Toniolo and Linder, Equation (7)

Average Error: 43.1 → 11.7

Time: 31.3s

Precision: binary64

Cost: 126728

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{2} \cdot t\\ t_2 := \frac{{\ell}^{2}}{x}\\ t_3 := {\ell}^{2} + 2 \cdot {t}^{2}\\ t_4 := 1 + \frac{1}{x}\\ t_5 := 2 \cdot \left(t_4 + \frac{1}{x}\right)\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-207}:\\ \;\;\;\;\frac{t_1}{\sqrt{\left(t_2 + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(-\frac{\left(-t_3\right) - t_3}{{x}^{2}}\right)\right)\right)\right) + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} - -1 \cdot \left(\frac{t_3}{x} + \frac{t_3}{{x}^{3}}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-249}:\\ \;\;\;\;\frac{t_1}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-165}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{t_5} + \sqrt{\frac{1}{t_5}} \cdot \frac{{\ell}^{2}}{x \cdot t}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t_2 + {t}^{2} \cdot t_4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \]

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 (* (sqrt 2.0) t))
        (t_2 (/ (pow l 2.0) x))
        (t_3 (+ (pow l 2.0) (* 2.0 (pow t 2.0))))
        (t_4 (+ 1.0 (/ 1.0 x)))
        (t_5 (* 2.0 (+ t_4 (/ 1.0 x)))))
   (if (<= t -2.9e-40)
     (- (/ 1.0 x) 1.0)
     (if (<= t -6e-207)
       (/
        t_1
        (sqrt
         (+
          (+
           t_2
           (+
            (/ (pow l 2.0) (pow x 3.0))
            (+
             (* 2.0 (+ (pow t 2.0) (/ (pow t 2.0) x)))
             (- (/ (- (- t_3) t_3) (pow x 2.0))))))
          (-
           (* 2.0 (/ (pow t 2.0) (pow x 3.0)))
           (* -1.0 (+ (/ t_3 x) (/ t_3 (pow x 3.0))))))))
       (if (<= t -3.4e-221)
         -1.0
         (if (<= t 3.6e-249)
           (/ t_1 (sqrt (* (pow l 2.0) (+ (/ 1.0 (+ -1.0 x)) (/ 1.0 x)))))
           (if (<= t 4e-165)
             (/
              t_1
              (+
               (* t (sqrt t_5))
               (* (sqrt (/ 1.0 t_5)) (/ (pow l 2.0) (* x t)))))
             (if (<= t 3.4e-17)
               (/ t_1 (sqrt (* 2.0 (+ t_2 (* (pow t 2.0) t_4)))))
               (/ t_1 (* t_1 (sqrt (/ (+ 1.0 x) (- x 1.0)))))))))))))

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 = sqrt(2.0) * t;
	double t_2 = pow(l, 2.0) / x;
	double t_3 = pow(l, 2.0) + (2.0 * pow(t, 2.0));
	double t_4 = 1.0 + (1.0 / x);
	double t_5 = 2.0 * (t_4 + (1.0 / x));
	double tmp;
	if (t <= -2.9e-40) {
		tmp = (1.0 / x) - 1.0;
	} else if (t <= -6e-207) {
		tmp = t_1 / sqrt(((t_2 + ((pow(l, 2.0) / pow(x, 3.0)) + ((2.0 * (pow(t, 2.0) + (pow(t, 2.0) / x))) + -((-t_3 - t_3) / pow(x, 2.0))))) + ((2.0 * (pow(t, 2.0) / pow(x, 3.0))) - (-1.0 * ((t_3 / x) + (t_3 / pow(x, 3.0)))))));
	} else if (t <= -3.4e-221) {
		tmp = -1.0;
	} else if (t <= 3.6e-249) {
		tmp = t_1 / sqrt((pow(l, 2.0) * ((1.0 / (-1.0 + x)) + (1.0 / x))));
	} else if (t <= 4e-165) {
		tmp = t_1 / ((t * sqrt(t_5)) + (sqrt((1.0 / t_5)) * (pow(l, 2.0) / (x * t))));
	} else if (t <= 3.4e-17) {
		tmp = t_1 / sqrt((2.0 * (t_2 + (pow(t, 2.0) * t_4))));
	} else {
		tmp = t_1 / (t_1 * sqrt(((1.0 + x) / (x - 1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

↓

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = sqrt(2.0d0) * t
    t_2 = (l ** 2.0d0) / x
    t_3 = (l ** 2.0d0) + (2.0d0 * (t ** 2.0d0))
    t_4 = 1.0d0 + (1.0d0 / x)
    t_5 = 2.0d0 * (t_4 + (1.0d0 / x))
    if (t <= (-2.9d-40)) then
        tmp = (1.0d0 / x) - 1.0d0
    else if (t <= (-6d-207)) then
        tmp = t_1 / sqrt(((t_2 + (((l ** 2.0d0) / (x ** 3.0d0)) + ((2.0d0 * ((t ** 2.0d0) + ((t ** 2.0d0) / x))) + -((-t_3 - t_3) / (x ** 2.0d0))))) + ((2.0d0 * ((t ** 2.0d0) / (x ** 3.0d0))) - ((-1.0d0) * ((t_3 / x) + (t_3 / (x ** 3.0d0)))))))
    else if (t <= (-3.4d-221)) then
        tmp = -1.0d0
    else if (t <= 3.6d-249) then
        tmp = t_1 / sqrt(((l ** 2.0d0) * ((1.0d0 / ((-1.0d0) + x)) + (1.0d0 / x))))
    else if (t <= 4d-165) then
        tmp = t_1 / ((t * sqrt(t_5)) + (sqrt((1.0d0 / t_5)) * ((l ** 2.0d0) / (x * t))))
    else if (t <= 3.4d-17) then
        tmp = t_1 / sqrt((2.0d0 * (t_2 + ((t ** 2.0d0) * t_4))))
    else
        tmp = t_1 / (t_1 * sqrt(((1.0d0 + x) / (x - 1.0d0))))
    end if
    code = tmp
end function

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 = Math.sqrt(2.0) * t;
	double t_2 = Math.pow(l, 2.0) / x;
	double t_3 = Math.pow(l, 2.0) + (2.0 * Math.pow(t, 2.0));
	double t_4 = 1.0 + (1.0 / x);
	double t_5 = 2.0 * (t_4 + (1.0 / x));
	double tmp;
	if (t <= -2.9e-40) {
		tmp = (1.0 / x) - 1.0;
	} else if (t <= -6e-207) {
		tmp = t_1 / Math.sqrt(((t_2 + ((Math.pow(l, 2.0) / Math.pow(x, 3.0)) + ((2.0 * (Math.pow(t, 2.0) + (Math.pow(t, 2.0) / x))) + -((-t_3 - t_3) / Math.pow(x, 2.0))))) + ((2.0 * (Math.pow(t, 2.0) / Math.pow(x, 3.0))) - (-1.0 * ((t_3 / x) + (t_3 / Math.pow(x, 3.0)))))));
	} else if (t <= -3.4e-221) {
		tmp = -1.0;
	} else if (t <= 3.6e-249) {
		tmp = t_1 / Math.sqrt((Math.pow(l, 2.0) * ((1.0 / (-1.0 + x)) + (1.0 / x))));
	} else if (t <= 4e-165) {
		tmp = t_1 / ((t * Math.sqrt(t_5)) + (Math.sqrt((1.0 / t_5)) * (Math.pow(l, 2.0) / (x * t))));
	} else if (t <= 3.4e-17) {
		tmp = t_1 / Math.sqrt((2.0 * (t_2 + (Math.pow(t, 2.0) * t_4))));
	} else {
		tmp = t_1 / (t_1 * Math.sqrt(((1.0 + x) / (x - 1.0))));
	}
	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 = math.sqrt(2.0) * t
	t_2 = math.pow(l, 2.0) / x
	t_3 = math.pow(l, 2.0) + (2.0 * math.pow(t, 2.0))
	t_4 = 1.0 + (1.0 / x)
	t_5 = 2.0 * (t_4 + (1.0 / x))
	tmp = 0
	if t <= -2.9e-40:
		tmp = (1.0 / x) - 1.0
	elif t <= -6e-207:
		tmp = t_1 / math.sqrt(((t_2 + ((math.pow(l, 2.0) / math.pow(x, 3.0)) + ((2.0 * (math.pow(t, 2.0) + (math.pow(t, 2.0) / x))) + -((-t_3 - t_3) / math.pow(x, 2.0))))) + ((2.0 * (math.pow(t, 2.0) / math.pow(x, 3.0))) - (-1.0 * ((t_3 / x) + (t_3 / math.pow(x, 3.0)))))))
	elif t <= -3.4e-221:
		tmp = -1.0
	elif t <= 3.6e-249:
		tmp = t_1 / math.sqrt((math.pow(l, 2.0) * ((1.0 / (-1.0 + x)) + (1.0 / x))))
	elif t <= 4e-165:
		tmp = t_1 / ((t * math.sqrt(t_5)) + (math.sqrt((1.0 / t_5)) * (math.pow(l, 2.0) / (x * t))))
	elif t <= 3.4e-17:
		tmp = t_1 / math.sqrt((2.0 * (t_2 + (math.pow(t, 2.0) * t_4))))
	else:
		tmp = t_1 / (t_1 * math.sqrt(((1.0 + x) / (x - 1.0))))
	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(sqrt(2.0) * t)
	t_2 = Float64((l ^ 2.0) / x)
	t_3 = Float64((l ^ 2.0) + Float64(2.0 * (t ^ 2.0)))
	t_4 = Float64(1.0 + Float64(1.0 / x))
	t_5 = Float64(2.0 * Float64(t_4 + Float64(1.0 / x)))
	tmp = 0.0
	if (t <= -2.9e-40)
		tmp = Float64(Float64(1.0 / x) - 1.0);
	elseif (t <= -6e-207)
		tmp = Float64(t_1 / sqrt(Float64(Float64(t_2 + Float64(Float64((l ^ 2.0) / (x ^ 3.0)) + Float64(Float64(2.0 * Float64((t ^ 2.0) + Float64((t ^ 2.0) / x))) + Float64(-Float64(Float64(Float64(-t_3) - t_3) / (x ^ 2.0)))))) + Float64(Float64(2.0 * Float64((t ^ 2.0) / (x ^ 3.0))) - Float64(-1.0 * Float64(Float64(t_3 / x) + Float64(t_3 / (x ^ 3.0))))))));
	elseif (t <= -3.4e-221)
		tmp = -1.0;
	elseif (t <= 3.6e-249)
		tmp = Float64(t_1 / sqrt(Float64((l ^ 2.0) * Float64(Float64(1.0 / Float64(-1.0 + x)) + Float64(1.0 / x)))));
	elseif (t <= 4e-165)
		tmp = Float64(t_1 / Float64(Float64(t * sqrt(t_5)) + Float64(sqrt(Float64(1.0 / t_5)) * Float64((l ^ 2.0) / Float64(x * t)))));
	elseif (t <= 3.4e-17)
		tmp = Float64(t_1 / sqrt(Float64(2.0 * Float64(t_2 + Float64((t ^ 2.0) * t_4)))));
	else
		tmp = Float64(t_1 / Float64(t_1 * sqrt(Float64(Float64(1.0 + x) / Float64(x - 1.0)))));
	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 = sqrt(2.0) * t;
	t_2 = (l ^ 2.0) / x;
	t_3 = (l ^ 2.0) + (2.0 * (t ^ 2.0));
	t_4 = 1.0 + (1.0 / x);
	t_5 = 2.0 * (t_4 + (1.0 / x));
	tmp = 0.0;
	if (t <= -2.9e-40)
		tmp = (1.0 / x) - 1.0;
	elseif (t <= -6e-207)
		tmp = t_1 / sqrt(((t_2 + (((l ^ 2.0) / (x ^ 3.0)) + ((2.0 * ((t ^ 2.0) + ((t ^ 2.0) / x))) + -((-t_3 - t_3) / (x ^ 2.0))))) + ((2.0 * ((t ^ 2.0) / (x ^ 3.0))) - (-1.0 * ((t_3 / x) + (t_3 / (x ^ 3.0)))))));
	elseif (t <= -3.4e-221)
		tmp = -1.0;
	elseif (t <= 3.6e-249)
		tmp = t_1 / sqrt(((l ^ 2.0) * ((1.0 / (-1.0 + x)) + (1.0 / x))));
	elseif (t <= 4e-165)
		tmp = t_1 / ((t * sqrt(t_5)) + (sqrt((1.0 / t_5)) * ((l ^ 2.0) / (x * t))));
	elseif (t <= 3.4e-17)
		tmp = t_1 / sqrt((2.0 * (t_2 + ((t ^ 2.0) * t_4))));
	else
		tmp = t_1 / (t_1 * sqrt(((1.0 + x) / (x - 1.0))));
	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[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[l, 2.0], $MachinePrecision] + N[(2.0 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(t$95$4 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.9e-40], N[(N[(1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[t, -6e-207], N[(t$95$1 / N[Sqrt[N[(N[(t$95$2 + N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] + N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[((-t$95$3) - t$95$3), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t, 2.0], $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(t$95$3 / x), $MachinePrecision] + N[(t$95$3 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.4e-221], -1.0, If[LessEqual[t, 3.6e-249], N[(t$95$1 / N[Sqrt[N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-165], N[(t$95$1 / N[(N[(t * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$5), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-17], N[(t$95$1 / N[Sqrt[N[(2.0 * N[(t$95$2 + N[(N[Power[t, 2.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$1 * N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -2.8999999999999999e-40

   1. Initial program 39.6

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

   2. Taylor expanded in x around inf 35.6

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

   3. Simplified 35.6

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

      [Start] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational_best_45_simplify-109 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational_best_45_simplify-73 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\color{blue}{{t}^{2} \cdot 2} + 2 \cdot \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-71 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational_best_45_simplify-16 [=>] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   4. Taylor expanded in t around -inf 6.7

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

   5. Simplified 6.7

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

      [Start] 6.7	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \]
      rational_best_45_simplify-25 [=>] 6.7	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \left(-1 \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \]
      rational_best_45_simplify-91 [=>] 6.7	\[ \frac{\sqrt{2} \cdot t}{t \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot -1\right)}} \]
      rational_best_45_simplify-16 [=>] 6.7	\[ \frac{\sqrt{2} \cdot t}{t \cdot \color{blue}{\left(-\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \]
      rational_best_45_simplify-91 [=>] 6.7	\[ \frac{\sqrt{2} \cdot t}{t \cdot \left(-\sqrt{\color{blue}{\left(1 + \frac{1}{x}\right) \cdot 2} + 2 \cdot \frac{1}{x}}\right)} \]
      rational_best_45_simplify-71 [=>] 6.7	\[ \frac{\sqrt{2} \cdot t}{t \cdot \left(-\sqrt{\color{blue}{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}}\right)} \]

   6. Taylor expanded in x around inf 6.6

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

   if -2.8999999999999999e-40 < t < -5.9999999999999999e-207

   1. Initial program 42.0

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

   2. Taylor expanded in x around -inf 17.9

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

   3. Simplified 17.9

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

      [Start] 17.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right)\right) - \left(-1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}} + -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-80 [=>] 17.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(\frac{{\ell}^{2}}{x} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right)\right)} - \left(-1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}} + -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-109 [=>] 17.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right) + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} - \left(-1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}} + -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)\right)}}} \]

   if -5.9999999999999999e-207 < t < -3.4000000000000001e-221

   1. Initial program 64.0

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

   2. Taylor expanded in x around inf 30.7

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

   3. Simplified 30.7

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

      [Start] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational_best_45_simplify-109 [=>] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational_best_45_simplify-73 [=>] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\color{blue}{{t}^{2} \cdot 2} + 2 \cdot \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-71 [=>] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational_best_45_simplify-16 [=>] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   4. Taylor expanded in t around -inf 40.0

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

   5. Simplified 40.0

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

      [Start] 40.0	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \]
      rational_best_45_simplify-25 [=>] 40.0	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \left(-1 \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \]
      rational_best_45_simplify-91 [=>] 40.0	\[ \frac{\sqrt{2} \cdot t}{t \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot -1\right)}} \]
      rational_best_45_simplify-16 [=>] 40.0	\[ \frac{\sqrt{2} \cdot t}{t \cdot \color{blue}{\left(-\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \]
      rational_best_45_simplify-91 [=>] 40.0	\[ \frac{\sqrt{2} \cdot t}{t \cdot \left(-\sqrt{\color{blue}{\left(1 + \frac{1}{x}\right) \cdot 2} + 2 \cdot \frac{1}{x}}\right)} \]
      rational_best_45_simplify-71 [=>] 40.0	\[ \frac{\sqrt{2} \cdot t}{t \cdot \left(-\sqrt{\color{blue}{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}}\right)} \]

   6. Taylor expanded in x around inf 40.0

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

   if -3.4000000000000001e-221 < t < 3.59999999999999995e-249

   1. Initial program 63.2

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

   2. Taylor expanded in l around inf 62.6

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

   3. Simplified 45.1

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

      [Start] 62.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\left(\frac{x}{x - 1} + \frac{1}{x - 1}\right) - 1\right) \cdot {\ell}^{2}}} \]
      rational_best_45_simplify-91 [=>] 62.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{{\ell}^{2} \cdot \left(\left(\frac{x}{x - 1} + \frac{1}{x - 1}\right) - 1\right)}}} \]
      rational_best_45_simplify-109 [=>] 45.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{{\ell}^{2} \cdot \color{blue}{\left(\frac{1}{x - 1} + \left(\frac{x}{x - 1} - 1\right)\right)}}} \]
      rational_best_45_simplify-55 [<=] 45.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{x + -1}} + \left(\frac{x}{x - 1} - 1\right)\right)}} \]
      rational_best_45_simplify-73 [=>] 45.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{\color{blue}{-1 + x}} + \left(\frac{x}{x - 1} - 1\right)\right)}} \]
      rational_best_45_simplify-55 [<=] 45.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \left(\frac{x}{\color{blue}{x + -1}} - 1\right)\right)}} \]
      rational_best_45_simplify-73 [=>] 45.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \left(\frac{x}{\color{blue}{-1 + x}} - 1\right)\right)}} \]

   4. Taylor expanded in x around inf 32.6

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

   if 3.59999999999999995e-249 < t < 4e-165

   1. Initial program 63.3

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

   2. Taylor expanded in x around inf 36.3

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

   3. Simplified 36.3

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

      [Start] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational_best_45_simplify-109 [=>] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational_best_45_simplify-73 [=>] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\color{blue}{{t}^{2} \cdot 2} + 2 \cdot \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-71 [=>] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational_best_45_simplify-16 [=>] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   4. Taylor expanded in t around inf 22.6

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

   5. Simplified 22.6

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

      [Start] 22.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational_best_45_simplify-91 [=>] 22.6	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational_best_45_simplify-91 [=>] 22.6	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\color{blue}{\left(1 + \frac{1}{x}\right) \cdot 2} + 2 \cdot \frac{1}{x}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational_best_45_simplify-71 [=>] 22.6	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\color{blue}{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational_best_45_simplify-91 [=>] 22.6	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)} + \sqrt{\frac{1}{\color{blue}{\left(1 + \frac{1}{x}\right) \cdot 2} + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational_best_45_simplify-71 [=>] 22.6	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)} + \sqrt{\frac{1}{\color{blue}{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational_best_45_simplify-91 [=>] 22.6	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)} + \sqrt{\frac{1}{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{\color{blue}{x \cdot t}}} \]

   if 4e-165 < t < 3.3999999999999998e-17

   1. Initial program 33.4

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

   2. Taylor expanded in x around inf 10.7

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

   3. Simplified 10.7

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

      [Start] 10.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational_best_45_simplify-109 [=>] 10.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational_best_45_simplify-73 [=>] 10.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 10.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\color{blue}{{t}^{2} \cdot 2} + 2 \cdot \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-71 [=>] 10.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational_best_45_simplify-91 [=>] 10.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational_best_45_simplify-16 [=>] 10.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(\frac{{\ell}^{2}}{x} - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   4. Taylor expanded in l around inf 10.9

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

   5. Applied egg-rr 10.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + \left({t}^{2} + \frac{{t}^{2}}{x}\right)\right)} + 0}} \]

   6. Simplified 10.9

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

      [Start] 10.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + \left({t}^{2} + \frac{{t}^{2}}{x}\right)\right)} + 0} \]
      rational_best_45_simplify-3 [=>] 10.9	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + \left({t}^{2} + \frac{{t}^{2}}{x}\right)\right)}}} \]
      rational_best_45_simplify-80 [=>] 10.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} + \left(\frac{{\ell}^{2}}{x} + \frac{{t}^{2}}{x}\right)\right)}}} \]

   7. Taylor expanded in t around 0 10.9

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

   if 3.3999999999999998e-17 < t

   1. Initial program 40.4

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

   2. Taylor expanded in l around 0 5.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-207}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(\frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right) + \left(-\frac{\left(-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}}\right)\right)\right)\right) + \left(2 \cdot \frac{{t}^{2}}{{x}^{3}} - -1 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{{x}^{3}}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-249}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)} + \sqrt{\frac{1}{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{x \cdot t}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	80264

\[\begin{array}{l} t_1 := {\ell}^{2} + 2 \cdot {t}^{2}\\ t_2 := \frac{{\ell}^{2}}{x}\\ t_3 := 1 + \frac{1}{x}\\ t_4 := 2 \cdot \left(t_3 + \frac{1}{x}\right)\\ t_5 := \sqrt{2} \cdot t\\ \mathbf{if}\;t \leq -1.66 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-207}:\\ \;\;\;\;\frac{t_5}{\sqrt{\left(t_2 + 2 \cdot \left({t}^{2} + \frac{{t}^{2}}{x}\right)\right) + \left(\left(-\frac{\left(-t_1\right) - t_1}{{x}^{2}}\right) - \left(-\frac{t_1}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-250}:\\ \;\;\;\;\frac{t_5}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;\frac{t_5}{t \cdot \sqrt{t_4} + \sqrt{\frac{1}{t_4}} \cdot \frac{{\ell}^{2}}{x \cdot t}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-18}:\\ \;\;\;\;\frac{t_5}{\sqrt{2 \cdot \left(t_2 + {t}^{2} \cdot t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_5}{t_5 \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \]

Alternative 2
Error	11.5
Cost	28692

\[\begin{array}{l} t_1 := 1 + \frac{1}{x}\\ t_2 := \sqrt{2} \cdot t\\ t_3 := \frac{t_2}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot t_1\right)}}\\ t_4 := 2 \cdot \left(t_1 + \frac{1}{x}\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-242}:\\ \;\;\;\;\frac{t_2}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-165}:\\ \;\;\;\;\frac{t_2}{t \cdot \sqrt{t_4} + \sqrt{\frac{1}{t_4}} \cdot \frac{{\ell}^{2}}{x \cdot t}}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-18}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \]

Alternative 3
Error	11.9
Cost	27608

\[\begin{array}{l} t_1 := 1 + \frac{1}{x}\\ t_2 := \sqrt{2} \cdot t\\ t_3 := \frac{t_2}{\sqrt{2 \cdot \left(\frac{{\ell}^{2}}{x} + {t}^{2} \cdot t_1\right)}}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-221}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{t_2}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-166}:\\ \;\;\;\;\frac{t_2}{t \cdot \sqrt{2 \cdot \left(t_1 + \frac{1}{x}\right)}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	20488

\[\begin{array}{l} t_1 := \sqrt{2} \cdot t\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-231}:\\ \;\;\;\;\frac{t_1}{\sqrt{{\ell}^{2} \cdot \left(\frac{1}{-1 + x} + \frac{1}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \]

Alternative 5
Error	14.3
Cost	20424

\[\begin{array}{l} t_1 := \sqrt{2} \cdot t\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-226}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \sqrt{\frac{1 + x}{x - 1}}}\\ \end{array} \]

Alternative 6
Error	14.7
Cost	20168

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\\ \end{array} \]

Alternative 7
Error	14.5
Cost	20104

\[\begin{array}{l} t_1 := \sqrt{2} \cdot t\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-217}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}}\\ \end{array} \]

Alternative 8
Error	15.2
Cost	14152

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-229}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \left(\left(1 + \frac{1}{x}\right) + \frac{1}{x}\right)}}\\ \end{array} \]

Alternative 9
Error	15.5
Cost	13640

\[\begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{-220}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \left(1 - \frac{1}{x}\right)\\ \end{array} \]

Alternative 10
Error	15.4
Cost	13512

\[\begin{array}{l} t_1 := \sqrt{2} \cdot t\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-206}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1}\\ \end{array} \]

Alternative 11
Error	15.7
Cost	13256

\[\begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-223}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 12
Error	36.1
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+197}:\\ \;\;\;\;\frac{t}{\ell} \cdot \left(-\sqrt{x}\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]

Alternative 13
Error	35.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-221}:\\ \;\;\;\;\frac{1}{x} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]

Alternative 14
Error	39.4
Cost	320

\[\frac{1}{x} - 1 \]

Alternative 15
Error	39.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023098 
(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)))))