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

↓

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ t_2 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-258}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(\ell, \ell, t \cdot \left(t + t\right)\right)}{x \cdot \mathsf{hypot}\left(t, t\right)}, \mathsf{hypot}\left(t, t\right)\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, t_2, \frac{\frac{\ell}{x} \cdot \frac{\ell}{t}}{t_2}\right)}\\ \end{array} \]

(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
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (/ (* l l) x)
             (+
              (* 2.0 (+ (* t t) (/ (* t t) x)))
              (/ (fma 2.0 (* t t) (* l l)) x)))))))
        (t_2 (sqrt (+ 2.0 (/ 4.0 x)))))
   (if (<= t -8e+45)
     (/ (- (sqrt 2.0)) (sqrt (/ (+ 2.0 (* 2.0 x)) (+ x -1.0))))
     (if (<= t -3.1e-258)
       t_1
       (if (<= t 1.8e-159)
         (*
          t
          (/
           (sqrt 2.0)
           (fma
            0.5
            (/ (* 2.0 (fma l l (* t (+ t t)))) (* x (hypot t t)))
            (hypot t t))))
         (if (<= t 2.1e+55)
           t_1
           (* (sqrt 2.0) (/ t (fma t t_2 (/ (* (/ l x) (/ l t)) 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) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x)))));
	double t_2 = sqrt((2.0 + (4.0 / x)));
	double tmp;
	if (t <= -8e+45) {
		tmp = -sqrt(2.0) / sqrt(((2.0 + (2.0 * x)) / (x + -1.0)));
	} else if (t <= -3.1e-258) {
		tmp = t_1;
	} else if (t <= 1.8e-159) {
		tmp = t * (sqrt(2.0) / fma(0.5, ((2.0 * fma(l, l, (t * (t + t)))) / (x * hypot(t, t))), hypot(t, t)));
	} else if (t <= 2.1e+55) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) * (t / fma(t, t_2, (((l / x) * (l / t)) / 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 * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))))
	t_2 = sqrt(Float64(2.0 + Float64(4.0 / x)))
	tmp = 0.0
	if (t <= -8e+45)
		tmp = Float64(Float64(-sqrt(2.0)) / sqrt(Float64(Float64(2.0 + Float64(2.0 * x)) / Float64(x + -1.0))));
	elseif (t <= -3.1e-258)
		tmp = t_1;
	elseif (t <= 1.8e-159)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(2.0 * fma(l, l, Float64(t * Float64(t + t)))) / Float64(x * hypot(t, t))), hypot(t, t))));
	elseif (t <= 2.1e+55)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) * Float64(t / fma(t, t_2, Float64(Float64(Float64(l / x) * Float64(l / t)) / 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[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -8e+45], N[((-N[Sqrt[2.0], $MachinePrecision]) / N[Sqrt[N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.1e-258], t$95$1, If[LessEqual[t, 1.8e-159], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(2.0 * N[(l * l + N[(t * N[(t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[Sqrt[t ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[t ^ 2 + t ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+55], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(t * t$95$2 + N[(N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $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 \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\
t_2 := \sqrt{2 + \frac{4}{x}}\\
\mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\
\;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\

\mathbf{elif}\;t \leq -3.1 \cdot 10^{-258}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t \leq 1.8 \cdot 10^{-159}:\\
\;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(\ell, \ell, t \cdot \left(t + t\right)\right)}{x \cdot \mathsf{hypot}\left(t, t\right)}, \mathsf{hypot}\left(t, t\right)\right)}\\

\mathbf{elif}\;t \leq 2.1 \cdot 10^{+55}:\\
\;\;\;\;t_1\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, t_2, \frac{\frac{\ell}{x} \cdot \frac{\ell}{t}}{t_2}\right)}\\


\end{array}

Derivation

Split input into 4 regimes
if t < -7.9999999999999994e45
1. Initial program 43.4
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Taylor expanded in t around -inf 3.6
  \[\leadsto \color{blue}{-1 \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{1 + x}{x - 1}}}} \]
3. Simplified3.6
  \[\leadsto \color{blue}{\frac{-\sqrt{2}}{\sqrt{\frac{2 + x \cdot 2}{-1 + x}}}} \]
  Proof
  (/.f64 (neg.f64 (sqrt.f64 2)) (sqrt.f64 (/.f64 (+.f64 2 (*.f64 x 2)) (+.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
  (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (sqrt.f64 2))) (sqrt.f64 (/.f64 (+.f64 2 (*.f64 x 2)) (+.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
  (/.f64 (*.f64 -1 (sqrt.f64 2)) (sqrt.f64 (/.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 1 2)) (*.f64 x 2)) (+.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
  (/.f64 (*.f64 -1 (sqrt.f64 2)) (sqrt.f64 (/.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 2 (+.f64 1 x))) (+.f64 -1 x)))): 0 points increase in error, 0 points decrease in error
  (/.f64 (*.f64 -1 (sqrt.f64 2)) (sqrt.f64 (/.f64 (*.f64 2 (+.f64 1 x)) (Rewrite<= +-commutative_binary64 (+.f64 x -1))))): 0 points increase in error, 0 points decrease in error
  (/.f64 (*.f64 -1 (sqrt.f64 2)) (sqrt.f64 (/.f64 (*.f64 2 (+.f64 1 x)) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))))): 0 points increase in error, 0 points decrease in error
  (/.f64 (*.f64 -1 (sqrt.f64 2)) (sqrt.f64 (/.f64 (*.f64 2 (+.f64 1 x)) (Rewrite<= sub-neg_binary64 (-.f64 x 1))))): 0 points increase in error, 0 points decrease in error
  (/.f64 (*.f64 -1 (sqrt.f64 2)) (sqrt.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (+.f64 1 x) (-.f64 x 1)))))): 1 points increase in error, 0 points decrease in error
  (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (sqrt.f64 2) (sqrt.f64 (*.f64 2 (/.f64 (+.f64 1 x) (-.f64 x 1))))))): 0 points increase in error, 0 points decrease in error
if -7.9999999999999994e45 < t < -3.09999999999999999e-258 or 1.80000000000000011e-159 < t < 2.1000000000000001e55
1. Initial program 34.0
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Simplified35.1
  \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}} \cdot t} \]
  Proof
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (fma.f64 (*.f64 2 t) t (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 2 t) t) (*.f64 l l)))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l)))) t): 22 points increase in error, 32 points decrease in error
  (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 19 points increase in error, 4 points decrease in error
3. Taylor expanded in x around inf 13.8
  \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
4. Simplified13.8
  \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}}} \cdot t \]
  Proof
  (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (neg.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (neg.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))) (neg.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (neg.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (neg.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (neg.f64 (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (neg.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
if -3.09999999999999999e-258 < t < 1.80000000000000011e-159
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. Simplified61.2
  \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}} \cdot t} \]
  Proof
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (fma.f64 (*.f64 2 t) t (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 2 t) t) (*.f64 l l)))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
  (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l)))) t): 22 points increase in error, 32 points decrease in error
  (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 19 points increase in error, 4 points decrease in error
3. Taylor expanded in x around inf 48.2
  \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 \cdot {t}^{2}} + 0.5 \cdot \frac{\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2 \cdot {t}^{2}} \cdot x}}} \cdot t \]
4. Simplified24.8
  \[\leadsto \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(\ell, \ell, t \cdot \left(t + t\right)\right)}{\mathsf{hypot}\left(t, t\right) \cdot x}, \mathsf{hypot}\left(t, t\right)\right)}} \cdot t \]
  Proof
  (fma.f64 1/2 (/.f64 (*.f64 2 (fma.f64 l l (*.f64 t (+.f64 t t)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (fma.f64 l l (*.f64 t (+.f64 t t)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (fma.f64 l l (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 t t) (*.f64 t t))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (fma.f64 l l (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 t t)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (fma.f64 l l (+.f64 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (fma.f64 l l (Rewrite=> count-2_binary64 (*.f64 2 (pow.f64 t 2))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (+.f64 (*.f64 2 (Rewrite=> unpow2_binary64 (*.f64 t t))) (pow.f64 l 2))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (+.f64 (*.f64 2 (*.f64 t t)) (Rewrite=> unpow2_binary64 (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= fma-udef_binary64 (fma.f64 2 (*.f64 t t) (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) (*.f64 1 (fma.f64 2 (*.f64 t t) (*.f64 l l))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (fma.f64 2 (*.f64 t t) (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) (*.f64 -1 (fma.f64 2 (*.f64 t t) (*.f64 l l))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))) (*.f64 -1 (fma.f64 2 (*.f64 t t) (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l)) (*.f64 -1 (fma.f64 2 (*.f64 t t) (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) (*.f64 -1 (fma.f64 2 (*.f64 t t) (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (fma.f64 2 (*.f64 t t) (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l)))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 2))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (hypot.f64 t t) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 t t) (*.f64 t t)))) x)) (hypot.f64 t t)): 37 points increase in error, 1 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 t t))) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (+.f64 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (Rewrite=> count-2_binary64 (*.f64 2 (pow.f64 t 2)))) x)) (hypot.f64 t t)): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (*.f64 2 (pow.f64 t 2))) x)) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 t t) (*.f64 t t))))): 6 points increase in error, 8 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (*.f64 2 (pow.f64 t 2))) x)) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 t t)))): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (*.f64 2 (pow.f64 t 2))) x)) (sqrt.f64 (+.f64 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
  (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (*.f64 2 (pow.f64 t 2))) x)) (sqrt.f64 (Rewrite=> count-2_binary64 (*.f64 2 (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (*.f64 2 (pow.f64 t 2))) x))) (sqrt.f64 (*.f64 2 (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= +-commutative_binary64 (+.f64 (sqrt.f64 (*.f64 2 (pow.f64 t 2))) (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
if 2.1000000000000001e55 < t
1. Initial program 45.2
  \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Simplified45.3
  \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
  Proof
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
  (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 4 points increase in error, 13 points decrease in error
  (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 2 points increase in error, 8 points decrease in error
3. Taylor expanded in x around inf 43.9
  \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]
4. Simplified43.9
  \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \left(t \cdot t\right) + \ell \cdot \ell}{x}\right)\right)}}} \]
  Proof
  (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l)) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (neg.f64 (/.f64 (+.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (neg.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x)))): 0 points increase in error, 0 points decrease in error
  (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
5. Taylor expanded in t around -inf 63.0
  \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{-1 \cdot \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot x\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \]
6. Simplified62.8
  \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{-1 \cdot \left(t \cdot \sqrt{\left(2 + \frac{2}{x}\right) + \frac{2}{x}} + \frac{\frac{\ell \cdot \ell}{t}}{x \cdot \sqrt{\left(2 + \frac{2}{x}\right) + \frac{2}{x}}}\right)}} \]
  Proof
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x)))) (/.f64 (/.f64 (*.f64 l l) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (/.f64 2 x)) (/.f64 2 x)))) (/.f64 (/.f64 (*.f64 l l) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (+.f64 (*.f64 2 1) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)) (/.f64 2 x)))) (/.f64 (/.f64 (*.f64 l l) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (+.f64 (*.f64 2 1) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))) (/.f64 2 x)))) (/.f64 (/.f64 (*.f64 l l) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 1 (/.f64 1 x)))) (/.f64 2 x)))) (/.f64 (/.f64 (*.f64 l l) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (/.f64 (/.f64 (*.f64 l l) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (/.f64 (/.f64 (*.f64 l l) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 2 (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (pow.f64 l 2) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (/.f64 2 x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (pow.f64 l 2) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 (*.f64 2 1) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (pow.f64 l 2) t) (*.f64 x (sqrt.f64 (+.f64 (+.f64 (*.f64 2 1) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (pow.f64 l 2) t) (*.f64 x (sqrt.f64 (+.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 1 (/.f64 1 x)))) (/.f64 2 x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (pow.f64 l 2) t) (*.f64 x (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (pow.f64 l 2) t) (*.f64 x (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))))))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (pow.f64 l 2) t) (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) x))))): 0 points increase in error, 0 points decrease in error
  (*.f64 -1 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (Rewrite<= associate-/r*_binary64 (/.f64 (pow.f64 l 2) (*.f64 t (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) x)))))): 1 points increase in error, 12 points decrease in error
  (*.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) (*.f64 t (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) x))) (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
  (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 l 2) (*.f64 t (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) x)))) (*.f64 -1 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
7. Applied egg-rr14.5
  \[\leadsto \sqrt{2} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{t}{\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \frac{\frac{\ell \cdot \ell}{x \cdot t}}{\sqrt{2 + \frac{4}{x}}}\right)}\right)} - 1\right)} \]
8. Simplified4.5
  \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{t}{\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \frac{\frac{\ell}{x} \cdot \frac{\ell}{t}}{\sqrt{2 + \frac{4}{x}}}\right)}} \]
  Proof
  (/.f64 t (fma.f64 t (sqrt.f64 (+.f64 2 (/.f64 4 x))) (/.f64 (*.f64 (/.f64 l x) (/.f64 l t)) (sqrt.f64 (+.f64 2 (/.f64 4 x)))))): 0 points increase in error, 0 points decrease in error
  (/.f64 t (fma.f64 t (sqrt.f64 (+.f64 2 (/.f64 4 x))) (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 x t))) (sqrt.f64 (+.f64 2 (/.f64 4 x)))))): 32 points increase in error, 10 points decrease in error
  (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 t (fma.f64 t (sqrt.f64 (+.f64 2 (/.f64 4 x))) (/.f64 (/.f64 (*.f64 l l) (*.f64 x t)) (sqrt.f64 (+.f64 2 (/.f64 4 x))))))))): 12 points increase in error, 7 points decrease in error
  (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 t (fma.f64 t (sqrt.f64 (+.f64 2 (/.f64 4 x))) (/.f64 (/.f64 (*.f64 l l) (*.f64 x t)) (sqrt.f64 (+.f64 2 (/.f64 4 x)))))))) 1)): 13 points increase in error, 6 points decrease in error
Recombined 4 regimes into one program.
Final simplification10.5
\[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-258}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(\ell, \ell, t \cdot \left(t + t\right)\right)}{x \cdot \mathsf{hypot}\left(t, t\right)}, \mathsf{hypot}\left(t, t\right)\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2 + \frac{4}{x}}, \frac{\frac{\ell}{x} \cdot \frac{\ell}{t}}{\sqrt{2 + \frac{4}{x}}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.1
Cost	27600

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ t_2 := \sqrt{2 + \frac{4}{x}}\\ t_3 := \sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, t_2, \frac{\frac{\ell}{x} \cdot \frac{\ell}{t}}{t_2}\right)}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-159}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	11.6
Cost	21976

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right) - \frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-233}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	11.6
Cost	21260

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-259}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t \cdot \left(-\sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}\right) - \frac{\ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 6.7 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	11.6
Cost	15440

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{t_1 + \left(t_2 + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{t_1 + \left(t_2 + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	11.5
Cost	15440

Alternative 6
Error	11.6
Cost	15056

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	15.6
Cost	13772

\[\begin{array}{l} \mathbf{if}\;\ell \leq 7600000000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;\ell \leq 4.8 \cdot 10^{+225}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \end{array} \]

Alternative 8
Error	14.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + {\left(\frac{-1}{x}\right)}^{2} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-233}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	14.4
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + {\left(\frac{-1}{x}\right)}^{2} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-232}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell}{x} \cdot \left(2 \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	14.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{\frac{2 + 2 \cdot x}{x + -1}}}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-233}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell}{x} \cdot \left(2 \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	15.5
Cost	13708

\[\begin{array}{l} \mathbf{if}\;\ell \leq 42000000000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+222}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \frac{1}{\mathsf{hypot}\left(t, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 12
Error	15.6
Cost	13580

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{if}\;\ell \leq 42000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 13
Error	14.8
Cost	7556

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-173}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + {\left(\frac{-1}{x}\right)}^{2} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-234}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	15.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-173}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-233}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	14.8
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-174}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-230}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	15.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	15.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-307}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	15.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-307}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	39.0
Cost	64

\[-1 \]

Error

Derivation

`if t < -7.9999999999999994e45`

`if -7.9999999999999994e45 < t < -3.09999999999999999e-258 or 1.80000000000000011e-159 < t < 2.1000000000000001e55`

`if -3.09999999999999999e-258 < t < 1.80000000000000011e-159`

`if 2.1000000000000001e55 < t`

Alternatives

Error

Reproduce