Toniolo and Linder, Equation (7)

Average Accuracy: 33.2% → 83.3%

Time: 31.1s

Precision: binary64

Cost: 28172

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{+53}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq -3.35 \cdot 10^{-164}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(2 \cdot \left(t \cdot t\right) + \frac{\ell \cdot \ell}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{-275}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+81}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ l (/ x l))) (t_2 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x)))))
   (if (<= t -1.85e+53)
     (+ (+ (/ 1.0 x) (/ -0.5 (* x x))) -1.0)
     (if (<= t -3.35e-164)
       (/ t (sqrt (/ (+ t_1 (+ (* 2.0 (* t t)) (/ (* l l) x))) 2.0)))
       (if (<= t -6.6e-275)
         (*
          t
          (/
           (sqrt 2.0)
           (- (fma t (sqrt t_2) (* (sqrt (/ 1.0 t_2)) (* (/ l x) (/ l t)))))))
         (if (<= t 1.16e+81)
           (/
            t
            (sqrt
             (/
              (+
               t_1
               (+
                (* 2.0 (+ (* t t) (/ t (/ x t))))
                (/ (fma 2.0 (* t t) (* l l)) x)))
              2.0)))
           (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 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 = l / (x / l);
	double t_2 = 2.0 + ((2.0 / x) + (2.0 / x));
	double tmp;
	if (t <= -1.85e+53) {
		tmp = ((1.0 / x) + (-0.5 / (x * x))) + -1.0;
	} else if (t <= -3.35e-164) {
		tmp = t / sqrt(((t_1 + ((2.0 * (t * t)) + ((l * l) / x))) / 2.0));
	} else if (t <= -6.6e-275) {
		tmp = t * (sqrt(2.0) / -fma(t, sqrt(t_2), (sqrt((1.0 / t_2)) * ((l / x) * (l / t)))));
	} else if (t <= 1.16e+81) {
		tmp = t / sqrt(((t_1 + ((2.0 * ((t * t) + (t / (x / t)))) + (fma(2.0, (t * t), (l * l)) / x))) / 2.0));
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / 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(l / Float64(x / l))
	t_2 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	tmp = 0.0
	if (t <= -1.85e+53)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(-0.5 / Float64(x * x))) + -1.0);
	elseif (t <= -3.35e-164)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(Float64(2.0 * Float64(t * t)) + Float64(Float64(l * l) / x))) / 2.0)));
	elseif (t <= -6.6e-275)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(-fma(t, sqrt(t_2), Float64(sqrt(Float64(1.0 / t_2)) * Float64(Float64(l / x) * Float64(l / t)))))));
	elseif (t <= 1.16e+81)
		tmp = Float64(t / sqrt(Float64(Float64(t_1 + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t)))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))) / 2.0)));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / 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[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.85e+53], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, -3.35e-164], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6.6e-275], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-N[(t * N[Sqrt[t$95$2], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.16e+81], N[(t / N[Sqrt[N[(N[(t$95$1 + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.85e53

   1. Initial program 30.5%

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

   2. Taylor expanded in t around -inf 93.4%

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

   3. Simplified 93.4%

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

      [Start] 93.4	\[ \frac{\sqrt{2} \cdot t}{-1 \cdot \left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}\right)} \]
      mul-1-neg [=>] 93.4	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
      *-commutative [<=] 93.4	\[ \frac{\sqrt{2} \cdot t}{-\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]
      sub-neg [=>] 93.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 93.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 93.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x + -1}}} \]
      +-commutative [=>] 93.4	\[ \frac{\sqrt{2} \cdot t}{-\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   4. Taylor expanded in x around inf 92.9%

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

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right) - 1} \]
      Proof

      [Start] 92.9	\[ \frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) \]
      +-commutative [=>] 92.9	\[ \frac{1}{x} - \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + 1\right)} \]
      associate--r+ [=>] 92.9	\[ \color{blue}{\left(\frac{1}{x} - 0.5 \cdot \frac{1}{{x}^{2}}\right) - 1} \]
      associate-*r/ [=>] 92.9	\[ \left(\frac{1}{x} - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - 1 \]
      metadata-eval [=>] 92.9	\[ \left(\frac{1}{x} - \frac{\color{blue}{0.5}}{{x}^{2}}\right) - 1 \]
      unpow2 [=>] 92.9	\[ \left(\frac{1}{x} - \frac{0.5}{\color{blue}{x \cdot x}}\right) - 1 \]

   if -1.85e53 < t < -3.35e-164

   1. Initial program 54.0%

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

   2. Simplified 54.0%

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

      [Start] 54.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 54.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 54.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 54.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 54.3%

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

   4. Taylor expanded in x around inf 84.9%

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

   5. Simplified 84.9%

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

      [Start] 84.9	\[ \frac{t}{\sqrt{\frac{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}{2}}} \]
      cancel-sign-sub-inv [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{2}}} \]
      associate-+l+ [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{2}}} \]
      unpow2 [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      distribute-lft-out [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{1} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)}{2}}} \]
      unpow2 [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      fma-def [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}{2}}} \]
      unpow2 [=>] 84.9	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}{2}}} \]

   6. Taylor expanded in t around 0 84.3%

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

   7. Simplified 84.3%

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

      [Start] 84.3	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 84.3	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{\color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]

   8. Taylor expanded in x around inf 84.2%

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

   9. Simplified 84.2%

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

      [Start] 84.2	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot {t}^{2} + \frac{\ell \cdot \ell}{x}\right)}{2}}} \]
      unpow2 [=>] 84.2	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \frac{\ell \cdot \ell}{x}\right)}{2}}} \]

   if -3.35e-164 < t < -6.600000000000001e-275

   1. Initial program 2.1%

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

   2. Simplified 2.1%

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

      [Start] 2.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 2.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 2.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 2.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 48.9%

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

   4. Simplified 48.9%

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

      [Start] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 48.9	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 58.9%

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

   6. Simplified 58.7%

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

      [Start] 58.9	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 58.9	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 58.9	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 58.9	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      +-commutative [<=] 58.9	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      mul-1-neg [=>] 58.9	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]

   if -6.600000000000001e-275 < t < 1.15999999999999994e81

   1. Initial program 36.9%

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

   2. Simplified 37.0%

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

      [Start] 36.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 37.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 37.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 37.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 37.2%

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

   4. Taylor expanded in x around inf 71.8%

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

   5. Simplified 71.8%

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

      [Start] 71.8	\[ \frac{t}{\sqrt{\frac{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}{2}}} \]
      cancel-sign-sub-inv [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{2}}} \]
      associate-+l+ [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{2}}} \]
      unpow2 [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      distribute-lft-out [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{1} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)}{2}}} \]
      unpow2 [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      fma-def [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}{2}}} \]
      unpow2 [=>] 71.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}{2}}} \]

   if 1.15999999999999994e81 < t

   1. Initial program 25.5%

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

   2. Simplified 25.6%

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

      [Start] 25.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 25.6	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 25.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 25.6	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around inf 9.2%

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

   4. Simplified 30.9%

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

      [Start] 9.2	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \cdot t \]
      associate-/l* [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}} \cdot t \]
      +-commutative [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}} \cdot t \]
      sub-neg [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}} \cdot t \]
      metadata-eval [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}} \cdot t \]
      +-commutative [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{\color{blue}{-1 + x}}{{t}^{2}}}}} \cdot t \]
      unpow2 [=>] 30.9	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{x + 1}{\frac{-1 + x}{\color{blue}{t \cdot t}}}}} \cdot t \]

   5. Applied egg-rr 94.9%

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

   6. Simplified 95.3%

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

      [Start] 94.9	\[ \left(\left(-\sqrt{2}\right) \cdot \frac{1}{\sqrt{2} \cdot \left(\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t\right)}\right) \cdot t \]
      associate-*r/ [=>] 95.1	\[ \color{blue}{\frac{\left(-\sqrt{2}\right) \cdot 1}{\sqrt{2} \cdot \left(\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t\right)}} \cdot t \]
      *-rgt-identity [=>] 95.1	\[ \frac{\color{blue}{-\sqrt{2}}}{\sqrt{2} \cdot \left(\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t\right)} \cdot t \]
      distribute-neg-frac [<=] 95.1	\[ \color{blue}{\left(-\frac{\sqrt{2}}{\sqrt{2} \cdot \left(\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t\right)}\right)} \cdot t \]
      associate-/r* [=>] 95.3	\[ \left(-\color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t}}\right) \cdot t \]
      distribute-neg-frac [=>] 95.3	\[ \color{blue}{\frac{-\frac{\sqrt{2}}{\sqrt{2}}}{\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t}} \cdot t \]
      *-inverses [=>] 95.3	\[ \frac{-\color{blue}{1}}{\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t} \cdot t \]
      metadata-eval [=>] 95.3	\[ \frac{\color{blue}{-1}}{\left(-\sqrt{\frac{x + 1}{x + -1}}\right) \cdot t} \cdot t \]
      distribute-lft-neg-out [=>] 95.3	\[ \frac{-1}{\color{blue}{-\sqrt{\frac{x + 1}{x + -1}} \cdot t}} \cdot t \]
      distribute-rgt-neg-in [=>] 95.3	\[ \frac{-1}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-t\right)}} \cdot t \]
      +-commutative [=>] 95.3	\[ \frac{-1}{\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}} \cdot \left(-t\right)} \cdot t \]

   7. Taylor expanded in x around inf 95.0%

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

   8. Simplified 95.0%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
      Proof

      [Start] 95.0	\[ \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x} \]
      associate--l+ [=>] 95.0	\[ \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]
      associate-*r/ [=>] 95.0	\[ 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]
      metadata-eval [=>] 95.0	\[ 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]
      unpow2 [=>] 95.0	\[ 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+53}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq -3.35 \cdot 10^{-164}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + \frac{\ell \cdot \ell}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{-275}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+81}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.6%
Cost	15312

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-164}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(2 \cdot \left(t \cdot t\right) + \frac{\ell \cdot \ell}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-233}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	82.4%
Cost	8656

\[\begin{array}{l} t_1 := \frac{\ell}{\frac{x}{\ell}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+55}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-164}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(2 \cdot \left(t \cdot t\right) + t_2\right)}{2}}}\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-240}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+81}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_1 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 3
Accuracy	82.4%
Cost	8272

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + \frac{\ell \cdot \ell}{x}\right)}{2}}}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{+56}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.4 \cdot 10^{-239}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 4
Accuracy	77.5%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-211}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{1}{x}} \cdot \left(-\ell\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 5
Accuracy	77.0%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-235}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\ell} \cdot \left(-\sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 6
Accuracy	77.6%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{-236}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-276}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 7
Accuracy	75.8%
Cost	1224

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-241}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-212}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\frac{\frac{x}{-4}}{x + \ell \cdot \frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 8
Accuracy	76.3%
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-241}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-281}:\\ \;\;\;\;\frac{t}{\frac{\ell}{x} \cdot \left(\ell + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 9
Accuracy	76.5%
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-239}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{-0.5}{x \cdot x}\right) + -1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-280}:\\ \;\;\;\;\frac{t}{\frac{\ell}{x} \cdot \left(\ell + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Accuracy	76.3%
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-241}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-281}:\\ \;\;\;\;\frac{t}{\frac{\ell}{x} \cdot \left(\ell + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11
Accuracy	75.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-281}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Accuracy	75.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13
Accuracy	74.9%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{-281}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Accuracy	38.4%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023122 
(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)))))