Toniolo and Linder, Equation (7)

Average Error: 42.9 → 9.0

Time: 27.7s

Precision: binary64

Cost: 39760

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ t_3 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + t_3\right)}}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-204}:\\ \;\;\;\;\frac{t_1}{\frac{2 \cdot \frac{\ell \cdot \ell}{x}}{t} \cdot \left(\sqrt{\frac{1}{t_2}} \cdot -0.5\right) - t \cdot \sqrt{t_2}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{{\left(\sqrt{\sqrt{2} \cdot \mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, \frac{\ell}{x}\right)}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-236}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot {\left(2 \cdot \left(t_3 + t \cdot \left(t + \frac{t}{x}\right)\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

FPCore

(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 (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))
        (t_3 (* l (/ l x))))
   (if (<= t -2e+112)
     (+ (/ 1.0 x) -1.0)
     (if (<= t -3.6e-138)
       (/
        t_1
        (sqrt (+ (/ l (/ x l)) (+ (* 2.0 (+ (* t t) (/ t (/ x t)))) t_3))))
       (if (<= t -1.7e-204)
         (/
          t_1
          (-
           (* (/ (* 2.0 (/ (* l l) x)) t) (* (sqrt (/ 1.0 t_2)) -0.5))
           (* t (sqrt t_2))))
         (if (<= t 1.05e-276)
           (*
            (sqrt 2.0)
            (/
             t
             (pow (sqrt (* (sqrt 2.0) (hypot (/ l (sqrt x)) (/ l x)))) 2.0)))
           (if (<= t 5.5e-236)
             1.0
             (if (<= t 6.8e+54)
               (*
                t
                (* (sqrt 2.0) (pow (* 2.0 (+ t_3 (* t (+ t (/ t x))))) -0.5)))
               (+ 1.0 (/ -1.0 x))))))))))

C

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 = (2.0 / x) + (2.0 + (2.0 / x));
	double t_3 = l * (l / x);
	double tmp;
	if (t <= -2e+112) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -3.6e-138) {
		tmp = t_1 / sqrt(((l / (x / l)) + ((2.0 * ((t * t) + (t / (x / t)))) + t_3)));
	} else if (t <= -1.7e-204) {
		tmp = t_1 / ((((2.0 * ((l * l) / x)) / t) * (sqrt((1.0 / t_2)) * -0.5)) - (t * sqrt(t_2)));
	} else if (t <= 1.05e-276) {
		tmp = sqrt(2.0) * (t / pow(sqrt((sqrt(2.0) * hypot((l / sqrt(x)), (l / x)))), 2.0));
	} else if (t <= 5.5e-236) {
		tmp = 1.0;
	} else if (t <= 6.8e+54) {
		tmp = t * (sqrt(2.0) * pow((2.0 * (t_3 + (t * (t + (t / x))))), -0.5));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

↓

public static double code(double x, double l, double t) {
	double t_1 = t * Math.sqrt(2.0);
	double t_2 = (2.0 / x) + (2.0 + (2.0 / x));
	double t_3 = l * (l / x);
	double tmp;
	if (t <= -2e+112) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= -3.6e-138) {
		tmp = t_1 / Math.sqrt(((l / (x / l)) + ((2.0 * ((t * t) + (t / (x / t)))) + t_3)));
	} else if (t <= -1.7e-204) {
		tmp = t_1 / ((((2.0 * ((l * l) / x)) / t) * (Math.sqrt((1.0 / t_2)) * -0.5)) - (t * Math.sqrt(t_2)));
	} else if (t <= 1.05e-276) {
		tmp = Math.sqrt(2.0) * (t / Math.pow(Math.sqrt((Math.sqrt(2.0) * Math.hypot((l / Math.sqrt(x)), (l / x)))), 2.0));
	} else if (t <= 5.5e-236) {
		tmp = 1.0;
	} else if (t <= 6.8e+54) {
		tmp = t * (Math.sqrt(2.0) * Math.pow((2.0 * (t_3 + (t * (t + (t / x))))), -0.5));
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

↓

def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	t_2 = (2.0 / x) + (2.0 + (2.0 / x))
	t_3 = l * (l / x)
	tmp = 0
	if t <= -2e+112:
		tmp = (1.0 / x) + -1.0
	elif t <= -3.6e-138:
		tmp = t_1 / math.sqrt(((l / (x / l)) + ((2.0 * ((t * t) + (t / (x / t)))) + t_3)))
	elif t <= -1.7e-204:
		tmp = t_1 / ((((2.0 * ((l * l) / x)) / t) * (math.sqrt((1.0 / t_2)) * -0.5)) - (t * math.sqrt(t_2)))
	elif t <= 1.05e-276:
		tmp = math.sqrt(2.0) * (t / math.pow(math.sqrt((math.sqrt(2.0) * math.hypot((l / math.sqrt(x)), (l / x)))), 2.0))
	elif t <= 5.5e-236:
		tmp = 1.0
	elif t <= 6.8e+54:
		tmp = t * (math.sqrt(2.0) * math.pow((2.0 * (t_3 + (t * (t + (t / x))))), -0.5))
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

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 = Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x)))
	t_3 = Float64(l * Float64(l / x))
	tmp = 0.0
	if (t <= -2e+112)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= -3.6e-138)
		tmp = Float64(t_1 / sqrt(Float64(Float64(l / Float64(x / l)) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t)))) + t_3))));
	elseif (t <= -1.7e-204)
		tmp = Float64(t_1 / Float64(Float64(Float64(Float64(2.0 * Float64(Float64(l * l) / x)) / t) * Float64(sqrt(Float64(1.0 / t_2)) * -0.5)) - Float64(t * sqrt(t_2))));
	elseif (t <= 1.05e-276)
		tmp = Float64(sqrt(2.0) * Float64(t / (sqrt(Float64(sqrt(2.0) * hypot(Float64(l / sqrt(x)), Float64(l / x)))) ^ 2.0)));
	elseif (t <= 5.5e-236)
		tmp = 1.0;
	elseif (t <= 6.8e+54)
		tmp = Float64(t * Float64(sqrt(2.0) * (Float64(2.0 * Float64(t_3 + Float64(t * Float64(t + Float64(t / x))))) ^ -0.5)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

↓

function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	t_2 = (2.0 / x) + (2.0 + (2.0 / x));
	t_3 = l * (l / x);
	tmp = 0.0;
	if (t <= -2e+112)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= -3.6e-138)
		tmp = t_1 / sqrt(((l / (x / l)) + ((2.0 * ((t * t) + (t / (x / t)))) + t_3)));
	elseif (t <= -1.7e-204)
		tmp = t_1 / ((((2.0 * ((l * l) / x)) / t) * (sqrt((1.0 / t_2)) * -0.5)) - (t * sqrt(t_2)));
	elseif (t <= 1.05e-276)
		tmp = sqrt(2.0) * (t / (sqrt((sqrt(2.0) * hypot((l / sqrt(x)), (l / x)))) ^ 2.0));
	elseif (t <= 5.5e-236)
		tmp = 1.0;
	elseif (t <= 6.8e+54)
		tmp = t * (sqrt(2.0) * ((2.0 * (t_3 + (t * (t + (t / x))))) ^ -0.5));
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

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[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+112], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -3.6e-138], N[(t$95$1 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.7e-204], N[(t$95$1 / N[(N[(N[(N[(2.0 * N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-276], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Power[N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(l / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-236], 1.0, If[LessEqual[t, 6.8e+54], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[(2.0 * N[(t$95$3 + N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -1.9999999999999999e112

   1. Initial program 52.8

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

   2. Simplified 52.7

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

      [Start] 52.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 52.7	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 52.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 52.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around inf 63.0

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

   4. Simplified 63.0

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
      Proof

      [Start] 63.0	\[ \left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      associate-*l* [=>] 63.0	\[ \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      sub-neg [=>] 63.0	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]
      metadata-eval [=>] 63.0	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]
      +-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]
      +-commutative [=>] 63.0	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   5. Taylor expanded in x around -inf 64.0

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)}\right) \]

   6. Simplified 4.0

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\left(-1 + \frac{1}{x}\right)}\right) \]
      Proof

      [Start] 64.0	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right) \]
      unpow2 [=>] 64.0	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x}\right)\right) \]
      rem-square-sqrt [=>] 4.0	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(\color{blue}{-1} + \frac{1}{x}\right)\right) \]

   7. Applied egg-rr 3.1

      \[\leadsto \color{blue}{\frac{1}{x} + -1} \]

   if -1.9999999999999999e112 < t < -3.60000000000000018e-138

   1. Initial program 25.5

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

   2. Taylor expanded in x around inf 11.2

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

   3. Simplified 11.2

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

      [Start] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 11.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 11.7

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

   5. Simplified 6.0

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

      [Start] 11.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      mul-1-neg [=>] 11.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]
      unpow2 [=>] 11.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \]
      associate-*r/ [<=] 6.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\color{blue}{\ell \cdot \frac{\ell}{x}}\right)\right)}} \]
      distribute-lft-neg-in [=>] 6.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\ell\right) \cdot \frac{\ell}{x}}\right)}} \]
      *-commutative [=>] 6.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \]

   if -3.60000000000000018e-138 < t < -1.7000000000000001e-204

   1. Initial program 55.1

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

   2. Taylor expanded in x around inf 26.6

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

   3. Simplified 26.6

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

      [Start] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 26.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around -inf 24.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \]

   5. Simplified 24.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{2 \cdot \frac{\ell \cdot \ell}{x}}{t} \cdot \left(\sqrt{\frac{1}{\left(2 + \frac{2}{x}\right) + \frac{2}{x}}} \cdot -0.5\right) - t \cdot \sqrt{\left(2 + \frac{2}{x}\right) + \frac{2}{x}}}} \]
      Proof

      [Start] 24.1	\[ \frac{\sqrt{2} \cdot t}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \]
      *-commutative [=>] 24.1	\[ \frac{\sqrt{2} \cdot t}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \]
      mul-1-neg [=>] 24.1	\[ \frac{\sqrt{2} \cdot t}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + \color{blue}{\left(-\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \]
      unsub-neg [=>] 24.1	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) - \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t}} \]

   if -1.7000000000000001e-204 < t < 1.05e-276

   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. Simplified 60.6

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

      [Start] 63.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 63.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 60.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 60.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around 0 60.6

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

   4. Simplified 60.6

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

      [Start] 60.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{{\ell}^{2}}{x - 1}, -\ell \cdot \ell\right)}} \]
      unpow2 [=>] 60.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\ell \cdot \ell}}{x - 1}, -\ell \cdot \ell\right)}} \]
      sub-neg [=>] 60.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\ell \cdot \ell}{\color{blue}{x + \left(-1\right)}}, -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 60.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\ell \cdot \ell}{x + \color{blue}{-1}}, -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 60.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\ell \cdot \ell}{\color{blue}{-1 + x}}, -\ell \cdot \ell\right)}} \]

   5. Taylor expanded in x around inf 35.3

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

   6. Simplified 35.3

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

      [Start] 35.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot \frac{{\ell}^{2}}{{x}^{2}}}} \]
      distribute-lft-out [=>] 35.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{{x}^{2}}\right)}}} \]
      unpow2 [=>] 35.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + \frac{{\ell}^{2}}{{x}^{2}}\right)}} \]
      unpow2 [=>] 35.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{x} + \frac{\color{blue}{\ell \cdot \ell}}{{x}^{2}}\right)}} \]
      unpow2 [=>] 35.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{\color{blue}{x \cdot x}}\right)}} \]

   7. Applied egg-rr 21.4

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

   if 1.05e-276 < t < 5.49999999999999959e-236

   1. Initial program 62.6

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

   2. Simplified 60.7

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

      [Start] 62.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.6	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 59.9	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 62.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 62.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 62.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 62.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 60.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around inf 46.3

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   4. Applied egg-rr 46.1

      \[\leadsto \color{blue}{1} \]

   if 5.49999999999999959e-236 < t < 6.8000000000000001e54

   1. Initial program 36.3

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

   2. Taylor expanded in x around inf 16.0

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

   3. Simplified 16.0

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

      [Start] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 16.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 16.2

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

   5. Simplified 10.8

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

      [Start] 16.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      mul-1-neg [=>] 16.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]
      unpow2 [=>] 16.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \]
      associate-*r/ [<=] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\color{blue}{\ell \cdot \frac{\ell}{x}}\right)\right)}} \]
      distribute-lft-neg-in [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\ell\right) \cdot \frac{\ell}{x}}\right)}} \]
      *-commutative [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \]

   6. Applied egg-rr 38.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \ell \cdot \frac{\ell}{x}\right)\right)}\right)} - 1}} \]

   7. Simplified 10.8

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

      [Start] 38.2	\[ \frac{\sqrt{2} \cdot t}{e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \ell \cdot \frac{\ell}{x}\right)\right)}\right)} - 1} \]
      expm1-def [=>] 12.0	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \ell \cdot \frac{\ell}{x}\right)\right)}\right)\right)}} \]
      expm1-log1p [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \ell \cdot \frac{\ell}{x}\right)\right)}}} \]
      fma-udef [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \ell \cdot \frac{\ell}{x}\right)}}} \]
      fma-udef [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \frac{\ell}{x} + \color{blue}{\left(2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}} \]
      +-commutative [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \frac{\ell}{x} + \color{blue}{\left(\ell \cdot \frac{\ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right)}}} \]
      associate-+r+ [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\ell \cdot \frac{\ell}{x} + \ell \cdot \frac{\ell}{x}\right) + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)}}} \]
      count-2 [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)}} \]
      associate-*r* [=>] 10.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right) + \color{blue}{\left(2 \cdot t\right) \cdot \left(t + \frac{t}{x}\right)}}} \]

   8. Applied egg-rr 10.7

      \[\leadsto \color{blue}{t \cdot \left(\sqrt{2} \cdot {\left(2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)\right)}^{-0.5}\right)} \]

   if 6.8000000000000001e54 < t

   1. Initial program 45.0

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

   2. Simplified 45.0

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

      [Start] 45.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 45.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 45.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 45.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around inf 4.5

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]

   4. Simplified 4.5

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
      Proof

      [Start] 4.5	\[ \left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      associate-*l* [=>] 4.5	\[ \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      sub-neg [=>] 4.5	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]
      metadata-eval [=>] 4.5	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]
      +-commutative [=>] 4.5	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]
      +-commutative [=>] 4.5	\[ \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   5. Applied egg-rr 3.6

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1} \]

   6. Simplified 3.6

      \[\leadsto \color{blue}{\sqrt{\frac{-1 + x}{x + 1}}} \]
      Proof

      [Start] 3.6	\[ e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1 \]
      expm1-def [=>] 3.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]
      expm1-log1p [=>] 3.6	\[ \color{blue}{\sqrt{\frac{-1 + x}{x + 1}}} \]

   7. Taylor expanded in x around inf 4.0

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-204}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2 \cdot \frac{\ell \cdot \ell}{x}}{t} \cdot \left(\sqrt{\frac{1}{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}} \cdot -0.5\right) - t \cdot \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{{\left(\sqrt{\sqrt{2} \cdot \mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, \frac{\ell}{x}\right)}\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-236}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot {\left(2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.6
Cost	22092

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell}{\frac{x}{\ell}}\\ t_3 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ t_4 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+116}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_2 + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + t_4\right)}}\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-209}:\\ \;\;\;\;\frac{t_1}{\frac{2 \cdot \frac{\ell \cdot \ell}{x}}{t} \cdot \left(\sqrt{\frac{1}{t_3}} \cdot -0.5\right) - t \cdot \sqrt{t_3}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot t_2}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-234}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot {\left(2 \cdot \left(t_4 + t \cdot \left(t + \frac{t}{x}\right)\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	14872

\[\begin{array}{l} t_1 := t \cdot \left(\sqrt{2} \cdot {\left(2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)\right)}^{-0.5}\right)\\ \mathbf{if}\;t \leq -1.62 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-199}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-236}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 3
Error	10.0
Cost	14872

\[\begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+117}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{t_1}^{0.5}}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-201}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 10^{-276}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-234}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot {t_1}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	14872

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -1.36 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + t_2\right)}}\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-200}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot t_1}}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-236}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot {\left(2 \cdot \left(t_2 + t \cdot \left(t + \frac{t}{x}\right)\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 5
Error	10.1
Cost	14808

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-197}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-236}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 6
Error	12.7
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-234}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(t \cdot t\right)}{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7
Error	12.9
Cost	14032

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-202}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-277}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-236}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot t_1}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(t \cdot t\right)}{2 \cdot \left(t_1 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 8
Error	16.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	15.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 10
Error	16.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	39.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023053 
(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)))))