (* t t) (* l (/ l x))))))
               (/ t_1 (* t_1 t_2))))))))))

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 t_1 = t * sqrt(2.0);
	double t_2 = sqrt(((x + 1.0) / (x + -1.0)));
	double t_3 = t_1 / (t_2 * (t * -sqrt(2.0)));
	double tmp;
	if (t <= -1.3682754575575214e+144) {
		tmp = t_3;
	} else if (t <= -1.2e-164) {
		tmp = t_1 / sqrt((2.0 * ((l / (x / l)) + (t * t))));
	} else if (t <= -3.2e-201) {
		tmp = t_3;
	} else if (t <= 1.55e-252) {
		tmp = t_1 / (sqrt(2.0) * hypot(t, (l / sqrt(x))));
	} else if (t <= 9.5e-178) {
		tmp = t_1 / fma(sqrt(2.0), t, ((sqrt(2.0) / ((t / l) / l)) * (0.5 / x)));
	} else if (t <= 1.301854491159175e+143) {
		tmp = t_1 / sqrt((2.0 * ((t * t) + (l * (l / x)))));
	} else {
		tmp = t_1 / (t_1 * t_2);
	}
	return tmp;
}

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	t_3 = Float64(t_1 / Float64(t_2 * Float64(t * Float64(-sqrt(2.0)))))
	tmp = 0.0
	if (t <= -1.3682754575575214e+144)
		tmp = t_3;
	elseif (t <= -1.2e-164)
		tmp = Float64(t_1 / sqrt(Float64(2.0 * Float64(Float64(l / Float64(x / l)) + Float64(t * t)))));
	elseif (t <= -3.2e-201)
		tmp = t_3;
	elseif (t <= 1.55e-252)
		tmp = Float64(t_1 / Float64(sqrt(2.0) * hypot(t, Float64(l / sqrt(x)))));
	elseif (t <= 9.5e-178)
		tmp = Float64(t_1 / fma(sqrt(2.0), t, Float64(Float64(sqrt(2.0) / Float64(Float64(t / l) / l)) * Float64(0.5 / x))));
	elseif (t <= 1.301854491159175e+143)
		tmp = Float64(t_1 / sqrt(Float64(2.0 * Float64(Float64(t * t) + Float64(l * Float64(l / x))))));
	else
		tmp = Float64(t_1 / Float64(t_1 * t_2));
	end
	return tmp
end

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(t$95$2 * N[(t * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.3682754575575214e+144], t$95$3, If[LessEqual[t, -1.2e-164], N[(t$95$1 / N[Sqrt[N[(2.0 * N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.2e-201], t$95$3, If[LessEqual[t, 1.55e-252], N[(t$95$1 / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t ^ 2 + N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-178], N[(t$95$1 / N[(N[Sqrt[2.0], $MachinePrecision] * t + N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.301854491159175e+143], N[(t$95$1 / N[Sqrt[N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

\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}
t_1 := t \cdot \sqrt{2}\\
t_2 := \sqrt{\frac{x + 1}{x + -1}}\\
t_3 := \frac{t_1}{t_2 \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\
\mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;t \leq -1.2 \cdot 10^{-164}:\\
\;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\

\mathbf{elif}\;t \leq -3.2 \cdot 10^{-201}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\
\;\;\;\;\frac{t_1}{\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\

\mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\
\;\;\;\;\frac{t_1}{\mathsf{fma}\left(\sqrt{2}, t, \frac{\sqrt{2}}{\frac{\frac{t}{\ell}}{\ell}} \cdot \frac{0.5}{x}\right)}\\

\mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\
\;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{t_1}{t_1 \cdot t_2}\\


\end{array}

Derivation

Split input into 6 regimes
if t < -1.36827545755752137e144 or -1.19999999999999992e-164 < t < -3.2000000000000001e-201
1. Initial program 59.9
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Taylor expanded in t around -inf 5.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
3. Simplified5.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-t \cdot \sqrt{2}\right)}} \]
  Proof
  (*.f64 (sqrt.f64 (/.f64 (+.f64 x 1) (+.f64 x -1))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 x)) (+.f64 x -1))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1))) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1))) (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
  (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
if -1.36827545755752137e144 < t < -1.19999999999999992e-164
1. Initial program 24.5
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Applied egg-rr30.5
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{1}, \frac{\mathsf{fma}\left(\ell, \ell, t \cdot \left(2 \cdot t\right)\right)}{x + -1}, \ell \cdot \left(-\ell\right)\right)}}} \]
3. Taylor expanded in x around -inf 25.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{{\ell}^{2} + \left(2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(2 \cdot {t}^{2} + -1 \cdot {\ell}^{2}\right)\right)}}} \]
4. Simplified11.0
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + t \cdot t\right) + 0}}} \]
  Proof
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= mul0-lft_binary64 (*.f64 0 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 (pow.f64 l 2)) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (pow.f64 l 2))) (pow.f64 l 2))): 13 points increase in error, 32 points decrease in error
  (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2))))) (pow.f64 l 2)): 0 points increase in error, 0 points decrease in error
  (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2)))))): 0 points increase in error, 0 points decrease in error
5. Taylor expanded in t around 0 11.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{x}} + t \cdot t\right) + 0}} \]
6. Simplified6.0
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{x} \cdot \ell} + t \cdot t\right) + 0}} \]
  Proof
  (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 21 points increase in error, 22 points decrease in error
  (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 18 points decrease in error
  (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error
7. Applied egg-rr6.0
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + t \cdot t\right) + 0}} \]
if -3.2000000000000001e-201 < t < 1.5499999999999999e-252
1. Initial program 63.0
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Applied egg-rr60.8
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{1}, \frac{\mathsf{fma}\left(\ell, \ell, t \cdot \left(2 \cdot t\right)\right)}{x + -1}, \ell \cdot \left(-\ell\right)\right)}}} \]
3. Taylor expanded in x around -inf 63.0
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{{\ell}^{2} + \left(2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(2 \cdot {t}^{2} + -1 \cdot {\ell}^{2}\right)\right)}}} \]
4. Simplified32.0
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + t \cdot t\right) + 0}}} \]
  Proof
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= mul0-lft_binary64 (*.f64 0 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 (pow.f64 l 2)) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (pow.f64 l 2))) (pow.f64 l 2))): 13 points increase in error, 32 points decrease in error
  (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2))))) (pow.f64 l 2)): 0 points increase in error, 0 points decrease in error
  (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2)))))): 0 points increase in error, 0 points decrease in error
5. Taylor expanded in t around 0 32.0
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{x}} + t \cdot t\right) + 0}} \]
6. Simplified30.8
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{x} \cdot \ell} + t \cdot t\right) + 0}} \]
  Proof
  (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 21 points increase in error, 22 points decrease in error
  (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 18 points decrease in error
  (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error
7. Applied egg-rr11.6
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right) \cdot \sqrt{2}}} \]
if 1.5499999999999999e-252 < t < 9.50000000000000009e-178
1. Initial program 63.0
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Applied egg-rr62.2
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{1}, \frac{\mathsf{fma}\left(\ell, \ell, t \cdot \left(2 \cdot t\right)\right)}{x + -1}, \ell \cdot \left(-\ell\right)\right)}}} \]
3. Taylor expanded in x around -inf 63.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{{\ell}^{2} + \left(2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(2 \cdot {t}^{2} + -1 \cdot {\ell}^{2}\right)\right)}}} \]
4. Simplified34.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + t \cdot t\right) + 0}}} \]
  Proof
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= mul0-lft_binary64 (*.f64 0 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 (pow.f64 l 2)) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (pow.f64 l 2))) (pow.f64 l 2))): 13 points increase in error, 32 points decrease in error
  (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2))))) (pow.f64 l 2)): 0 points increase in error, 0 points decrease in error
  (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2)))))): 0 points increase in error, 0 points decrease in error
5. Taylor expanded in t around 0 34.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{x}} + t \cdot t\right) + 0}} \]
6. Simplified32.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{x} \cdot \ell} + t \cdot t\right) + 0}} \]
  Proof
  (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 21 points increase in error, 22 points decrease in error
  (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 18 points decrease in error
  (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error
7. Taylor expanded in l around 0 22.5
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{0.5 \cdot \frac{\sqrt{2} \cdot {\ell}^{2}}{t \cdot x} + \sqrt{2} \cdot t}} \]
8. Simplified22.3
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(\sqrt{2}, t, \frac{\sqrt{2}}{\frac{\frac{t}{\ell}}{\ell}} \cdot \frac{0.5}{x}\right)}} \]
  Proof
  (fma.f64 (sqrt.f64 2) t (*.f64 (/.f64 (sqrt.f64 2) (/.f64 (/.f64 t l) l)) (/.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
  (fma.f64 (sqrt.f64 2) t (*.f64 (/.f64 (sqrt.f64 2) (Rewrite<= associate-/r*_binary64 (/.f64 t (*.f64 l l)))) (/.f64 1/2 x))): 24 points increase in error, 3 points decrease in error
  (fma.f64 (sqrt.f64 2) t (*.f64 (/.f64 (sqrt.f64 2) (/.f64 t (Rewrite<= unpow2_binary64 (pow.f64 l 2)))) (/.f64 1/2 x))): 0 points increase in error, 0 points decrease in error
  (fma.f64 (sqrt.f64 2) t (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 2) (pow.f64 l 2)) t)) (/.f64 1/2 x))): 2 points increase in error, 3 points decrease in error
  (fma.f64 (sqrt.f64 2) t (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1/2 x) (/.f64 (*.f64 (sqrt.f64 2) (pow.f64 l 2)) t)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 (sqrt.f64 2) t (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1/2 (*.f64 (sqrt.f64 2) (pow.f64 l 2))) (*.f64 x t)))): 5 points increase in error, 16 points decrease in error
  (fma.f64 (sqrt.f64 2) t (/.f64 (*.f64 1/2 (*.f64 (sqrt.f64 2) (pow.f64 l 2))) (Rewrite<= *-commutative_binary64 (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 (sqrt.f64 2) t (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (*.f64 (sqrt.f64 2) (pow.f64 l 2)) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sqrt.f64 2) t) (*.f64 1/2 (/.f64 (*.f64 (sqrt.f64 2) (pow.f64 l 2)) (*.f64 t x))))): 0 points increase in error, 1 points decrease in error
  (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (*.f64 (sqrt.f64 2) (pow.f64 l 2)) (*.f64 t x))) (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
if 9.50000000000000009e-178 < t < 1.301854491159175e143
1. Initial program 26.3
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Applied egg-rr32.4
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{1}, \frac{\mathsf{fma}\left(\ell, \ell, t \cdot \left(2 \cdot t\right)\right)}{x + -1}, \ell \cdot \left(-\ell\right)\right)}}} \]
3. Taylor expanded in x around -inf 26.8
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{{\ell}^{2} + \left(2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \left(2 \cdot {t}^{2} + -1 \cdot {\ell}^{2}\right)\right)}}} \]
4. Simplified11.7
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + t \cdot t\right) + 0}}} \]
  Proof
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x) (*.f64 t t))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (*.f64 2 (+.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2)))) 0): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= mul0-lft_binary64 (*.f64 0 (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error
  (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 (pow.f64 l 2)) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (pow.f64 l 2))) (pow.f64 l 2))): 13 points increase in error, 32 points decrease in error
  (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2))))) (pow.f64 l 2)): 0 points increase in error, 0 points decrease in error
  (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (+.f64 (*.f64 2 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2)))))): 0 points increase in error, 0 points decrease in error
5. Taylor expanded in t around 0 12.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{{\ell}^{2}}{x}} + t \cdot t\right) + 0}} \]
6. Simplified7.1
  \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\frac{\ell}{x} \cdot \ell} + t \cdot t\right) + 0}} \]
  Proof
  (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 21 points increase in error, 22 points decrease in error
  (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 18 points decrease in error
  (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error
if 1.301854491159175e143 < t
1. Initial program 59.3
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Taylor expanded in l around 0 1.5
  \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
Recombined 6 regimes into one program.
Final simplification6.7
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(\sqrt{2}, t, \frac{\sqrt{2}}{\frac{\frac{t}{\ell}}{\ell}} \cdot \frac{0.5}{x}\right)}\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.2
Cost	26768

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ t_3 := \frac{t_1}{t_2 \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_2}\\ \end{array} \]

Alternative 2
Error	9.2
Cost	20952

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ t_3 := \sqrt{\frac{x + 1}{x + -1}}\\ t_4 := \frac{t_1}{t_3 \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot t_2}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + t_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_3}\\ \end{array} \]

Alternative 3
Error	9.5
Cost	20688

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -6.539986134950287 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-\sqrt{\frac{x + -1}{x + 1}}\right)\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 4
Error	9.6
Cost	20560

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -6.539986134950287 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-\sqrt{\frac{x + -1}{x + 1}}\right)\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\sqrt{2} \cdot \left(t \cdot \sqrt{1 + \frac{2}{x}}\right)}\\ \end{array} \]

Alternative 5
Error	9.6
Cost	20100

Alternative 6
Error	9.5
Cost	14288

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ t_3 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\ \;\;\;\;\frac{t_1}{t_3 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot t_3}\\ \end{array} \]

Alternative 7
Error	9.5
Cost	14288

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\ \;\;\;\;\frac{t_1}{t_2 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot t_2}\\ \end{array} \]

Alternative 8
Error	9.5
Cost	14288

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.3682754575575214 \cdot 10^{+144}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\frac{\ell}{\frac{x}{\ell}} + t \cdot t\right)}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.301854491159175 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 9
Error	14.6
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-272}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{t} \cdot \frac{t}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 10
Error	14.5
Cost	13768

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-272}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 11
Error	14.2
Cost	13768

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 12
Error	14.0
Cost	13768

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;\frac{-\sqrt{2}}{\frac{t_2}{\frac{t}{t}}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot t_2}\\ \end{array} \]

Alternative 13
Error	14.7
Cost	13444

Alternative 14
Error	15.0
Cost	13188

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-272}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	34.3
Cost	7180

\[\begin{array}{l} t_1 := \frac{-t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	34.3
Cost	7180

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{-t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-272}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	34.3
Cost	7180

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-40}:\\ \;\;\;\;\frac{-t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-272}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	35.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	34.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-252}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	38.9
Cost	64

\[1 \]

Error

Derivation

`if t < -1.36827545755752137e144 or -1.19999999999999992e-164 < t < -3.2000000000000001e-201`

`if -1.36827545755752137e144 < t < -1.19999999999999992e-164`

`if -3.2000000000000001e-201 < t < 1.5499999999999999e-252`

`if 1.5499999999999999e-252 < t < 9.50000000000000009e-178`

`if 9.50000000000000009e-178 < t < 1.301854491159175e143`

`if 1.301854491159175e143 < t`

Alternatives

Error

Reproduce