Toniolo and Linder, Equation (7)

Average Error: 43.4 → 11.8

Time: 35.9s

Precision: binary64

Cost: 40728

Math

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

↓

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{x}\\ t_2 := 1 + \left(\frac{1}{x} + \frac{1}{x}\right)\\ t_3 := t \cdot \sqrt{2 \cdot t_2} + \sqrt{\frac{0.5}{t_2}} \cdot \frac{{\ell}^{2}}{t \cdot x}\\ t_4 := \sqrt{2} \cdot t\\ t_5 := t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \left(-t_1\right)\right)}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{t_4}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-162}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-283}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-t_3}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-208}:\\ \;\;\;\;\frac{t_4}{t_3}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

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 (/ (pow l 2.0) x))
        (t_2 (+ 1.0 (+ (/ 1.0 x) (/ 1.0 x))))
        (t_3
         (+
          (* t (sqrt (* 2.0 t_2)))
          (* (sqrt (/ 0.5 t_2)) (/ (pow l 2.0) (* t x)))))
        (t_4 (* (sqrt 2.0) t))
        (t_5
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+ t_1 (- (* 2.0 (+ (/ (pow t 2.0) x) (pow t 2.0))) (- t_1))))))))
   (if (<= t -5e-94)
     (/ t_4 (* (* t (sqrt 2.0)) (- (sqrt (/ (+ x 1.0) (+ -1.0 x))))))
     (if (<= t -7.5e-162)
       t_5
       (if (<= t -3.9e-283)
         (* t (/ (sqrt 2.0) (- t_3)))
         (if (<= t 1.32e-257)
           (* t (* (/ 1.0 l) (sqrt x)))
           (if (<= t 1.65e-208)
             (/ t_4 t_3)
             (if (<= t 8.5e-39) t_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 = pow(l, 2.0) / x;
	double t_2 = 1.0 + ((1.0 / x) + (1.0 / x));
	double t_3 = (t * sqrt((2.0 * t_2))) + (sqrt((0.5 / t_2)) * (pow(l, 2.0) / (t * x)));
	double t_4 = sqrt(2.0) * t;
	double t_5 = t * (sqrt(2.0) / sqrt((t_1 + ((2.0 * ((pow(t, 2.0) / x) + pow(t, 2.0))) - -t_1))));
	double tmp;
	if (t <= -5e-94) {
		tmp = t_4 / ((t * sqrt(2.0)) * -sqrt(((x + 1.0) / (-1.0 + x))));
	} else if (t <= -7.5e-162) {
		tmp = t_5;
	} else if (t <= -3.9e-283) {
		tmp = t * (sqrt(2.0) / -t_3);
	} else if (t <= 1.32e-257) {
		tmp = t * ((1.0 / l) * sqrt(x));
	} else if (t <= 1.65e-208) {
		tmp = t_4 / t_3;
	} else if (t <= 8.5e-39) {
		tmp = t_5;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	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 = (l ** 2.0d0) / x
    t_2 = 1.0d0 + ((1.0d0 / x) + (1.0d0 / x))
    t_3 = (t * sqrt((2.0d0 * t_2))) + (sqrt((0.5d0 / t_2)) * ((l ** 2.0d0) / (t * x)))
    t_4 = sqrt(2.0d0) * t
    t_5 = t * (sqrt(2.0d0) / sqrt((t_1 + ((2.0d0 * (((t ** 2.0d0) / x) + (t ** 2.0d0))) - -t_1))))
    if (t <= (-5d-94)) then
        tmp = t_4 / ((t * sqrt(2.0d0)) * -sqrt(((x + 1.0d0) / ((-1.0d0) + x))))
    else if (t <= (-7.5d-162)) then
        tmp = t_5
    else if (t <= (-3.9d-283)) then
        tmp = t * (sqrt(2.0d0) / -t_3)
    else if (t <= 1.32d-257) then
        tmp = t * ((1.0d0 / l) * sqrt(x))
    else if (t <= 1.65d-208) then
        tmp = t_4 / t_3
    else if (t <= 8.5d-39) then
        tmp = t_5
    else
        tmp = 1.0d0 - (1.0d0 / x)
    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.pow(l, 2.0) / x;
	double t_2 = 1.0 + ((1.0 / x) + (1.0 / x));
	double t_3 = (t * Math.sqrt((2.0 * t_2))) + (Math.sqrt((0.5 / t_2)) * (Math.pow(l, 2.0) / (t * x)));
	double t_4 = Math.sqrt(2.0) * t;
	double t_5 = t * (Math.sqrt(2.0) / Math.sqrt((t_1 + ((2.0 * ((Math.pow(t, 2.0) / x) + Math.pow(t, 2.0))) - -t_1))));
	double tmp;
	if (t <= -5e-94) {
		tmp = t_4 / ((t * Math.sqrt(2.0)) * -Math.sqrt(((x + 1.0) / (-1.0 + x))));
	} else if (t <= -7.5e-162) {
		tmp = t_5;
	} else if (t <= -3.9e-283) {
		tmp = t * (Math.sqrt(2.0) / -t_3);
	} else if (t <= 1.32e-257) {
		tmp = t * ((1.0 / l) * Math.sqrt(x));
	} else if (t <= 1.65e-208) {
		tmp = t_4 / t_3;
	} else if (t <= 8.5e-39) {
		tmp = t_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 = math.pow(l, 2.0) / x
	t_2 = 1.0 + ((1.0 / x) + (1.0 / x))
	t_3 = (t * math.sqrt((2.0 * t_2))) + (math.sqrt((0.5 / t_2)) * (math.pow(l, 2.0) / (t * x)))
	t_4 = math.sqrt(2.0) * t
	t_5 = t * (math.sqrt(2.0) / math.sqrt((t_1 + ((2.0 * ((math.pow(t, 2.0) / x) + math.pow(t, 2.0))) - -t_1))))
	tmp = 0
	if t <= -5e-94:
		tmp = t_4 / ((t * math.sqrt(2.0)) * -math.sqrt(((x + 1.0) / (-1.0 + x))))
	elif t <= -7.5e-162:
		tmp = t_5
	elif t <= -3.9e-283:
		tmp = t * (math.sqrt(2.0) / -t_3)
	elif t <= 1.32e-257:
		tmp = t * ((1.0 / l) * math.sqrt(x))
	elif t <= 1.65e-208:
		tmp = t_4 / t_3
	elif t <= 8.5e-39:
		tmp = t_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((l ^ 2.0) / x)
	t_2 = Float64(1.0 + Float64(Float64(1.0 / x) + Float64(1.0 / x)))
	t_3 = Float64(Float64(t * sqrt(Float64(2.0 * t_2))) + Float64(sqrt(Float64(0.5 / t_2)) * Float64((l ^ 2.0) / Float64(t * x))))
	t_4 = Float64(sqrt(2.0) * t)
	t_5 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_1 + Float64(Float64(2.0 * Float64(Float64((t ^ 2.0) / x) + (t ^ 2.0))) - Float64(-t_1))))))
	tmp = 0.0
	if (t <= -5e-94)
		tmp = Float64(t_4 / Float64(Float64(t * sqrt(2.0)) * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))))));
	elseif (t <= -7.5e-162)
		tmp = t_5;
	elseif (t <= -3.9e-283)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-t_3)));
	elseif (t <= 1.32e-257)
		tmp = Float64(t * Float64(Float64(1.0 / l) * sqrt(x)));
	elseif (t <= 1.65e-208)
		tmp = Float64(t_4 / t_3);
	elseif (t <= 8.5e-39)
		tmp = t_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 = (l ^ 2.0) / x;
	t_2 = 1.0 + ((1.0 / x) + (1.0 / x));
	t_3 = (t * sqrt((2.0 * t_2))) + (sqrt((0.5 / t_2)) * ((l ^ 2.0) / (t * x)));
	t_4 = sqrt(2.0) * t;
	t_5 = t * (sqrt(2.0) / sqrt((t_1 + ((2.0 * (((t ^ 2.0) / x) + (t ^ 2.0))) - -t_1))));
	tmp = 0.0;
	if (t <= -5e-94)
		tmp = t_4 / ((t * sqrt(2.0)) * -sqrt(((x + 1.0) / (-1.0 + x))));
	elseif (t <= -7.5e-162)
		tmp = t_5;
	elseif (t <= -3.9e-283)
		tmp = t * (sqrt(2.0) / -t_3);
	elseif (t <= 1.32e-257)
		tmp = t * ((1.0 / l) * sqrt(x));
	elseif (t <= 1.65e-208)
		tmp = t_4 / t_3;
	elseif (t <= 8.5e-39)
		tmp = t_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[(N[Power[l, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(0.5 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$5 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[(N[(2.0 * N[(N[(N[Power[t, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - (-t$95$1)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e-94], N[(t$95$4 / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.5e-162], t$95$5, If[LessEqual[t, -3.9e-283], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-t$95$3)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.32e-257], N[(t * N[(N[(1.0 / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65e-208], N[(t$95$4 / t$95$3), $MachinePrecision], If[LessEqual[t, 8.5e-39], t$95$5, N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.9999999999999995e-94

   1. Initial program 38.6

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

   2. Taylor expanded in t around -inf 7.9

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

   3. Simplified 7.9

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

      [Start] 7.9	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      rational.json-simplify-43 [=>] 7.9	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \left(\sqrt{\frac{1 + x}{x - 1}} \cdot -1\right)}} \]
      rational.json-simplify-2 [<=] 7.9	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \left(\sqrt{\frac{1 + x}{x - 1}} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 7.9	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \color{blue}{\left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]
      rational.json-simplify-1 [=>] 7.9	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]
      rational.json-simplify-15 [<=] 7.9	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + -1}}}\right)} \]
      rational.json-simplify-1 [=>] 7.9	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   if -4.9999999999999995e-94 < t < -7.49999999999999972e-162 or 1.65000000000000003e-208 < t < 8.5000000000000005e-39

   1. Initial program 41.2

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

   2. Simplified 41.2

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

      [Start] 41.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      rational.json-simplify-49 [=>] 41.2	\[ \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      rational.json-simplify-16 [=>] 41.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf 15.7

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

   4. Simplified 15.7

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

      [Start] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-1 [=>] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-48 [=>] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational.json-simplify-1 [=>] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot {t}^{2} + \color{blue}{\frac{{t}^{2}}{x} \cdot 2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-51 [=>] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational.json-simplify-9 [=>] 15.7	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   5. Taylor expanded in l around inf 15.9

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

   if -7.49999999999999972e-162 < t < -3.9000000000000002e-283

   1. Initial program 63.1

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

   2. Simplified 63.1

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

      [Start] 63.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      rational.json-simplify-49 [=>] 63.1	\[ \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      rational.json-simplify-16 [=>] 63.1	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf 34.6

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

   4. Simplified 34.6

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

      [Start] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-1 [=>] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-48 [=>] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational.json-simplify-1 [=>] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot {t}^{2} + \color{blue}{\frac{{t}^{2}}{x} \cdot 2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-51 [=>] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational.json-simplify-9 [=>] 34.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   5. Taylor expanded in t around -inf 25.2

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

   6. Simplified 25.2

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

      [Start] 25.2	\[ t \cdot \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \]
      rational.json-simplify-2 [=>] 25.2	\[ t \cdot \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \]
      rational.json-simplify-2 [=>] 25.2	\[ t \cdot \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \]
      rational.json-simplify-2 [=>] 25.2	\[ t \cdot \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right) \cdot -1}} \]
      rational.json-simplify-51 [=>] 25.2	\[ t \cdot \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \]
      rational.json-simplify-2 [=>] 25.2	\[ t \cdot \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) \cdot -1}} \]
      rational.json-simplify-9 [=>] 25.2	\[ t \cdot \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \]

   if -3.9000000000000002e-283 < t < 1.32e-257

   1. Initial program 62.0

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

   2. Simplified 62.0

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

      [Start] 62.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      rational.json-simplify-49 [=>] 62.0	\[ \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      rational.json-simplify-16 [=>] 62.0	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf 30.2

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

   4. Simplified 30.2

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

      [Start] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-1 [=>] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-48 [=>] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational.json-simplify-1 [=>] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot {t}^{2} + \color{blue}{\frac{{t}^{2}}{x} \cdot 2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-51 [=>] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational.json-simplify-9 [=>] 30.2	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   5. Taylor expanded in l around inf 31.6

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

   6. Taylor expanded in l around 0 31.6

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

   if 1.32e-257 < t < 1.65000000000000003e-208

   1. Initial program 63.4

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

   2. Taylor expanded in x around inf 34.5

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

   3. Simplified 34.5

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

      [Start] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-1 [=>] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-48 [=>] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational.json-simplify-1 [=>] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot {t}^{2} + \color{blue}{\frac{{t}^{2}}{x} \cdot 2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-51 [=>] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational.json-simplify-9 [=>] 34.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   4. Taylor expanded in t around inf 22.9

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

   5. Simplified 22.9

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

      [Start] 22.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational.json-simplify-2 [<=] 22.9	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational.json-simplify-2 [=>] 22.9	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + \color{blue}{\frac{1}{x} \cdot 2}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational.json-simplify-51 [=>] 22.9	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{1}{x} + \left(1 + \frac{1}{x}\right)\right)}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]
      rational.json-simplify-41 [=>] 22.9	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \color{blue}{\left(1 + \left(\frac{1}{x} + \frac{1}{x}\right)\right)}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \]

   if 8.5000000000000005e-39 < 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. Simplified 40.4

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

      [Start] 40.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      rational.json-simplify-49 [=>] 40.4	\[ \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      rational.json-simplify-16 [=>] 40.4	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Taylor expanded in x around inf 36.6

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

   4. Simplified 36.6

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

      [Start] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-1 [=>] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      rational.json-simplify-48 [=>] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      rational.json-simplify-1 [=>] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{\left(2 \cdot {t}^{2} + 2 \cdot \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot {t}^{2} + \color{blue}{\frac{{t}^{2}}{x} \cdot 2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-51 [=>] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      rational.json-simplify-2 [=>] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} \cdot -1}\right)}} \]
      rational.json-simplify-9 [=>] 36.6	\[ t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \]

   5. Taylor expanded in t around inf 7.1

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

   6. Simplified 7.1

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

      [Start] 7.1	\[ t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t} \]
      rational.json-simplify-2 [<=] 7.1	\[ t \cdot \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}} \]
      rational.json-simplify-2 [=>] 7.1	\[ t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + \color{blue}{\frac{1}{x} \cdot 2}}} \]
      rational.json-simplify-51 [=>] 7.1	\[ t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{\color{blue}{2 \cdot \left(\frac{1}{x} + \left(1 + \frac{1}{x}\right)\right)}}} \]
      rational.json-simplify-41 [=>] 7.1	\[ t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 \cdot \color{blue}{\left(1 + \left(\frac{1}{x} + \frac{1}{x}\right)\right)}}} \]

   7. Taylor expanded in x around inf 6.7

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

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-283}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\left(t \cdot \sqrt{2 \cdot \left(1 + \left(\frac{1}{x} + \frac{1}{x}\right)\right)} + \sqrt{\frac{0.5}{1 + \left(\frac{1}{x} + \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{t \cdot \sqrt{2 \cdot \left(1 + \left(\frac{1}{x} + \frac{1}{x}\right)\right)} + \sqrt{\frac{0.5}{1 + \left(\frac{1}{x} + \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-39}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.1
Cost	28300

\[\begin{array}{l} t_1 := 1 + \left(\frac{1}{x} + \frac{1}{x}\right)\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-135}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 \cdot t_1} + \sqrt{\frac{0.5}{t_1}} \cdot \frac{{\ell}^{2}}{t \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 2
Error	13.9
Cost	28300

\[\begin{array}{l} t_1 := 1 + \left(\frac{1}{x} + \frac{1}{x}\right)\\ t_2 := \sqrt{2} \cdot t\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-282}:\\ \;\;\;\;\frac{t_2}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-258}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{t_2}{t \cdot \sqrt{2 \cdot t_1} + \sqrt{\frac{0.5}{t_1}} \cdot \frac{{\ell}^{2}}{t \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 3
Error	15.3
Cost	20356

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{-254}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-208}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 4
Error	15.2
Cost	20356

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-284}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-254}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 5
Error	15.4
Cost	14084

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-280}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \left(\frac{1}{x} + \frac{1}{x}\right)\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-255}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-172}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 6
Error	15.4
Cost	14084

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(1 + \left(\frac{1}{x} + \frac{1}{x}\right)\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-255}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-208}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 7
Error	15.3
Cost	14084

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(1 + \left(\frac{1}{x} + \frac{1}{x}\right)\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-254}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-209}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-173}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 8
Error	15.7
Cost	13572

\[\begin{array}{l} t_1 := 1 - \frac{1}{x}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{2} \cdot \left(t_1 \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-254}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-210}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	15.9
Cost	13188

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{-254}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-211}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-174}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 10
Error	33.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+144}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{-1}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+25}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 11
Error	34.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+142}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+25}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]

Alternative 12
Error	34.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+147}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{-1}{\ell}\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{+25}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]

Alternative 13
Error	37.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+24}:\\ \;\;\;\;1 - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \end{array} \]

Alternative 14
Error	39.1
Cost	320

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

Alternative 15
Error	39.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023064 
(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)))))