Toniolo and Linder, Equation (7)

Average Error: 43.2 → 10.0

Time: 1.8min

Precision: binary64

Cost: 28628

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := \sqrt{t_2}\\ t_4 := \sqrt{\frac{x + 1}{x + -1}}\\ t_5 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ t_6 := \sqrt{\frac{1}{t_2}}\\ \mathbf{if}\;t \leq -2600000:\\ \;\;\;\;\frac{t_1}{\sqrt{2} \cdot \left(t_4 \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + t_5\right)}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, t_3, t_6 \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-285}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(t, t_3, 0.5 \cdot \left(t_6 \cdot \frac{2 \cdot \frac{\ell}{\frac{x}{\ell}}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1060000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(t_5 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_4}\\ \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x))))
        (t_3 (sqrt t_2))
        (t_4 (sqrt (/ (+ x 1.0) (+ x -1.0))))
        (t_5 (/ (fma 2.0 (* t t) (* l l)) x))
        (t_6 (sqrt (/ 1.0 t_2))))
   (if (<= t -2600000.0)
     (/ t_1 (* (sqrt 2.0) (* t_4 (- t))))
     (if (<= t -2e-160)
       (* (sqrt 2.0) (/ t (sqrt (* 2.0 (+ (* t t) t_5)))))
       (if (<= t -1.4e-257)
         (* t (/ (sqrt 2.0) (- (fma t t_3 (* t_6 (* (/ l x) (/ l t)))))))
         (if (<= t 1.16e-285)
           (* t (- (/ (sqrt x) l)))
           (if (<= t 6.1e-155)
             (/ t_1 (fma t t_3 (* 0.5 (* t_6 (/ (* 2.0 (/ l (/ x l))) t)))))
             (if (<= t 1060000.0)
               (*
                t
                (/
                 (sqrt 2.0)
                 (sqrt
                  (+
                   (/ (* l l) x)
                   (+ t_5 (* 2.0 (+ (* t t) (/ (* t t) x))))))))
               (/ t_1 (* t_1 t_4))))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_3 = sqrt(t_2);
	double t_4 = sqrt(((x + 1.0) / (x + -1.0)));
	double t_5 = fma(2.0, (t * t), (l * l)) / x;
	double t_6 = sqrt((1.0 / t_2));
	double tmp;
	if (t <= -2600000.0) {
		tmp = t_1 / (sqrt(2.0) * (t_4 * -t));
	} else if (t <= -2e-160) {
		tmp = sqrt(2.0) * (t / sqrt((2.0 * ((t * t) + t_5))));
	} else if (t <= -1.4e-257) {
		tmp = t * (sqrt(2.0) / -fma(t, t_3, (t_6 * ((l / x) * (l / t)))));
	} else if (t <= 1.16e-285) {
		tmp = t * -(sqrt(x) / l);
	} else if (t <= 6.1e-155) {
		tmp = t_1 / fma(t, t_3, (0.5 * (t_6 * ((2.0 * (l / (x / l))) / t))));
	} else if (t <= 1060000.0) {
		tmp = t * (sqrt(2.0) / sqrt((((l * l) / x) + (t_5 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else {
		tmp = t_1 / (t_1 * t_4);
	}
	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(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_3 = sqrt(t_2)
	t_4 = sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))
	t_5 = Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)
	t_6 = sqrt(Float64(1.0 / t_2))
	tmp = 0.0
	if (t <= -2600000.0)
		tmp = Float64(t_1 / Float64(sqrt(2.0) * Float64(t_4 * Float64(-t))));
	elseif (t <= -2e-160)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(2.0 * Float64(Float64(t * t) + t_5)))));
	elseif (t <= -1.4e-257)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, t_3, Float64(t_6 * Float64(Float64(l / x) * Float64(l / t)))))));
	elseif (t <= 1.16e-285)
		tmp = Float64(t * Float64(-Float64(sqrt(x) / l)));
	elseif (t <= 6.1e-155)
		tmp = Float64(t_1 / fma(t, t_3, Float64(0.5 * Float64(t_6 * Float64(Float64(2.0 * Float64(l / Float64(x / l))) / t)))));
	elseif (t <= 1060000.0)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(t_5 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))));
	else
		tmp = Float64(t_1 / Float64(t_1 * t_4));
	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[Sqrt[2.0], $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[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2600000.0], N[(t$95$1 / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$4 * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2e-160], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 * N[(N[(t * t), $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.4e-257], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * t$95$3 + N[(t$95$6 * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.16e-285], N[(t * (-N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 6.1e-155], N[(t$95$1 / N[(t * t$95$3 + N[(0.5 * N[(t$95$6 * N[(N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1060000.0], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(t$95$5 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -2.6e6

   1. Initial program 42.5

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

   2. Taylor expanded in t around -inf 5.0

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

   3. Simplified 5.0

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

      [Start] 5.0	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 5.0	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      associate-*l* [=>] 5.0	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
      sub-neg [=>] 5.0	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}\right)} \]
      metadata-eval [=>] 5.0	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}\right)} \]
      +-commutative [=>] 5.0	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}\right)} \]
      +-commutative [=>] 5.0	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   if -2.6e6 < t < -2e-160

   1. Initial program 30.9

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

   2. Simplified 44.0

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

      [Start] 30.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 31.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 30.7	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 44.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 44.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 44.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 44.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 44.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 44.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 44.0	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 9.8

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

   4. Simplified 9.8

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

      [Start] 9.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 9.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 9.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 9.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 9.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 9.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 9.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   if -2e-160 < t < -1.40000000000000001e-257

   1. Initial program 62.8

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

   2. Simplified 62.8

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

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

   3. Taylor expanded in x around inf 32.9

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

   4. Simplified 32.9

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

      [Start] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 32.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 25.4

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

   6. Simplified 25.4

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

      [Start] 25.4	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 25.4	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 25.4	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 25.4	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      +-commutative [<=] 25.4	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      mul-1-neg [=>] 25.4	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]

   if -1.40000000000000001e-257 < t < 1.1599999999999999e-285

   1. Initial program 63.2

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

   2. Simplified 63.2

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

      [Start] 63.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 63.2	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 63.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 63.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 30.8

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

   4. Simplified 30.8

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

      [Start] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 30.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in l around -inf 30.9

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

   6. Simplified 30.9

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

      [Start] 30.9	\[ \frac{\sqrt{2}}{-1 \cdot \left(\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}\right)} \cdot t \]
      mul-1-neg [=>] 30.9	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}} \cdot t \]
      associate-*l* [=>] 30.9	\[ \frac{\sqrt{2}}{-\color{blue}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}} \cdot t \]

   7. Taylor expanded in l around 0 30.9

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

   8. Simplified 30.9

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

      [Start] 30.9	\[ \left(-1 \cdot \left(\frac{1}{\ell} \cdot \sqrt{x}\right)\right) \cdot t \]
      mul-1-neg [=>] 30.9	\[ \color{blue}{\left(-\frac{1}{\ell} \cdot \sqrt{x}\right)} \cdot t \]
      associate-*l/ [=>] 30.9	\[ \left(-\color{blue}{\frac{1 \cdot \sqrt{x}}{\ell}}\right) \cdot t \]
      *-lft-identity [=>] 30.9	\[ \left(-\frac{\color{blue}{\sqrt{x}}}{\ell}\right) \cdot t \]

   if 1.1599999999999999e-285 < t < 6.10000000000000023e-155

   1. Initial program 61.8

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

   2. Taylor expanded in x around inf 33.5

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

   3. Simplified 33.5

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

      [Start] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 33.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around inf 24.1

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

   5. Simplified 24.1

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

      [Start] 24.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)} \]
      *-commutative [<=] 24.1	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)} \]
      fma-def [=>] 24.1	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)}} \]

   if 6.10000000000000023e-155 < t < 1.06e6

   1. Initial program 30.9

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

   2. Simplified 30.9

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

      [Start] 30.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 30.9	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 9.6

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

   4. Simplified 9.6

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

      [Start] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 9.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   if 1.06e6 < t

   1. Initial program 42.6

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

   2. Taylor expanded in l around 0 5.2

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2600000:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2} \cdot \left(\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-285}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, 0.5 \cdot \left(\sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \frac{2 \cdot \frac{\ell}{\frac{x}{\ell}}}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1060000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.7
Cost	28372

\[\begin{array}{l} t_1 := \sqrt{\frac{x + 1}{x + -1}}\\ t_2 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ t_3 := t \cdot \sqrt{2}\\ t_4 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -60000:\\ \;\;\;\;\frac{t_3}{\sqrt{2} \cdot \left(t_1 \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -2.65 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + t_2\right)}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-285}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{t_4}, \sqrt{\frac{1}{t_4}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 220000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t_3 \cdot t_1}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	28372

\[\begin{array}{l} t_1 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ t_3 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ t_4 := t \cdot \sqrt{2}\\ t_5 := \mathsf{fma}\left(t, \sqrt{t_1}, \sqrt{\frac{1}{t_1}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)\\ \mathbf{if}\;t \leq -124000:\\ \;\;\;\;\frac{t_4}{\sqrt{2} \cdot \left(t_2 \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + t_3\right)}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-256}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-t_5}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-285}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_5}\\ \mathbf{elif}\;t \leq 50000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(t_3 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4}{t_4 \cdot t_2}\\ \end{array} \]

Alternative 3
Error	11.6
Cost	21976

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ t_3 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ \mathbf{if}\;t \leq -54000:\\ \;\;\;\;\frac{t_1}{\sqrt{2} \cdot \left(t_2 \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + t_3\right)}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-255}:\\ \;\;\;\;t \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-267}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 220000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(t_3 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_2}\\ \end{array} \]

Alternative 4
Error	11.6
Cost	20952

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \frac{-1}{t}\right)\\ t_3 := \frac{\ell \cdot \ell}{x}\\ t_4 := t \cdot \frac{\sqrt{2}}{\sqrt{t_3 + \left(t_3 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.12 \cdot 10^{-159}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-257}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 980000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 5
Error	11.6
Cost	20952

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \sqrt{\frac{x + 1}{x + -1}}\\ t_3 := t \cdot \sqrt{2}\\ t_4 := t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.26 \cdot 10^{+23}:\\ \;\;\;\;\frac{t_3}{\sqrt{2} \cdot \left(t_2 \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-161}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-256}:\\ \;\;\;\;t \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-266}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t_3 \cdot t_2}\\ \end{array} \]

Alternative 6
Error	11.6
Cost	20952

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \sqrt{\frac{x + 1}{x + -1}}\\ \mathbf{if}\;t \leq -440000:\\ \;\;\;\;\frac{t_1}{\sqrt{2} \cdot \left(t_3 \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-257}:\\ \;\;\;\;t \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 410000:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot t_3}\\ \end{array} \]

Alternative 7
Error	11.6
Cost	15320

\[\begin{array}{l} t_1 := \frac{x + -1}{x + 1}\\ t_2 := t \cdot \left(\sqrt{t_1} \cdot \frac{-1}{t}\right)\\ t_3 := \frac{\ell \cdot \ell}{x}\\ t_4 := t \cdot \frac{\sqrt{2}}{\sqrt{t_3 + \left(t_3 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-160}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -5.7 \cdot 10^{-257}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 900000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_1}}{{0.5}^{-0.5}}\\ \end{array} \]

Alternative 8
Error	14.3
Cost	13832

\[\begin{array}{l} t_1 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-256}:\\ \;\;\;\;t \cdot \left(\sqrt{t_1} \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_1}}{{0.5}^{-0.5}}\\ \end{array} \]

Alternative 9
Error	24.2
Cost	7496

\[\begin{array}{l} t_1 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \sqrt{\frac{t_1}{t \cdot t}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{t_1} \cdot \frac{1}{t}\right)\\ \end{array} \]

Alternative 10
Error	14.4
Cost	7496

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-256}:\\ \;\;\;\;t \cdot \left(t_1 \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(t_1 \cdot \frac{1}{t}\right)\\ \end{array} \]

Alternative 11
Error	24.4
Cost	7364

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x + -1}{x + 1}}{t \cdot t}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-265}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{\frac{-1}{t}}{x}\right)\right)\\ \end{array} \]

Alternative 12
Error	24.5
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{t \cdot t}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \left(-\frac{\sqrt{x}}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{\frac{-1}{t}}{x}\right)\right)\\ \end{array} \]

Alternative 13
Error	34.4
Cost	6984

\[\begin{array}{l} t_1 := \frac{\sqrt{x}}{\ell}\\ \mathbf{if}\;\ell \leq -750000000:\\ \;\;\;\;t \cdot \left(-t_1\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+78}:\\ \;\;\;\;t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{\frac{-1}{t}}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot t_1\\ \end{array} \]

Alternative 14
Error	35.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-270}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{\frac{-1}{t}}{x}\right)\right)\\ \end{array} \]

Alternative 15
Error	34.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-269}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{\frac{-1}{t}}{x}\right)\right)\\ \end{array} \]

Alternative 16
Error	39.4
Cost	1216

\[t \cdot \left(\frac{0.5}{t \cdot \left(x \cdot x\right)} + \left(\frac{1}{t} + \frac{\frac{-1}{t}}{x}\right)\right) \]

Alternative 17
Error	39.5
Cost	704

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

Alternative 18
Error	39.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023237 
(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)))))