Toniolo and Linder, Equation (7)

Percentage Accurate: 34.2% → 89.3%

Time: 35.8s

Precision: binary64

Cost: 39628

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{+153}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(t \cdot \sqrt{2}, t_3\right), t_3\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

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
         (*
          t
          (/ (sqrt 2.0) (sqrt (* 2.0 (+ (* l (/ l x)) (* t (+ t (/ t x)))))))))
        (t_2 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x))))
        (t_3 (/ l (sqrt x))))
   (if (<= t -2.45e+153)
     (*
      t
      (/
       (sqrt 2.0)
       (- (fma t (sqrt t_2) (* (sqrt (/ 1.0 t_2)) (* (/ l x) (/ l t)))))))
     (if (<= t -5.9e-145)
       t_1
       (if (<= t 8e-192)
         (* t (/ (sqrt 2.0) (hypot (hypot (* t (sqrt 2.0)) t_3) t_3)))
         (if (<= t 1.7e-12)
           t_1
           (/ (sqrt 2.0) (* (sqrt 2.0) (sqrt (/ (+ x 1.0) (+ 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 = t * (sqrt(2.0) / sqrt((2.0 * ((l * (l / x)) + (t * (t + (t / x)))))));
	double t_2 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_3 = l / sqrt(x);
	double tmp;
	if (t <= -2.45e+153) {
		tmp = t * (sqrt(2.0) / -fma(t, sqrt(t_2), (sqrt((1.0 / t_2)) * ((l / x) * (l / t)))));
	} else if (t <= -5.9e-145) {
		tmp = t_1;
	} else if (t <= 8e-192) {
		tmp = t * (sqrt(2.0) / hypot(hypot((t * sqrt(2.0)), t_3), t_3));
	} else if (t <= 1.7e-12) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) / (sqrt(2.0) * sqrt(((x + 1.0) / (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(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(l * Float64(l / x)) + Float64(t * Float64(t + Float64(t / x))))))))
	t_2 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_3 = Float64(l / sqrt(x))
	tmp = 0.0
	if (t <= -2.45e+153)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, sqrt(t_2), Float64(sqrt(Float64(1.0 / t_2)) * Float64(Float64(l / x) * Float64(l / t)))))));
	elseif (t <= -5.9e-145)
		tmp = t_1;
	elseif (t <= 8e-192)
		tmp = Float64(t * Float64(sqrt(2.0) / hypot(hypot(Float64(t * sqrt(2.0)), t_3), t_3)));
	elseif (t <= 1.7e-12)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) / Float64(sqrt(2.0) * sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	end
	return 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[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.45e+153], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * N[Sqrt[t$95$2], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.9e-145], t$95$1, If[LessEqual[t, 8e-192], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[Sqrt[N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e-12], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.45000000000000001e153

   1. Initial program 1.9%

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

   2. Simplified 1.9%

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

      [Start] 1.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 1.9	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 1.4%

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

   4. Simplified 1.4%

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

      [Start] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 1.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 60.5%

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

   6. Simplified 98.7%

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

      [Start] 60.5	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      distribute-lft-out [=>] 60.5	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \cdot t \]
      mul-1-neg [=>] 60.5	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \cdot t \]
      *-commutative [=>] 60.5	\[ \frac{\sqrt{2}}{-\left(\color{blue}{\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} + t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 60.5	\[ \frac{\sqrt{2}}{-\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \color{blue}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t}\right)} \cdot t \]
      +-commutative [<=] 60.5	\[ \frac{\sqrt{2}}{-\color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      *-commutative [<=] 60.5	\[ \frac{\sqrt{2}}{-\left(\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]

   if -2.45000000000000001e153 < t < -5.8999999999999998e-145 or 8.0000000000000008e-192 < t < 1.7e-12

   1. Initial program 43.1%

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

   2. Simplified 43.1%

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

      [Start] 43.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 43.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 78.0%

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

   4. Simplified 78.0%

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

      [Start] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 78.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 77.4%

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

   6. Simplified 77.4%

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

      [Start] 77.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 77.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 88.0%

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

      [Start] 77.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      pow1/2 [=>] 77.5	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{0.5}}} \cdot t \]
      sqr-pow [=>] 77.3	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot t \]

   8. Simplified 88.2%

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

      [Start] 88.0	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 88.2	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 88.2	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 88.2	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)}}} \cdot t \]
      fma-udef [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right) + \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right) + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}}} \cdot t \]
      fma-udef [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      count-2 [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}}} \cdot t \]
      *-commutative [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}} \cdot t \]
      fma-udef [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(\frac{t}{x} \cdot t + t \cdot t\right)}\right)}} \cdot t \]
      +-commutative [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right)}} \cdot t \]
      distribute-rgt-out [=>] 88.2	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{t \cdot \left(t + \frac{t}{x}\right)}\right)}} \cdot t \]

   if -5.8999999999999998e-145 < t < 8.0000000000000008e-192

   1. Initial program 5.1%

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

   2. Simplified 5.1%

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

      [Start] 5.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 5.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 49.6%

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

   4. Simplified 49.6%

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

      [Start] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 49.6%

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

   6. Simplified 49.6%

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

      [Start] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 77.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\mathsf{hypot}\left(\mathsf{hypot}\left(\mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right) \cdot \sqrt{2}, \frac{\ell}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}} \cdot t \]
      Step-by-step derivation

      [Start] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right) + \frac{\ell \cdot \ell}{x}}}} \cdot t \]
      add-sqr-sqrt [=>] 49.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}} + \frac{\ell \cdot \ell}{x}}} \cdot t \]
      add-sqr-sqrt [=>] 49.7	\[ \frac{\sqrt{2}}{\sqrt{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} + \color{blue}{\sqrt{\frac{\ell \cdot \ell}{x}} \cdot \sqrt{\frac{\ell \cdot \ell}{x}}}}} \cdot t \]
      hypot-def [=>] 49.6	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{hypot}\left(\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}, \sqrt{\frac{\ell \cdot \ell}{x}}\right)}} \cdot t \]

   8. Taylor expanded in x around inf 77.8%

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

   if 1.7e-12 < t

   1. Initial program 28.2%

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

   2. Simplified 28.2%

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

      [Start] 28.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-/l* [=>] 28.2	\[ \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]
      fma-neg [=>] 28.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]
      remove-double-neg [<=] 28.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]
      fma-neg [<=] 28.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]
      sub-neg [=>] 28.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      metadata-eval [=>] 28.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      remove-double-neg [=>] 28.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]
      fma-def [=>] 28.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Taylor expanded in l around 0 97.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   4. Simplified 97.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
      Step-by-step derivation

      [Start] 97.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      +-commutative [=>] 97.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]
      sub-neg [=>] 97.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 97.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 97.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{+153}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{-145}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(t \cdot \sqrt{2}, \frac{\ell}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.3%
Cost	28172

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+149}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-291}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 2
Accuracy	85.0%
Cost	20432

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+149}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-266}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 3
Accuracy	84.9%
Cost	14544

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+153}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-265}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{-1}{x}\right) - \frac{0.5}{{x}^{3}}\right) - \frac{\frac{-0.5}{x}}{x}\\ \end{array} \]

Alternative 4
Accuracy	77.6%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-273}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{\frac{0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{-1}{x}\right) - \frac{0.5}{{x}^{3}}\right) - \frac{\frac{-0.5}{x}}{x}\\ \end{array} \]

Alternative 5
Accuracy	77.1%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-270}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{\frac{0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x} - \frac{\frac{-0.5}{x}}{x}\right) - -1\\ \end{array} \]

Alternative 6
Accuracy	77.5%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-272}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{\frac{0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-186}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x} - \frac{\frac{-0.5}{x}}{x}\right) - -1\\ \end{array} \]

Alternative 7
Accuracy	77.5%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-276}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{\frac{0.5}{x}}{x}\right)\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x} - \frac{\frac{-0.5}{x}}{x}\right) - -1\\ \end{array} \]

Alternative 8
Accuracy	76.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-301}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x} - \frac{\frac{-0.5}{x}}{x}\right) - -1\\ \end{array} \]

Alternative 9
Accuracy	76.3%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-301}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{\frac{0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-1}{x} - \frac{\frac{-0.5}{x}}{x}\right) - -1\\ \end{array} \]

Alternative 10
Accuracy	75.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-301}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11
Accuracy	76.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-301}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Accuracy	75.3%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-301}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	38.6%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023160 
(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)))))