Toniolo and Linder, Equation (7)

Average Accuracy: 32.2% → 89.4%

Time: 39.1s

Precision: binary64

Cost: 52620

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), t_3\right), t_3\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ t (sqrt (fma t (+ t (/ t x)) (* l (/ l x))))))
        (t_2 (* t (sqrt 2.0)))
        (t_3 (/ l (sqrt x))))
   (if (<= t -2e+82)
     (/ t_2 (* t_2 (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t -2.8e-160)
       t_1
       (if (<= t 1.05e-179)
         (*
          t
          (/
           (sqrt 2.0)
           (hypot (hypot (* (sqrt 2.0) (hypot t (/ t (sqrt x)))) t_3) t_3)))
         (if (<= t 4.5e-8) t_1 (sqrt (/ 2.0 (+ 2.0 (/ 4.0 x))))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t / sqrt(fma(t, (t + (t / x)), (l * (l / x))));
	double t_2 = t * sqrt(2.0);
	double t_3 = l / sqrt(x);
	double tmp;
	if (t <= -2e+82) {
		tmp = t_2 / (t_2 * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= -2.8e-160) {
		tmp = t_1;
	} else if (t <= 1.05e-179) {
		tmp = t * (sqrt(2.0) / hypot(hypot((sqrt(2.0) * hypot(t, (t / sqrt(x)))), t_3), t_3));
	} else if (t <= 4.5e-8) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 / (2.0 + (4.0 / x))));
	}
	return tmp;
}

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

↓

function code(x, l, t)
	t_1 = Float64(t / sqrt(fma(t, Float64(t + Float64(t / x)), Float64(l * Float64(l / x)))))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(l / sqrt(x))
	tmp = 0.0
	if (t <= -2e+82)
		tmp = Float64(t_2 / Float64(t_2 * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= -2.8e-160)
		tmp = t_1;
	elseif (t <= 1.05e-179)
		tmp = Float64(t * Float64(sqrt(2.0) / hypot(hypot(Float64(sqrt(2.0) * hypot(t, Float64(t / sqrt(x)))), t_3), t_3)));
	elseif (t <= 4.5e-8)
		tmp = t_1;
	else
		tmp = sqrt(Float64(2.0 / Float64(2.0 + Float64(4.0 / 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[N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2e+82], N[(t$95$2 / N[(t$95$2 * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.8e-160], t$95$1, If[LessEqual[t, 1.05e-179], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t ^ 2 + N[(t / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e-8], t$95$1, N[Sqrt[N[(2.0 / N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.9999999999999999e82

   1. Initial program 24.3%

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

   2. Taylor expanded in t around -inf 94.8%

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

   3. Simplified 94.8%

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

      [Start] 94.8	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 94.8	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      distribute-rgt-neg-in [=>] 94.8	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]
      *-commutative [<=] 94.8	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)} \]
      +-commutative [<=] 94.8	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]
      sub-neg [=>] 94.8	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]
      metadata-eval [=>] 94.8	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]
      +-commutative [=>] 94.8	\[ \frac{\sqrt{2} \cdot t}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   if -1.9999999999999999e82 < t < -2.80000000000000016e-160 or 1.0499999999999999e-179 < t < 4.49999999999999993e-8

   1. Initial program 51.7%

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

   2. Simplified 51.8%

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

      [Start] 51.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 51.8	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 82.7%

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

   4. Simplified 82.7%

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

      [Start] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 82.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 82.2%

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

   6. Simplified 82.2%

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

      [Start] 82.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 82.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 89.8%

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

      [Start] 82.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      pow1/2 [=>] 82.2	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{0.5}}} \cdot t \]
      sqr-pow [=>] 82.0	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}} \cdot t \]

   8. Simplified 90.1%

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

      [Start] 89.8	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 90.1	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 90.1	\[ \frac{\sqrt{2}}{{\left(\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 90.1	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\mathsf{fma}\left(2, \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right), \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)\right)}}} \cdot t \]
      fma-udef [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right) + \mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, \frac{\ell}{x} \cdot \ell\right) + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}}} \cdot t \]
      fma-udef [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      count-2 [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)} + 2 \cdot \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}}} \cdot t \]
      *-commutative [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \mathsf{fma}\left(\frac{t}{x}, t, t \cdot t\right)\right)}} \cdot t \]
      fma-udef [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(\frac{t}{x} \cdot t + t \cdot t\right)}\right)}} \cdot t \]
      +-commutative [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)}\right)}} \cdot t \]
      distribute-rgt-out [=>] 90.1	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{t \cdot \left(t + \frac{t}{x}\right)}\right)}} \cdot t \]

   9. Applied egg-rr 90.3%

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

      [Start] 90.1	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}} \cdot t \]
      associate-*l/ [=>] 90.0	\[ \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}} \]
      sqrt-prod [=>] 90.4	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{2} \cdot \sqrt{\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)}}} \]
      associate-/r* [=>] 90.3	\[ \color{blue}{\frac{\frac{\sqrt{2} \cdot t}{\sqrt{2}}}{\sqrt{\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)}}} \]
      *-commutative [=>] 90.3	\[ \frac{\frac{\color{blue}{t \cdot \sqrt{2}}}{\sqrt{2}}}{\sqrt{\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)}} \]
      +-commutative [=>] 90.3	\[ \frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\sqrt{\color{blue}{t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}}}} \]
      fma-def [=>] 90.3	\[ \frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\sqrt{\color{blue}{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}} \]

   10. Simplified 90.5%

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

       [Start] 90.3	\[ \frac{\frac{t \cdot \sqrt{2}}{\sqrt{2}}}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}} \]
       associate-/l* [=>] 90.5	\[ \frac{\color{blue}{\frac{t}{\frac{\sqrt{2}}{\sqrt{2}}}}}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}} \]
       *-inverses [=>] 90.5	\[ \frac{\frac{t}{\color{blue}{1}}}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}} \]
       /-rgt-identity [=>] 90.5	\[ \frac{\color{blue}{t}}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}} \]

   if -2.80000000000000016e-160 < t < 1.0499999999999999e-179

   1. Initial program 2.0%

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

   2. Simplified 2.0%

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

      [Start] 2.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 2.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 47.1%

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

   4. Simplified 47.1%

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

      [Start] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 47.1%

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

   6. Simplified 47.1%

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

      [Start] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 78.3%

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

      [Start] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)\right) + \frac{\ell \cdot \ell}{x}}}} \cdot t \]
      add-sqr-sqrt [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}} + \frac{\ell \cdot \ell}{x}}} \cdot t \]
      add-sqr-sqrt [=>] 47.1	\[ \frac{\sqrt{2}}{\sqrt{\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} \cdot \sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)} + \color{blue}{\sqrt{\frac{\ell \cdot \ell}{x}} \cdot \sqrt{\frac{\ell \cdot \ell}{x}}}}} \cdot t \]
      hypot-def [=>] 47.1	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{hypot}\left(\sqrt{2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\ell \cdot \ell}{x}\right)}, \sqrt{\frac{\ell \cdot \ell}{x}}\right)}} \cdot t \]

   if 4.49999999999999993e-8 < t

   1. Initial program 35.8%

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

   2. Simplified 35.8%

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

      [Start] 35.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 35.8	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 40.2%

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

   4. Simplified 40.2%

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

      [Start] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 40.2	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 1.6%

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

   6. Simplified 1.6%

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

      [Start] 1.6	\[ \frac{\sqrt{2}}{-1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      mul-1-neg [=>] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{-t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}} \cdot t \]
      distribute-rgt-neg-in [=>] 1.6	\[ \frac{\sqrt{2}}{\color{blue}{t \cdot \left(-\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)}} \cdot t \]
      distribute-lft-in [=>] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      metadata-eval [=>] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      associate-+l+ [=>] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}\right)} \cdot t \]
      associate-*r/ [=>] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}\right)} \cdot t \]
      metadata-eval [=>] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}\right)} \cdot t \]
      associate-*r/ [=>] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}\right)} \cdot t \]
      metadata-eval [=>] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}\right)} \cdot t \]

   7. Applied egg-rr 40.3%

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

      [Start] 1.6	\[ \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)} \cdot t \]
      add-sqr-sqrt [=>] 0.0	\[ \color{blue}{\sqrt{\frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)} \cdot t} \cdot \sqrt{\frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)} \cdot t}} \]
      sqrt-unprod [=>] 90.4	\[ \color{blue}{\sqrt{\left(\frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)} \cdot t\right) \cdot \left(\frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)} \cdot t\right)}} \]
      *-commutative [=>] 90.4	\[ \sqrt{\color{blue}{\left(t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)}\right)} \cdot \left(\frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)} \cdot t\right)} \]
      *-commutative [=>] 90.4	\[ \sqrt{\left(t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)}\right) \cdot \color{blue}{\left(t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)}\right)}} \]
      swap-sqr [=>] 40.3	\[ \sqrt{\color{blue}{\left(t \cdot t\right) \cdot \left(\frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)} \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)}\right)}} \]
      frac-times [=>] 40.3	\[ \sqrt{\left(t \cdot t\right) \cdot \color{blue}{\frac{\sqrt{2} \cdot \sqrt{2}}{\left(t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)\right) \cdot \left(t \cdot \left(-\sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}\right)\right)}}} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{\sqrt{1 \cdot \frac{2}{2 + \frac{4}{x}}}} \]
      Proof

      [Start] 40.3	\[ \sqrt{\left(t \cdot t\right) \cdot \frac{2}{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}} \]
      associate-*r/ [=>] 40.4	\[ \sqrt{\color{blue}{\frac{\left(t \cdot t\right) \cdot 2}{\left(2 + \frac{4}{x}\right) \cdot \left(t \cdot t\right)}}} \]
      *-commutative [=>] 40.4	\[ \sqrt{\frac{\left(t \cdot t\right) \cdot 2}{\color{blue}{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}} \]
      times-frac [=>] 40.6	\[ \sqrt{\color{blue}{\frac{t \cdot t}{t \cdot t} \cdot \frac{2}{2 + \frac{4}{x}}}} \]
      *-inverses [=>] 90.9	\[ \sqrt{\color{blue}{1} \cdot \frac{2}{2 + \frac{4}{x}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\sqrt{2} \cdot \mathsf{hypot}\left(t, \frac{t}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.7%
Cost	28244

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-241}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-154}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t + t \cdot t\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_2\right)}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 2
Accuracy	85.2%
Cost	28172

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ t_3 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+78}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-292}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(\sqrt{t_3}, t, \sqrt{\frac{1}{t_3}} \cdot \frac{\ell \cdot \ell}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 3
Accuracy	84.0%
Cost	20356

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -1.42 \cdot 10^{+79}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 4
Accuracy	84.0%
Cost	20356

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 5
Accuracy	83.8%
Cost	20100

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 6
Accuracy	83.9%
Cost	14160

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{t_2}}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{t_2}}\\ \end{array} \]

Alternative 7
Accuracy	77.9%
Cost	14033

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-267}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{t_1}}\\ \mathbf{elif}\;t \leq 3.05 \cdot 10^{-238}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-112} \lor \neg \left(t \leq 9 \cdot 10^{-104}\right):\\ \;\;\;\;\sqrt{\frac{2}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \end{array} \]

Alternative 8
Accuracy	78.0%
Cost	13444

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-267}:\\ \;\;\;\;\frac{-\sqrt{2}}{\sqrt{t_1}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-241}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-114} \lor \neg \left(t \leq 9 \cdot 10^{-104}\right):\\ \;\;\;\;\sqrt{\frac{2}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot {\left(\frac{\ell}{\frac{1}{\frac{\ell}{x}}}\right)}^{-0.5}\\ \end{array} \]

Alternative 9
Accuracy	77.7%
Cost	7569

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-112} \lor \neg \left(t \leq 9 \cdot 10^{-104}\right):\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot {\left(\frac{\ell}{\frac{1}{\frac{\ell}{x}}}\right)}^{-0.5}\\ \end{array} \]

Alternative 10
Accuracy	45.7%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+24}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{\frac{t}{t + \frac{t}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \end{array} \]

Alternative 11
Accuracy	60.6%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-90}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{t \cdot t}}\\ \mathbf{elif}\;t \leq 10^{-237}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t + \frac{t}{x}}}\\ \end{array} \]

Alternative 12
Accuracy	77.5%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-239}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t + \frac{t}{x}}}\\ \end{array} \]

Alternative 13
Accuracy	77.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -3.75 \cdot 10^{-267}:\\ \;\;\;\;-\sqrt{\frac{1}{1 + \frac{1}{x}}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-241}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 14
Accuracy	45.7%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;\ell \leq -9.8 \cdot 10^{+23}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+207}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\ell \cdot {x}^{-0.5}}\\ \end{array} \]

Alternative 15
Accuracy	44.4%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.8 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+207}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 16
Accuracy	45.5%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+19}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+209}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 17
Accuracy	45.3%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Accuracy	46.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{-271}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Accuracy	39.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023136 
(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)))))