Toniolo and Linder, Equation (7)

Average Error: 43.2 → 12.4

Time: 27.1s

Precision: binary64

Cost: 21844

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ \mathbf{if}\;t \leq -9 \cdot 10^{-63}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + t_2\right)}}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-261}:\\ \;\;\;\;-\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-303}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{\left(2 + \frac{4}{x \cdot x}\right) + \left(\frac{2}{x} + \frac{2}{x}\right)}}\\ \end{array} \]

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 (/ (fma 2.0 (* t t) (* l l)) x)))
   (if (<= t -9e-63)
     (/ t_1 (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (* (sqrt 2.0) (- t))))
     (if (<= t -8e-186)
       (* (sqrt 2.0) (/ t (sqrt (* 2.0 (+ (* t t) t_2)))))
       (if (<= t -2.1e-261)
         (- (sqrt (/ 2.0 (+ 2.0 (/ 4.0 x)))))
         (if (<= t -5.6e-303)
           (* (/ t l) (sqrt x))
           (if (<= t 3.3e-19)
             (/
              t_1
              (sqrt
               (+ (/ l (/ x l)) (+ t_2 (* 2.0 (+ (* t t) (/ t (/ x t))))))))
             (/
              t_1
              (*
               t
               (sqrt
                (+ (+ 2.0 (/ 4.0 (* x x))) (+ (/ 2.0 x) (/ 2.0 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 = fma(2.0, (t * t), (l * l)) / x;
	double tmp;
	if (t <= -9e-63) {
		tmp = t_1 / (sqrt(((x + 1.0) / (x + -1.0))) * (sqrt(2.0) * -t));
	} else if (t <= -8e-186) {
		tmp = sqrt(2.0) * (t / sqrt((2.0 * ((t * t) + t_2))));
	} else if (t <= -2.1e-261) {
		tmp = -sqrt((2.0 / (2.0 + (4.0 / x))));
	} else if (t <= -5.6e-303) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= 3.3e-19) {
		tmp = t_1 / sqrt(((l / (x / l)) + (t_2 + (2.0 * ((t * t) + (t / (x / t)))))));
	} else {
		tmp = t_1 / (t * sqrt(((2.0 + (4.0 / (x * x))) + ((2.0 / x) + (2.0 / 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(fma(2.0, Float64(t * t), Float64(l * l)) / x)
	tmp = 0.0
	if (t <= -9e-63)
		tmp = Float64(t_1 / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(sqrt(2.0) * Float64(-t))));
	elseif (t <= -8e-186)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(2.0 * Float64(Float64(t * t) + t_2)))));
	elseif (t <= -2.1e-261)
		tmp = Float64(-sqrt(Float64(2.0 / Float64(2.0 + Float64(4.0 / x)))));
	elseif (t <= -5.6e-303)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= 3.3e-19)
		tmp = Float64(t_1 / sqrt(Float64(Float64(l / Float64(x / l)) + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t))))))));
	else
		tmp = Float64(t_1 / Float64(t * sqrt(Float64(Float64(2.0 + Float64(4.0 / Float64(x * x))) + Float64(Float64(2.0 / x) + Float64(2.0 / x))))));
	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[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -9e-63], N[(t$95$1 / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -8e-186], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 * N[(N[(t * t), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.1e-261], (-N[Sqrt[N[(2.0 / N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -5.6e-303], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e-19], N[(t$95$1 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t * N[Sqrt[N[(N[(2.0 + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -8.9999999999999999e-63

   1. Initial program 39.2

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

   2. Taylor expanded in t around -inf 6.4

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

   3. Simplified 6.4

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

      [Start] 6.4	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 6.4	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [<=] 6.4	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      sub-neg [=>] 6.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 6.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 6.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}} \]
      +-commutative [=>] 6.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   if -8.9999999999999999e-63 < t < -7.9999999999999993e-186

   1. Initial program 40.9

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

   2. Simplified 55.1

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

      [Start] 40.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 41.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 40.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 55.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 55.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 55.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 55.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 55.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 55.6	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 55.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around -inf 15.3

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

   4. Simplified 15.3

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

      [Start] 15.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + 2 \cdot {t}^{2}}} \]
      distribute-lft-out [=>] 15.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x} + {t}^{2}\right)}}} \]
      +-commutative [=>] 15.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 15.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x} + {t}^{2}\right)}} \]
      fma-udef [<=] 15.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 15.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x} + {t}^{2}\right)}} \]
      unpow2 [=>] 15.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \color{blue}{t \cdot t}\right)}} \]

   if -7.9999999999999993e-186 < t < -2.09999999999999996e-261

   1. Initial program 63.5

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

   2. Taylor expanded in x around inf 34.6

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

   3. Simplified 34.6

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

      [Start] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 34.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around -inf 36.8

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

   5. Simplified 36.8

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

      [Start] 36.8	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \]
      associate-*r* [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(-1 \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}} \]
      *-commutative [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot \left(-1 \cdot t\right)}} \]
      distribute-lft-in [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}} \cdot \left(-1 \cdot t\right)} \]
      metadata-eval [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot \left(-1 \cdot t\right)} \]
      associate-*r/ [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + 2 \cdot \frac{1}{x}} \cdot \left(-1 \cdot t\right)} \]
      metadata-eval [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{\color{blue}{2}}{x}\right) + 2 \cdot \frac{1}{x}} \cdot \left(-1 \cdot t\right)} \]
      associate-*r/ [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{2}{x}\right) + \color{blue}{\frac{2 \cdot 1}{x}}} \cdot \left(-1 \cdot t\right)} \]
      metadata-eval [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{2}{x}\right) + \frac{\color{blue}{2}}{x}} \cdot \left(-1 \cdot t\right)} \]
      mul-1-neg [=>] 36.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(2 + \frac{2}{x}\right) + \frac{2}{x}} \cdot \color{blue}{\left(-t\right)}} \]

   6. Applied egg-rr 36.8

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

   7. Simplified 36.8

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

      [Start] 36.8	\[ -1 \cdot \left(\sqrt{\frac{2}{2 + \frac{4}{x}}} \cdot \frac{t}{t}\right) \]
      mul-1-neg [=>] 36.8	\[ \color{blue}{-\sqrt{\frac{2}{2 + \frac{4}{x}}} \cdot \frac{t}{t}} \]
      distribute-lft-neg-in [=>] 36.8	\[ \color{blue}{\left(-\sqrt{\frac{2}{2 + \frac{4}{x}}}\right) \cdot \frac{t}{t}} \]
      *-inverses [=>] 36.8	\[ \left(-\sqrt{\frac{2}{2 + \frac{4}{x}}}\right) \cdot \color{blue}{1} \]
      *-rgt-identity [=>] 36.8	\[ \color{blue}{-\sqrt{\frac{2}{2 + \frac{4}{x}}}} \]

   if -2.09999999999999996e-261 < t < -5.6e-303

   1. Initial program 62.3

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

   2. Simplified 60.3

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

      [Start] 62.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 62.3	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 59.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 61.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 61.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 61.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 61.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 61.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 61.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 60.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around 0 59.4

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

   4. Simplified 61.8

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

      [Start] 59.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\left(1 + x\right) \cdot {\ell}^{2}}{x - 1} - {\ell}^{2}}} \]
      associate-/l* [=>] 61.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{1 + x}{\frac{x - 1}{{\ell}^{2}}}} - {\ell}^{2}}} \]
      +-commutative [=>] 61.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{x + 1}}{\frac{x - 1}{{\ell}^{2}}} - {\ell}^{2}}} \]
      sub-neg [=>] 61.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{\ell}^{2}}} - {\ell}^{2}}} \]
      metadata-eval [=>] 61.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + \color{blue}{-1}}{{\ell}^{2}}} - {\ell}^{2}}} \]
      +-commutative [=>] 61.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{{\ell}^{2}}} - {\ell}^{2}}} \]
      unpow2 [=>] 61.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{-1 + x}{\color{blue}{\ell \cdot \ell}}} - {\ell}^{2}}} \]
      unpow2 [=>] 61.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{-1 + x}{\ell \cdot \ell}} - \color{blue}{\ell \cdot \ell}}} \]

   5. Taylor expanded in x around inf 29.5

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}}}} \]

   6. Simplified 29.5

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

      [Start] 29.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2} - -1 \cdot {\ell}^{2}}{x}}} \]
      cancel-sign-sub-inv [=>] 29.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{{\ell}^{2} + \left(--1\right) \cdot {\ell}^{2}}}{x}}} \]
      unpow2 [=>] 29.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell} + \left(--1\right) \cdot {\ell}^{2}}{x}}} \]
      metadata-eval [=>] 29.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell + \color{blue}{1} \cdot {\ell}^{2}}{x}}} \]
      unpow2 [=>] 29.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell + 1 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{x}}} \]
      distribute-rgt1-in [=>] 29.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{\left(1 + 1\right) \cdot \left(\ell \cdot \ell\right)}}{x}}} \]
      metadata-eval [=>] 29.5	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{2} \cdot \left(\ell \cdot \ell\right)}{x}}} \]

   7. Taylor expanded in t around 0 36.2

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

   if -5.6e-303 < t < 3.2999999999999998e-19

   1. Initial program 47.4

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

   2. Taylor expanded in x around inf 21.3

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

   3. Simplified 21.3

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

      [Start] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 21.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   if 3.2999999999999998e-19 < t

   1. Initial program 40.2

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

   2. Taylor expanded in x around -inf 37.3

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

   3. Simplified 37.3

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

      [Start] 37.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]

   4. Taylor expanded in t around inf 5.8

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

   5. Simplified 5.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\left(\frac{4}{x \cdot x} + 2\right) + \left(\frac{2}{x} + \frac{2}{x}\right)}}} \]
      Proof

      [Start] 5.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + \left(2 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}\right)} \cdot t} \]
      *-commutative [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + \left(2 \cdot \frac{1}{x} + 4 \cdot \frac{1}{{x}^{2}}\right)}}} \]
      associate-+r+ [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\color{blue}{\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) + 4 \cdot \frac{1}{{x}^{2}}}}} \]
      +-commutative [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\color{blue}{4 \cdot \frac{1}{{x}^{2}} + \left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)}}} \]
      distribute-lft-in [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}\right)}} \]
      metadata-eval [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{4 \cdot \frac{1}{{x}^{2}} + \left(\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)}} \]
      associate-+l+ [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{4 \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)\right)}}} \]
      associate-+r+ [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\color{blue}{\left(4 \cdot \frac{1}{{x}^{2}} + 2\right) + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}} \]
      associate-*r/ [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\left(\color{blue}{\frac{4 \cdot 1}{{x}^{2}}} + 2\right) + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}} \]
      metadata-eval [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\left(\frac{\color{blue}{4}}{{x}^{2}} + 2\right) + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}} \]
      unpow2 [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\left(\frac{4}{\color{blue}{x \cdot x}} + 2\right) + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}} \]
      associate-*r/ [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\left(\frac{4}{x \cdot x} + 2\right) + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}} \]
      metadata-eval [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\left(\frac{4}{x \cdot x} + 2\right) + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}} \]
      associate-*r/ [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\left(\frac{4}{x \cdot x} + 2\right) + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}} \]
      metadata-eval [=>] 5.8	\[ \frac{\sqrt{2} \cdot t}{t \cdot \sqrt{\left(\frac{4}{x \cdot x} + 2\right) + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}} \]

3. Recombined 6 regimes into one program.

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-63}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-261}:\\ \;\;\;\;-\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-303}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\left(2 + \frac{4}{x \cdot x}\right) + \left(\frac{2}{x} + \frac{2}{x}\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.2
Cost	21076

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{-62}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-259}:\\ \;\;\;\;-\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-307}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{\left(2 + \frac{4}{x \cdot x}\right) + \left(\frac{2}{x} + \frac{2}{x}\right)}}\\ \end{array} \]

Alternative 2
Error	15.1
Cost	20356

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(\sqrt{2} \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-264}:\\ \;\;\;\;-\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{\left(2 + \frac{4}{x \cdot x}\right) + \left(\frac{2}{x} + \frac{2}{x}\right)}}\\ \end{array} \]

Alternative 3
Error	15.3
Cost	14672

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ t_2 := -\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{-68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-263}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\left(2 + \frac{4}{x \cdot x}\right) + \left(\frac{2}{x} + \frac{2}{x}\right)}}\\ \end{array} \]

Alternative 4
Error	15.2
Cost	14672

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ t_3 := \sqrt{\left(2 + \frac{4}{x \cdot x}\right) + \left(\frac{2}{x} + \frac{2}{x}\right)}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(-t_3\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-259}:\\ \;\;\;\;-\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot t_3}\\ \end{array} \]

Alternative 5
Error	15.3
Cost	7376

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ t_2 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{-68}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-259}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	33.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+175}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 7
Error	34.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+37}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+175}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 8
Error	32.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-144}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 9
Error	34.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	38.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023073 
(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)))))