Toniolo and Linder, Equation (7)

Average Accuracy: 32.2% → 85.6%

Time: 36.1s

Precision: binary64

Cost: 28628

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 2 \cdot \frac{\ell}{\frac{x}{\ell}}\\ t_3 := \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}{t}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.06 \cdot 10^{-162}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-199}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_2}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(t, \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}, \frac{t_2}{t} \cdot \left(0.5 \cdot \sqrt{\frac{0.5}{1 + \left(\frac{1}{x} + \frac{1}{x}\right)}}\right)\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+55}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (* 2.0 (/ l (/ x l))))
        (t_3
         (/
          (sqrt 2.0)
          (/
           (sqrt
            (+
             (* l (/ 1.0 (/ x l)))
             (+ (* 2.0 (+ (* t t) (/ (* t t) x))) (* l (/ l x)))))
           t))))
   (if (<= t -4e+16)
     -1.0
     (if (<= t -1.06e-162)
       t_3
       (if (<= t -4.1e-199)
         -1.0
         (if (<= t 3.8e-308)
           (/ t_1 (sqrt t_2))
           (if (<= t 1.35e-161)
             (/
              t_1
              (fma
               t
               (sqrt (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))
               (*
                (/ t_2 t)
                (* 0.5 (sqrt (/ 0.5 (+ 1.0 (+ (/ 1.0 x) (/ 1.0 x)))))))))
             (if (<= t 4.5e+55) t_3 (sqrt (/ (+ -1.0 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 = t * sqrt(2.0);
	double t_2 = 2.0 * (l / (x / l));
	double t_3 = sqrt(2.0) / (sqrt(((l * (1.0 / (x / l))) + ((2.0 * ((t * t) + ((t * t) / x))) + (l * (l / x))))) / t);
	double tmp;
	if (t <= -4e+16) {
		tmp = -1.0;
	} else if (t <= -1.06e-162) {
		tmp = t_3;
	} else if (t <= -4.1e-199) {
		tmp = -1.0;
	} else if (t <= 3.8e-308) {
		tmp = t_1 / sqrt(t_2);
	} else if (t <= 1.35e-161) {
		tmp = t_1 / fma(t, sqrt(((2.0 / x) + (2.0 + (2.0 / x)))), ((t_2 / t) * (0.5 * sqrt((0.5 / (1.0 + ((1.0 / x) + (1.0 / x))))))));
	} else if (t <= 4.5e+55) {
		tmp = t_3;
	} else {
		tmp = sqrt(((-1.0 + 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(t * sqrt(2.0))
	t_2 = Float64(2.0 * Float64(l / Float64(x / l)))
	t_3 = Float64(sqrt(2.0) / Float64(sqrt(Float64(Float64(l * Float64(1.0 / Float64(x / l))) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(l * Float64(l / x))))) / t))
	tmp = 0.0
	if (t <= -4e+16)
		tmp = -1.0;
	elseif (t <= -1.06e-162)
		tmp = t_3;
	elseif (t <= -4.1e-199)
		tmp = -1.0;
	elseif (t <= 3.8e-308)
		tmp = Float64(t_1 / sqrt(t_2));
	elseif (t <= 1.35e-161)
		tmp = Float64(t_1 / fma(t, sqrt(Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x)))), Float64(Float64(t_2 / t) * Float64(0.5 * sqrt(Float64(0.5 / Float64(1.0 + Float64(Float64(1.0 / x) + Float64(1.0 / x)))))))));
	elseif (t <= 4.5e+55)
		tmp = t_3;
	else
		tmp = sqrt(Float64(Float64(-1.0 + 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[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(l * N[(1.0 / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4e+16], -1.0, If[LessEqual[t, -1.06e-162], t$95$3, If[LessEqual[t, -4.1e-199], -1.0, If[LessEqual[t, 3.8e-308], N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-161], N[(t$95$1 / N[(t * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(t$95$2 / t), $MachinePrecision] * N[(0.5 * N[Sqrt[N[(0.5 / N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e+55], t$95$3, N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4e16 or -1.06000000000000003e-162 < t < -4.10000000000000022e-199

   1. Initial program 30.7%

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

   2. Simplified 30.8%

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

      [Start] 30.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 30.8	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around inf 1.6%

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

   4. Simplified 1.6%

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

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

   5. Applied egg-rr 36.3%

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

      [Start] 1.6	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)} \cdot t \]
      add-sqr-sqrt [=>] 0.0	\[ \color{blue}{\left(\sqrt{\frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}} \cdot \sqrt{\frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}}\right)} \cdot t \]
      sqrt-unprod [=>] 38.0	\[ \color{blue}{\sqrt{\frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)} \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}}} \cdot t \]
      associate-/r* [=>] 38.1	\[ \sqrt{\color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{t \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}} \cdot t \]
      *-inverses [=>] 38.1	\[ \sqrt{\frac{\color{blue}{1}}{t \cdot \sqrt{\frac{x + 1}{-1 + x}}} \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}} \cdot t \]
      associate-/r* [=>] 38.1	\[ \sqrt{\frac{1}{t \cdot \sqrt{\frac{x + 1}{-1 + x}}} \cdot \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2}}}{t \cdot \sqrt{\frac{x + 1}{-1 + x}}}}} \cdot t \]
      *-inverses [=>] 38.1	\[ \sqrt{\frac{1}{t \cdot \sqrt{\frac{x + 1}{-1 + x}}} \cdot \frac{\color{blue}{1}}{t \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \cdot t \]
      frac-times [=>] 36.3	\[ \sqrt{\color{blue}{\frac{1 \cdot 1}{\left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right) \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}}} \cdot t \]
      metadata-eval [=>] 36.3	\[ \sqrt{\frac{\color{blue}{1}}{\left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right) \cdot \left(t \cdot \sqrt{\frac{x + 1}{-1 + x}}\right)}} \cdot t \]

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]

   7. Simplified 87.6%

      \[\leadsto \color{blue}{-1} \]
      Proof

      [Start] 0.0	\[ {\left(\sqrt{-1}\right)}^{2} \]
      unpow2 [=>] 0.0	\[ \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]
      rem-square-sqrt [=>] 87.6	\[ \color{blue}{-1} \]

   if -4e16 < t < -1.06000000000000003e-162 or 1.35e-161 < t < 4.49999999999999998e55

   1. Initial program 53.0%

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

   2. Simplified 53.0%

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

      [Start] 53.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-/l* [=>] 53.0	\[ \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]
      fma-neg [=>] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]
      remove-double-neg [<=] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]
      fma-neg [<=] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]
      sub-neg [=>] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      metadata-eval [=>] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]
      remove-double-neg [=>] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]
      +-commutative [=>] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}}{t}} \]
      fma-def [=>] 53.0	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}}{t}} \]

   3. Taylor expanded in x around inf 84.1%

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

   4. Simplified 84.1%

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

      [Start] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{t}} \]
      associate--l+ [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}}{t}} \]
      unpow2 [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{t}} \]
      distribute-lft-out [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{t}} \]
      unpow2 [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{t}} \]
      unpow2 [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{t}} \]
      associate-*r/ [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}}{t}} \]
      mul-1-neg [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}}{t}} \]
      unpow2 [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}}{t}} \]
      +-commutative [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}}{t}} \]
      unpow2 [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}}{t}} \]
      fma-udef [<=] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}}{t}} \]

   5. Applied egg-rr 84.1%

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

      [Start] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}{t}} \]
      associate-/l* [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}{t}} \]
      div-inv [=>] 84.1	\[ \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\ell \cdot \frac{1}{\frac{x}{\ell}}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}{t}} \]

   6. Taylor expanded in t around 0 83.6%

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

   7. Simplified 92.2%

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

      [Start] 83.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}}{t}} \]
      mul-1-neg [=>] 83.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}}{t}} \]
      unpow2 [=>] 83.6	\[ \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}}{t}} \]
      associate-*r/ [<=] 92.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\color{blue}{\ell \cdot \frac{\ell}{x}}\right)\right)}}{t}} \]
      distribute-rgt-neg-in [=>] 92.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\ell \cdot \left(-\frac{\ell}{x}\right)}\right)}}{t}} \]
      distribute-neg-frac [=>] 92.2	\[ \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \ell \cdot \color{blue}{\frac{-\ell}{x}}\right)}}{t}} \]

   if -4.10000000000000022e-199 < t < 3.79999999999999975e-308

   1. Initial program 1.7%

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

   2. Taylor expanded in x around inf 50.7%

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

   3. Simplified 50.7%

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

      [Start] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + {\ell}^{2}\right)}{x}\right)}} \]
      unpow2 [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \]

   4. Taylor expanded in t around 0 50.7%

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

   5. Simplified 53.1%

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

      [Start] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}} \]
      cancel-sign-sub-inv [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(--1\right) \cdot \frac{{\ell}^{2}}{x}}}} \]
      unpow2 [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(--1\right) \cdot \frac{{\ell}^{2}}{x}}} \]
      associate-*l/ [<=] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(--1\right) \cdot \frac{{\ell}^{2}}{x}}} \]
      metadata-eval [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{x} \cdot \ell + \color{blue}{1} \cdot \frac{{\ell}^{2}}{x}}} \]
      unpow2 [=>] 50.7	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{x} \cdot \ell + 1 \cdot \frac{\color{blue}{\ell \cdot \ell}}{x}}} \]
      associate-*l/ [<=] 53.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{x} \cdot \ell + 1 \cdot \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)}}} \]
      distribute-rgt1-in [=>] 53.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(1 + 1\right) \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \]
      metadata-eval [=>] 53.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2} \cdot \left(\frac{\ell}{x} \cdot \ell\right)}} \]
      associate-/r/ [<=] 53.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\frac{\ell}{\frac{x}{\ell}}}}} \]

   if 3.79999999999999975e-308 < t < 1.35e-161

   1. Initial program 2.4%

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

   2. Taylor expanded in x around inf 46.2%

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

   3. Simplified 46.2%

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

      [Start] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + {\ell}^{2}\right)}{x}\right)}} \]
      unpow2 [=>] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 46.2	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \]

   4. Taylor expanded in t around inf 61.9%

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

   5. Simplified 61.9%

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{\left(2 + \frac{2}{x}\right) + \frac{2}{x}}, \frac{2 \cdot \frac{\ell}{\frac{x}{\ell}}}{t} \cdot \left(\sqrt{\frac{0.5}{1 + \left(\frac{1}{x} + \frac{1}{x}\right)}} \cdot 0.5\right)\right)}} \]
      Proof

      [Start] 61.9	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)} \]
      *-commutative [<=] 61.9	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)} \]
      fma-def [=>] 61.9	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)}} \]
      distribute-lft-in [=>] 61.9	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)} \]
      metadata-eval [=>] 61.9	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)} \]
      associate-*r/ [=>] 61.9	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{\left(2 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + 2 \cdot \frac{1}{x}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)} \]
      metadata-eval [=>] 61.9	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{\left(2 + \frac{\color{blue}{2}}{x}\right) + 2 \cdot \frac{1}{x}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)} \]
      associate-*r/ [=>] 61.9	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{\left(2 + \frac{2}{x}\right) + \color{blue}{\frac{2 \cdot 1}{x}}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)} \]
      metadata-eval [=>] 61.9	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(t, \sqrt{\left(2 + \frac{2}{x}\right) + \frac{\color{blue}{2}}{x}}, 0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right)\right)} \]

   if 4.49999999999999998e55 < t

   1. Initial program 27.8%

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

   2. Simplified 27.8%

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

      [Start] 27.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 27.8	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Taylor expanded in t around inf 93.0%

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

   4. Simplified 93.0%

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

      [Start] 93.0	\[ \frac{\sqrt{2}}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}} \cdot t \]
      associate-*l* [=>] 93.0	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \cdot t \]
      +-commutative [=>] 93.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \cdot t \]
      sub-neg [=>] 93.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \cdot t \]
      metadata-eval [=>] 93.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \cdot t \]
      +-commutative [=>] 93.0	\[ \frac{\sqrt{2}}{\sqrt{2} \cdot \left(t \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \cdot t \]

   5. Taylor expanded in t around 0 93.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.06 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}{t}}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-199}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}, \frac{2 \cdot \frac{\ell}{\frac{x}{\ell}}}{t} \cdot \left(0.5 \cdot \sqrt{\frac{0.5}{1 + \left(\frac{1}{x} + \frac{1}{x}\right)}}\right)\right)}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.5%
Cost	28628

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{x}{\ell}}\\ t_2 := \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}{t}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-201}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-305}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{t_1}}\\ \mathbf{elif}\;t \leq 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(t, \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}, \frac{t_1}{t} \cdot \left(0.5 \cdot \sqrt{\frac{0.5}{1 + \left(\frac{1}{x} + \frac{1}{x}\right)}}\right)\right)}{t}}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 2
Accuracy	81.1%
Cost	15448

\[\begin{array}{l} t_1 := \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + \frac{\ell \cdot \ell - \left(t \cdot t\right) \cdot -4}{x}\right)}}{t}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-199}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-273}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-192}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 3
Accuracy	83.8%
Cost	15448

\[\begin{array}{l} t_1 := \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}{t}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -2.15 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-201}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-264}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-192}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 4
Accuracy	80.9%
Cost	15320

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{t_2}{\sqrt{t_1 + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + t_1\right)}}\\ \mathbf{if}\;t \leq -3500000000:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-163}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_2}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-192}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+17}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 5
Accuracy	76.6%
Cost	15192

\[\begin{array}{l} t_1 := -1 + \frac{1}{x}\\ t_2 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.35 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-192}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + t \cdot \left(t \cdot \left(2 + \left(\frac{2}{x} + \frac{2}{x}\right)\right)\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 6
Accuracy	76.4%
Cost	14680

\[\begin{array}{l} t_1 := -1 + \frac{1}{x}\\ t_2 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-263}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-192}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{1}{\frac{x}{\ell}} + 2 \cdot \left(t \cdot t\right)}}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 7
Accuracy	76.6%
Cost	14032

\[\begin{array}{l} t_1 := -1 + \frac{1}{x}\\ t_2 := \frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}{t}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-271}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 8
Accuracy	76.4%
Cost	14032

\[\begin{array}{l} t_1 := -1 + \frac{1}{x}\\ t_2 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-263}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 9
Accuracy	76.3%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-267}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 7.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 10
Accuracy	77.0%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-197}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-263}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 11
Accuracy	76.3%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-267}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-280}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 12
Accuracy	76.0%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-270}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-280}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 13
Accuracy	75.2%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 14
Accuracy	74.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15
Accuracy	75.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 16
Accuracy	74.4%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Accuracy	38.5%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023151 
(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)))))