Toniolo and Linder, Equation (7)

Average Accuracy: 32.7% → 81.5%

Time: 29.0s

Precision: binary64

Cost: 33868

Math

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

↓

\[\begin{array}{l} t_1 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_2 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_1}, \sqrt{\frac{1}{t_1}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{t_2}{\sqrt{\ell \cdot \left(\ell \cdot \left(\frac{2}{x} + \frac{2}{x \cdot x}\right)\right)}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \left(t \cdot t\right) \cdot 4\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_2\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_2}{\sqrt{\mathsf{fma}\left(-1, \frac{-2}{\frac{x \cdot x}{\ell \cdot \ell}}, 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right) + \left(\frac{\ell}{\frac{x}{\ell}} + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x)))) (t_2 (* t (sqrt 2.0))))
   (if (<= t -5.8e-218)
     (*
      t
      (/
       (sqrt 2.0)
       (- (fma t (sqrt t_1) (* (sqrt (/ 1.0 t_1)) (* (/ l x) (/ l t)))))))
     (if (<= t 8.5e-233)
       (/ t_2 (sqrt (* l (* l (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
       (if (<= t 1.2e-186)
         (*
          (sqrt 2.0)
          (/
           t
           (fma
            0.5
            (/ (fma 2.0 (* l l) (* (* t t) 4.0)) (* (sqrt 2.0) (* t x)))
            t_2)))
         (if (<= t 1.6e-42)
           (/
            t_2
            (sqrt
             (+
              (fma
               -1.0
               (/ -2.0 (/ (* x x) (* l l)))
               (* 2.0 (+ (* t t) (/ t (/ x t)))))
              (+ (/ l (/ x l)) (/ (fma 2.0 (* t t) (* l l)) x)))))
           (sqrt (/ t (* t (/ (+ x 1.0) (+ x -1.0)))))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_2 = t * sqrt(2.0);
	double tmp;
	if (t <= -5.8e-218) {
		tmp = t * (sqrt(2.0) / -fma(t, sqrt(t_1), (sqrt((1.0 / t_1)) * ((l / x) * (l / t)))));
	} else if (t <= 8.5e-233) {
		tmp = t_2 / sqrt((l * (l * ((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 1.2e-186) {
		tmp = sqrt(2.0) * (t / fma(0.5, (fma(2.0, (l * l), ((t * t) * 4.0)) / (sqrt(2.0) * (t * x))), t_2));
	} else if (t <= 1.6e-42) {
		tmp = t_2 / sqrt((fma(-1.0, (-2.0 / ((x * x) / (l * l))), (2.0 * ((t * t) + (t / (x / t))))) + ((l / (x / l)) + (fma(2.0, (t * t), (l * l)) / x))));
	} else {
		tmp = sqrt((t / (t * ((x + 1.0) / (x + -1.0)))));
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_2 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (t <= -5.8e-218)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, sqrt(t_1), Float64(sqrt(Float64(1.0 / t_1)) * Float64(Float64(l / x) * Float64(l / t)))))));
	elseif (t <= 8.5e-233)
		tmp = Float64(t_2 / sqrt(Float64(l * Float64(l * Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 1.2e-186)
		tmp = Float64(sqrt(2.0) * Float64(t / fma(0.5, Float64(fma(2.0, Float64(l * l), Float64(Float64(t * t) * 4.0)) / Float64(sqrt(2.0) * Float64(t * x))), t_2)));
	elseif (t <= 1.6e-42)
		tmp = Float64(t_2 / sqrt(Float64(fma(-1.0, Float64(-2.0 / Float64(Float64(x * x) / Float64(l * l))), Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t))))) + Float64(Float64(l / Float64(x / l)) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)))));
	else
		tmp = sqrt(Float64(t / Float64(t * Float64(Float64(x + 1.0) / Float64(x + -1.0)))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.8e-218], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * N[Sqrt[t$95$1], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-233], N[(t$95$2 / N[Sqrt[N[(l * N[(l * N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e-186], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(0.5 * N[(N[(2.0 * N[(l * l), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e-42], N[(t$95$2 / N[Sqrt[N[(N[(-1.0 * N[(-2.0 / N[(N[(x * x), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(t / N[(t * N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.8000000000000004e-218

   1. Initial program 36.0%

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

   2. Simplified 36.0%

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

      [Start] 36.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 36.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 36.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 36.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 51.4%

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

   4. Simplified 51.4%

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

      [Start] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 51.4	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 74.2%

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

   6. Simplified 83.5%

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

      [Start] 74.2	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 74.2	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 74.2	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 74.2	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      +-commutative [<=] 74.2	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      mul-1-neg [=>] 74.2	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]

   if -5.8000000000000004e-218 < t < 8.5000000000000005e-233

   1. Initial program 2.7%

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

   2. Taylor expanded in x around -inf 44.2%

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

   3. Simplified 44.2%

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

      [Start] 44.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      +-commutative [=>] 44.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 44.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]

   4. Taylor expanded in l around inf 53.7%

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

   5. Simplified 54.6%

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

      [Start] 53.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}\right) \cdot {\ell}^{2}}} \]
      *-commutative [=>] 53.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{{\ell}^{2} \cdot \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}\right)}}} \]
      unpow2 [=>] 53.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}\right)}} \]
      associate-*l* [=>] 54.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\ell \cdot \left(\ell \cdot \left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{x}\right)\right)}}} \]
      +-commutative [=>] 54.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\ell \cdot \color{blue}{\left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}\right)}\right)}} \]
      associate-*r/ [=>] 54.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\ell \cdot \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]
      metadata-eval [=>] 54.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\ell \cdot \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{{x}^{2}}\right)\right)}} \]
      associate-*r/ [=>] 54.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\ell \cdot \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{{x}^{2}}}\right)\right)}} \]
      metadata-eval [=>] 54.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\ell \cdot \left(\frac{2}{x} + \frac{\color{blue}{2}}{{x}^{2}}\right)\right)}} \]
      unpow2 [=>] 54.6	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\ell \cdot \left(\ell \cdot \left(\frac{2}{x} + \frac{2}{\color{blue}{x \cdot x}}\right)\right)}} \]

   if 8.5000000000000005e-233 < t < 1.20000000000000002e-186

   1. Initial program 0.0%

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

   2. Simplified 1.4%

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

      [Start] 0.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 0.0	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 0.8	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 0.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 0.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 0.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 0.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 0.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 0.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 1.4	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in x around inf 57.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{0.5 \cdot \frac{2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} + \sqrt{2} \cdot t}} \]

   4. Simplified 57.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \left(t \cdot t\right) \cdot 4\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}} \]
      Proof

      [Start] 57.2	\[ \sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)} + \sqrt{2} \cdot t} \]
      fma-def [=>] 57.2	\[ \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{2 \cdot {\ell}^{2} + 4 \cdot {t}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)}} \]
      unpow2 [=>] 57.2	\[ \sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)} + 4 \cdot {t}^{2}}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      fma-def [=>] 57.2	\[ \sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{\color{blue}{\mathsf{fma}\left(2, \ell \cdot \ell, 4 \cdot {t}^{2}\right)}}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      *-commutative [=>] 57.2	\[ \sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \color{blue}{{t}^{2} \cdot 4}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      unpow2 [=>] 57.2	\[ \sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \color{blue}{\left(t \cdot t\right)} \cdot 4\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      *-commutative [=>] 57.2	\[ \sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \left(t \cdot t\right) \cdot 4\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \color{blue}{t \cdot \sqrt{2}}\right)} \]

   if 1.20000000000000002e-186 < t < 1.60000000000000012e-42

   1. Initial program 39.6%

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

   2. Taylor expanded in x around -inf 76.9%

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

   3. Simplified 76.8%

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

      [Start] 76.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      +-commutative [=>] 76.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(\left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 76.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(-1 \cdot \frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right) - \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]

   4. Taylor expanded in t around 0 76.8%

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

   5. Simplified 76.8%

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

      [Start] 76.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, -2 \cdot \frac{{\ell}^{2}}{{x}^{2}}, 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right) + \left(\frac{\ell}{\frac{x}{\ell}} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \]
      associate-*r/ [=>] 76.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{-2 \cdot {\ell}^{2}}{{x}^{2}}}, 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right) + \left(\frac{\ell}{\frac{x}{\ell}} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \]
      associate-/l* [=>] 76.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \color{blue}{\frac{-2}{\frac{{x}^{2}}{{\ell}^{2}}}}, 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right) + \left(\frac{\ell}{\frac{x}{\ell}} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \]
      unpow2 [=>] 76.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \frac{-2}{\frac{\color{blue}{x \cdot x}}{{\ell}^{2}}}, 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right) + \left(\frac{\ell}{\frac{x}{\ell}} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \]
      unpow2 [=>] 76.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(-1, \frac{-2}{\frac{x \cdot x}{\color{blue}{\ell \cdot \ell}}}, 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right) + \left(\frac{\ell}{\frac{x}{\ell}} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \]

   if 1.60000000000000012e-42 < t

   1. Initial program 37.7%

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

   2. Taylor expanded in t around -inf 1.6%

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

   3. Simplified 1.6%

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

      [Start] 1.6	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      associate-*l* [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
      sub-neg [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}\right)} \]
      metadata-eval [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}\right)} \]
      +-commutative [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}\right)} \]
      +-commutative [=>] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   4. Applied egg-rr 44.0%

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

      [Start] 1.6	\[ \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)} \]
      add-sqr-sqrt [=>] 0.0	\[ \color{blue}{\sqrt{\frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}} \cdot \sqrt{\frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}}} \]
      sqrt-unprod [=>] 90.3	\[ \color{blue}{\sqrt{\frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)} \cdot \frac{\sqrt{2} \cdot t}{-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}}} \]
      frac-times [=>] 44.0	\[ \sqrt{\color{blue}{\frac{\left(\sqrt{2} \cdot t\right) \cdot \left(\sqrt{2} \cdot t\right)}{\left(-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right) \cdot \left(-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right)}}} \]
      swap-sqr [=>] 43.9	\[ \sqrt{\frac{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(t \cdot t\right)}}{\left(-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right) \cdot \left(-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right)}} \]
      add-sqr-sqrt [<=] 43.5	\[ \sqrt{\frac{\color{blue}{2} \cdot \left(t \cdot t\right)}{\left(-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right) \cdot \left(-\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right)}} \]
      sqr-neg [=>] 43.5	\[ \sqrt{\frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\left(\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right)}}} \]
      swap-sqr [=>] 43.3	\[ \sqrt{\frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(\left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right) \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right)}}} \]
      add-sqr-sqrt [<=] 44.0	\[ \sqrt{\frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{2} \cdot \left(\left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right) \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)\right)}} \]

   5. Simplified 90.3%

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

      [Start] 44.0	\[ \sqrt{\frac{2 \cdot \left(t \cdot t\right)}{2 \cdot \left(\frac{x + 1}{x + -1} \cdot \left(t \cdot t\right)\right)}} \]
      times-frac [=>] 44.0	\[ \sqrt{\color{blue}{\frac{2}{2} \cdot \frac{t \cdot t}{\frac{x + 1}{x + -1} \cdot \left(t \cdot t\right)}}} \]
      metadata-eval [=>] 44.0	\[ \sqrt{\color{blue}{1} \cdot \frac{t \cdot t}{\frac{x + 1}{x + -1} \cdot \left(t \cdot t\right)}} \]
      *-lft-identity [=>] 44.0	\[ \sqrt{\color{blue}{\frac{t \cdot t}{\frac{x + 1}{x + -1} \cdot \left(t \cdot t\right)}}} \]
      associate-*r* [=>] 44.0	\[ \sqrt{\frac{t \cdot t}{\color{blue}{\left(\frac{x + 1}{x + -1} \cdot t\right) \cdot t}}} \]
      times-frac [=>] 90.3	\[ \sqrt{\color{blue}{\frac{t}{\frac{x + 1}{x + -1} \cdot t} \cdot \frac{t}{t}}} \]
      *-commutative [=>] 90.3	\[ \sqrt{\frac{t}{\color{blue}{t \cdot \frac{x + 1}{x + -1}}} \cdot \frac{t}{t}} \]
      +-commutative [=>] 90.3	\[ \sqrt{\frac{t}{t \cdot \frac{\color{blue}{1 + x}}{x + -1}} \cdot \frac{t}{t}} \]
      *-inverses [=>] 90.3	\[ \sqrt{\frac{t}{t \cdot \frac{1 + x}{x + -1}} \cdot \color{blue}{1}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \left(\ell \cdot \left(\frac{2}{x} + \frac{2}{x \cdot x}\right)\right)}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, \ell \cdot \ell, \left(t \cdot t\right) \cdot 4\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(-1, \frac{-2}{\frac{x \cdot x}{\ell \cdot \ell}}, 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right) + \left(\frac{\ell}{\frac{x}{\ell}} + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.4%
Cost	27908

\[\begin{array}{l} t_1 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-240}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_1}, \sqrt{\frac{1}{t_1}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 2
Accuracy	80.3%
Cost	21448

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-218}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 3
Accuracy	80.3%
Cost	20680

\[\begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-218}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 4
Accuracy	80.7%
Cost	14792

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-218}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 5
Accuracy	79.0%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-218}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-151}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\ell \cdot \left(\ell \cdot \left(\frac{2}{x} + \frac{2}{x \cdot x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 6
Accuracy	78.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-218}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{t}{t \cdot \frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 7
Accuracy	78.8%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-218}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{x + 1}{x + -1}}}\\ \end{array} \]

Alternative 8
Accuracy	78.3%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-218}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-153}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Accuracy	77.4%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-216}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Accuracy	77.7%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-240}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-230}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Accuracy	78.1%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-240}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-229}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Accuracy	76.0%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-299}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) + \frac{-0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Accuracy	75.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-299}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	75.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	38.5%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023138 
(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)))))