Toniolo and Linder, Equation (7)

Percentage Accurate: 33.3% → 87.1%

Time: 32.5s

Precision: binary64

Cost: 39752

• Unsound rule application detected in e-graph. Results may not be sound.

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\sqrt{x}}\\ t_2 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(t_1, t \cdot \sqrt{t_2}\right), t_1\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, t_2, 2 \cdot \frac{\ell}{\frac{x}{\ell}}\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ l (sqrt x))) (t_2 (+ 2.0 (/ 4.0 x))))
   (if (<= t -6.5e-107)
     (/ (sqrt 2.0) (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (- (sqrt 2.0))))
     (if (<= t 7e-161)
       (* t (/ (sqrt 2.0) (hypot (hypot t_1 (* t (sqrt t_2))) t_1)))
       (if (<= t 2.9e+55)
         (*
          t
          (/
           (sqrt 2.0)
           (pow (pow (fma (* t t) t_2 (* 2.0 (/ l (/ x l)))) 0.25) 2.0)))
         1.0)))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = l / sqrt(x);
	double t_2 = 2.0 + (4.0 / x);
	double tmp;
	if (t <= -6.5e-107) {
		tmp = sqrt(2.0) / (sqrt(((x + 1.0) / (x + -1.0))) * -sqrt(2.0));
	} else if (t <= 7e-161) {
		tmp = t * (sqrt(2.0) / hypot(hypot(t_1, (t * sqrt(t_2))), t_1));
	} else if (t <= 2.9e+55) {
		tmp = t * (sqrt(2.0) / pow(pow(fma((t * t), t_2, (2.0 * (l / (x / l)))), 0.25), 2.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(l / sqrt(x))
	t_2 = Float64(2.0 + Float64(4.0 / x))
	tmp = 0.0
	if (t <= -6.5e-107)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(-sqrt(2.0))));
	elseif (t <= 7e-161)
		tmp = Float64(t * Float64(sqrt(2.0) / hypot(hypot(t_1, Float64(t * sqrt(t_2))), t_1)));
	elseif (t <= 2.9e+55)
		tmp = Float64(t * Float64(sqrt(2.0) / ((fma(Float64(t * t), t_2, Float64(2.0 * Float64(l / Float64(x / l)))) ^ 0.25) ^ 2.0)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.5e-107], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7e-161], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[Sqrt[t$95$1 ^ 2 + N[(t * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e+55], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[Power[N[(N[(t * t), $MachinePrecision] * t$95$2 + N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -6.5000000000000002e-107

   1. Initial program 36.3%

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

   2. Simplified 36.3%

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

      [Start] 36.3	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-/l* [=>] 36.3	\[ \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]
      fma-neg [=>] 36.3	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]
      remove-double-neg [<=] 36.3	\[ \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]
      fma-neg [<=] 36.3	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]
      sub-neg [=>] 36.3	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      metadata-eval [=>] 36.3	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      remove-double-neg [=>] 36.3	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]
      fma-def [=>] 36.3	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Taylor expanded in t around -inf 91.7%

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

   4. Simplified 91.7%

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

      [Start] 91.7	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 91.7	\[ \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [=>] 91.7	\[ \frac{\sqrt{2}}{-\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \sqrt{2}}} \]
      distribute-rgt-neg-in [=>] 91.7	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{1 + x}{x - 1}} \cdot \left(-\sqrt{2}\right)}} \]
      +-commutative [=>] 91.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{x + 1}}{x - 1}} \cdot \left(-\sqrt{2}\right)} \]
      sub-neg [=>] 91.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}} \cdot \left(-\sqrt{2}\right)} \]
      metadata-eval [=>] 91.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}}} \cdot \left(-\sqrt{2}\right)} \]
      +-commutative [=>] 91.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-\sqrt{2}\right)} \]

   if -6.5000000000000002e-107 < t < 7.00000000000000039e-161

   1. Initial program 7.4%

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

   2. Simplified 7.4%

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

      [Start] 7.4	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 7.4	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 56.1%

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

   4. Simplified 56.1%

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

      [Start] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Applied egg-rr 56.0%

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

      [Start] 56.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]
      add-sqr-sqrt [=>] 55.9	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot \sqrt{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}}}} \cdot t \]
      pow2 [=>] 55.9	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\sqrt{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}}\right)}^{2}}} \cdot t \]

   6. Taylor expanded in t around 0 56.0%

      \[\leadsto \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\left(\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]

   7. Simplified 61.3%

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

      [Start] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \left(\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      *-commutative [<=] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \left(\color{blue}{{t}^{2} \cdot \left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)} + \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      fma-def [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\mathsf{fma}\left({t}^{2}, 2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]
      unpow2 [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(\color{blue}{t \cdot t}, 2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      distribute-lft-in [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, \color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      metadata-eval [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, \left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-+l+ [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, \color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-*r/ [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      metadata-eval [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-*r/ [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      metadata-eval [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      unpow2 [=>] 56.0	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-/l* [=>] 61.3	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]

   8. Applied egg-rr 86.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, t \cdot \sqrt{2 + \frac{4}{x}}\right), \frac{\ell}{\sqrt{x}}\right)}} \cdot t \]
      Step-by-step derivation

      [Start] 61.3	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      pow-pow [=>] 61.5	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)\right)}^{\left(0.25 \cdot 2\right)}}} \cdot t \]
      metadata-eval [=>] 61.5	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      pow1/2 [<=] 61.5	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)}}} \cdot t \]
      +-commutative [=>] 61.5	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right) + \frac{\ell}{x} \cdot \ell}}} \cdot t \]
      add-sqr-sqrt [=>] 61.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)} \cdot \sqrt{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)}} + \frac{\ell}{x} \cdot \ell}} \cdot t \]
      add-sqr-sqrt [=>] 61.4	\[ \frac{\sqrt{2}}{\sqrt{\sqrt{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)} \cdot \sqrt{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)} + \color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell} \cdot \sqrt{\frac{\ell}{x} \cdot \ell}}}} \cdot t \]
      hypot-def [=>] 61.3	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{hypot}\left(\sqrt{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)}, \sqrt{\frac{\ell}{x} \cdot \ell}\right)}} \cdot t \]

   if 7.00000000000000039e-161 < t < 2.8999999999999999e55

   1. Initial program 59.8%

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

   2. Simplified 60.1%

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

      [Start] 59.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 60.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 79.7%

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

   4. Simplified 79.7%

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

      [Start] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      unpow2 [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}}{x}\right)\right)}} \cdot t \]
      +-commutative [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + \ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]

   5. Applied egg-rr 79.2%

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

      [Start] 79.7	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]
      add-sqr-sqrt [=>] 79.2	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot \sqrt{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}}}} \cdot t \]
      pow2 [=>] 79.2	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\sqrt{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}}\right)}^{2}}} \cdot t \]

   6. Taylor expanded in t around 0 79.2%

      \[\leadsto \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\left(\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]

   7. Simplified 87.4%

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

      [Start] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \left(\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      *-commutative [<=] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \left(\color{blue}{{t}^{2} \cdot \left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)} + \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      fma-def [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\mathsf{fma}\left({t}^{2}, 2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]
      unpow2 [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(\color{blue}{t \cdot t}, 2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      distribute-lft-in [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, \color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      metadata-eval [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, \left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-+l+ [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, \color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}, \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-*r/ [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      metadata-eval [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-*r/ [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      metadata-eval [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right), \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      unpow2 [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-/l* [=>] 87.4	\[ \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]

   8. Taylor expanded in l around 0 79.2%

      \[\leadsto \frac{\sqrt{2}}{{\left({\color{blue}{\left({t}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right) + 2 \cdot \frac{{\ell}^{2}}{x}\right)}}^{0.25}\right)}^{2}} \cdot t \]

   9. Simplified 87.4%

      \[\leadsto \frac{\sqrt{2}}{{\left({\color{blue}{\left(\mathsf{fma}\left(t \cdot t, 2 + \frac{4}{x}, 2 \cdot \frac{\ell}{\frac{x}{\ell}}\right)\right)}}^{0.25}\right)}^{2}} \cdot t \]
      Step-by-step derivation

      [Start] 79.2	\[ \frac{\sqrt{2}}{{\left({\left({t}^{2} \cdot \left(2 + 4 \cdot \frac{1}{x}\right) + 2 \cdot \frac{{\ell}^{2}}{x}\right)}^{0.25}\right)}^{2}} \cdot t \]
      fma-def [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\color{blue}{\left(\mathsf{fma}\left({t}^{2}, 2 + 4 \cdot \frac{1}{x}, 2 \cdot \frac{{\ell}^{2}}{x}\right)\right)}}^{0.25}\right)}^{2}} \cdot t \]
      unpow2 [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(\color{blue}{t \cdot t}, 2 + 4 \cdot \frac{1}{x}, 2 \cdot \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-*r/ [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, 2 + \color{blue}{\frac{4 \cdot 1}{x}}, 2 \cdot \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      metadata-eval [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, 2 + \frac{\color{blue}{4}}{x}, 2 \cdot \frac{{\ell}^{2}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      unpow2 [=>] 79.2	\[ \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, 2 + \frac{4}{x}, 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]
      associate-/l* [=>] 87.4	\[ \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, 2 + \frac{4}{x}, 2 \cdot \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)\right)}^{0.25}\right)}^{2}} \cdot t \]

   if 2.8999999999999999e55 < t

   1. Initial program 35.6%

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

   2. Simplified 35.6%

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

      [Start] 35.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 35.6	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in x around inf 94.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   4. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{1} \]
      Step-by-step derivation

      [Start] 94.4	\[ \sqrt{2} \cdot \sqrt{0.5} \]
      sqrt-unprod [=>] 95.9	\[ \color{blue}{\sqrt{2 \cdot 0.5}} \]
      metadata-eval [=>] 95.9	\[ \sqrt{\color{blue}{1}} \]
      metadata-eval [=>] 95.9	\[ \color{blue}{1} \]

3. Recombined 4 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(\frac{\ell}{\sqrt{x}}, t \cdot \sqrt{2 + \frac{4}{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, 2 + \frac{4}{x}, 2 \cdot \frac{\ell}{\frac{x}{\ell}}\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.8%
Cost	27720

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+111}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-193}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\ell \cdot \frac{\ell}{x} + \mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), t_1\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-166}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+55}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, 2 + \frac{4}{x}, 2 \cdot t_1\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	85.8%
Cost	27473

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+112}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-193} \lor \neg \left(t \leq 4.4 \cdot 10^{-161}\right) \land t \leq 2.9 \cdot 10^{+55}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\mathsf{fma}\left(t \cdot t, 2 + \frac{4}{x}, 2 \cdot \frac{\ell}{\frac{x}{\ell}}\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	80.3%
Cost	21712

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-193}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+54}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	80.1%
Cost	20100

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -7 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.36 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Accuracy	80.0%
Cost	15056

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -7 \cdot 10^{-120}:\\ \;\;\;\;t \cdot \frac{-\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-193}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-160}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+47}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	77.7%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-120}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	78.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-120}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-193}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{\frac{x + -1}{x + 1}}}{t}\\ \end{array} \]

Alternative 8
Accuracy	78.6%
Cost	7368

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-119}:\\ \;\;\;\;t \cdot \frac{-t_1}{t}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-192}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \left(\ell \cdot \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{t_1}{t}\\ \end{array} \]

Alternative 9
Accuracy	76.6%
Cost	7180

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-176}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-286}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	76.2%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-176}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	77.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-175}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-193}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	75.9%
Cost	1348

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-301}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \left(\frac{\frac{0.5}{t}}{x \cdot x} + \frac{\frac{-1}{x}}{t}\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	75.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-301}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{1}{t} + \frac{\frac{-1}{x}}{t}\right)\\ \end{array} \]

Alternative 14
Accuracy	75.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-301}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	37.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023161 
(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)))))