Toniolo and Linder, Equation (7)

Average Error: 43.1 → 10.0

Time: 32.4s

Precision: binary64

Cost: 20620

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{t_1 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ t_3 := \frac{t_1}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-201}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+21}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

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 (/ t_1 (* t_1 (- (sqrt (/ (+ x 1.0) (+ x -1.0)))))))
        (t_3
         (/ t_1 (sqrt (+ (* 2.0 (* t (+ t (/ t x)))) (* 2.0 (* l (/ l x))))))))
   (if (<= t -6.4e+93)
     t_2
     (if (<= t -8.6e-201)
       t_3
       (if (<= t -3.8e-285)
         t_2
         (if (<= t 1.65e+21) t_3 (/ (sqrt 2.0) (sqrt (+ 2.0 (/ 4.0 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 = t_1 / (t_1 * -sqrt(((x + 1.0) / (x + -1.0))));
	double t_3 = t_1 / sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x)))));
	double tmp;
	if (t <= -6.4e+93) {
		tmp = t_2;
	} else if (t <= -8.6e-201) {
		tmp = t_3;
	} else if (t <= -3.8e-285) {
		tmp = t_2;
	} else if (t <= 1.65e+21) {
		tmp = t_3;
	} else {
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / 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) :: tmp
    t_1 = t * sqrt(2.0d0)
    t_2 = t_1 / (t_1 * -sqrt(((x + 1.0d0) / (x + (-1.0d0)))))
    t_3 = t_1 / sqrt(((2.0d0 * (t * (t + (t / x)))) + (2.0d0 * (l * (l / x)))))
    if (t <= (-6.4d+93)) then
        tmp = t_2
    else if (t <= (-8.6d-201)) then
        tmp = t_3
    else if (t <= (-3.8d-285)) then
        tmp = t_2
    else if (t <= 1.65d+21) then
        tmp = t_3
    else
        tmp = sqrt(2.0d0) / sqrt((2.0d0 + (4.0d0 / 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 = t * Math.sqrt(2.0);
	double t_2 = t_1 / (t_1 * -Math.sqrt(((x + 1.0) / (x + -1.0))));
	double t_3 = t_1 / Math.sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x)))));
	double tmp;
	if (t <= -6.4e+93) {
		tmp = t_2;
	} else if (t <= -8.6e-201) {
		tmp = t_3;
	} else if (t <= -3.8e-285) {
		tmp = t_2;
	} else if (t <= 1.65e+21) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 + (4.0 / 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 = t_1 / (t_1 * -math.sqrt(((x + 1.0) / (x + -1.0))))
	t_3 = t_1 / math.sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x)))))
	tmp = 0
	if t <= -6.4e+93:
		tmp = t_2
	elif t <= -8.6e-201:
		tmp = t_3
	elif t <= -3.8e-285:
		tmp = t_2
	elif t <= 1.65e+21:
		tmp = t_3
	else:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 + (4.0 / 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(t_1 / Float64(t_1 * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))))
	t_3 = Float64(t_1 / sqrt(Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(2.0 * Float64(l * Float64(l / x))))))
	tmp = 0.0
	if (t <= -6.4e+93)
		tmp = t_2;
	elseif (t <= -8.6e-201)
		tmp = t_3;
	elseif (t <= -3.8e-285)
		tmp = t_2;
	elseif (t <= 1.65e+21)
		tmp = t_3;
	else
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 + Float64(4.0 / 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 = t_1 / (t_1 * -sqrt(((x + 1.0) / (x + -1.0))));
	t_3 = t_1 / sqrt(((2.0 * (t * (t + (t / x)))) + (2.0 * (l * (l / x)))));
	tmp = 0.0;
	if (t <= -6.4e+93)
		tmp = t_2;
	elseif (t <= -8.6e-201)
		tmp = t_3;
	elseif (t <= -3.8e-285)
		tmp = t_2;
	elseif (t <= 1.65e+21)
		tmp = t_3;
	else
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / 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[(t$95$1 / N[(t$95$1 * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.4e+93], t$95$2, If[LessEqual[t, -8.6e-201], t$95$3, If[LessEqual[t, -3.8e-285], t$95$2, If[LessEqual[t, 1.65e+21], t$95$3, N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.4000000000000003e93 or -8.5999999999999995e-201 < t < -3.8000000000000002e-285

   1. Initial program 53.0

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

   2. Taylor expanded in t around -inf 10.3

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

   3. Simplified 10.3

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

      [Start] 10.3	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 10.3	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [<=] 10.3	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      sub-neg [=>] 10.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 10.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 10.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}} \]
      +-commutative [=>] 10.3	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   if -6.4000000000000003e93 < t < -8.5999999999999995e-201 or -3.8000000000000002e-285 < t < 1.65e21

   1. Initial program 37.5

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

   2. Taylor expanded in x around inf 17.0

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

   3. Simplified 17.0

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

      [Start] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 17.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 17.2

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

   5. Simplified 17.2

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

      [Start] 17.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 17.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot {\ell}^{2}}{x}}\right)}} \]
      mul-1-neg [=>] 17.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-{\ell}^{2}}}{x}\right)}} \]
      unpow2 [=>] 17.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\ell \cdot \ell}}{x}\right)}} \]

   6. Applied egg-rr 13.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}}} \]

   7. Simplified 13.0

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

      [Start] 13.1	\[ \frac{\sqrt{2} \cdot t}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}} \]
      pow-sqr [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\left(2 \cdot 0.25\right)}}} \]
      metadata-eval [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\color{blue}{0.5}}} \]
      unpow1/2 [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)}}} \]
      +-commutative [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right) + \frac{\ell}{x} \cdot \ell}}} \]
      fma-udef [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \frac{\ell}{x} \cdot \ell\right)} + \frac{\ell}{x} \cdot \ell}} \]
      associate-+l+ [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}}} \]
      fma-udef [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \]
      distribute-rgt-out [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \]
      count-2 [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \]
      *-commutative [=>] 13.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{x}\right)}}} \]

   if 1.65e21 < t

   1. Initial program 43.0

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

   2. Simplified 43.0

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

      [Start] 43.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 43.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 43.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 43.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 40.3

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

   4. Simplified 40.3

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

      [Start] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 40.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around inf 4.9

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

   6. Simplified 4.9

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

      [Start] 4.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t} \cdot t \]
      *-commutative [<=] 4.9	\[ \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}} \cdot t \]
      distribute-lft-in [=>] 4.9	\[ \frac{\sqrt{2}}{t \cdot \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}} \cdot t \]
      metadata-eval [=>] 4.9	\[ \frac{\sqrt{2}}{t \cdot \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot t \]
      associate-*r/ [=>] 4.9	\[ \frac{\sqrt{2}}{t \cdot \sqrt{\left(2 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + 2 \cdot \frac{1}{x}}} \cdot t \]
      metadata-eval [=>] 4.9	\[ \frac{\sqrt{2}}{t \cdot \sqrt{\left(2 + \frac{\color{blue}{2}}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot t \]
      associate-*r/ [=>] 4.9	\[ \frac{\sqrt{2}}{t \cdot \sqrt{\left(2 + \frac{2}{x}\right) + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]
      metadata-eval [=>] 4.9	\[ \frac{\sqrt{2}}{t \cdot \sqrt{\left(2 + \frac{2}{x}\right) + \frac{\color{blue}{2}}{x}}} \cdot t \]

   7. Applied egg-rr 4.6

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\right)} - 1} \]

   8. Simplified 4.5

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}} \]
      Proof

      [Start] 4.6	\[ e^{\mathsf{log1p}\left(t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\right)} - 1 \]
      expm1-def [=>] 4.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\right)\right)} \]
      expm1-log1p [=>] 4.9	\[ \color{blue}{t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}} \]
      associate-*r/ [=>] 4.6	\[ \color{blue}{\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}} \]
      times-frac [=>] 4.5	\[ \color{blue}{\frac{t}{t} \cdot \frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}} \]
      *-inverses [=>] 4.5	\[ \color{blue}{1} \cdot \frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}} \]
      associate-*r/ [=>] 4.5	\[ \color{blue}{\frac{1 \cdot \sqrt{2}}{\sqrt{2 + \frac{4}{x}}}} \]
      *-lft-identity [=>] 4.5	\[ \frac{\color{blue}{\sqrt{2}}}{\sqrt{2 + \frac{4}{x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-201}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-285}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+21}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.0
Cost	20620

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{\sqrt{2} \cdot \sqrt{\frac{x + -1}{x + 1}}}{\frac{t}{\sqrt{0.5}}}\right)\\ t_2 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.42 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 2
Error	10.0
Cost	20620

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \frac{\sqrt{2}}{t_1 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ t_3 := \frac{t_1}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-201}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+21}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 3
Error	10.1
Cost	20364

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \left(-\sqrt{0.5}\right)\right)\\ t_2 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-202}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 4
Error	10.1
Cost	14672

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ t_3 := \frac{t_1}{\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)} \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{+93}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-274}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 5
Error	15.0
Cost	14220

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\left(-t\right) \cdot \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}}\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-202}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 6
Error	14.9
Cost	14220

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)} \cdot \left(-t\right)}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-202}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq -2.35 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 7
Error	24.0
Cost	13768

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{2}{t \cdot t}}{t_1}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{t_1}}\\ \end{array} \]

Alternative 8
Error	24.0
Cost	13768

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-153}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{2}{t \cdot t}}{t_1}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{t_1}}\\ \end{array} \]

Alternative 9
Error	24.0
Cost	13512

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{2}{t \cdot t}}{t_1}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-109}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{t_1}}\\ \end{array} \]

Alternative 10
Error	24.2
Cost	7364

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-154}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{2}{t \cdot t}}{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-109}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	35.9
Cost	7248

\[\begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{x} \cdot \left(-\frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-109}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	34.2
Cost	7244

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{if}\;t \leq -9.6 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-275}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	35.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{x} \cdot \left(-\frac{t}{\ell}\right)\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-295}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	35.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Error	35.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-304}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16
Error	35.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	35.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{-295}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	39.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022356 
(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)))))