Toniolo and Linder, Equation (7)

Average Error: 43.4 → 8.2

Time: 26.4s

Precision: binary64

Cost: 28240

Math

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

↓

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}{2}}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := \mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)\\ t_4 := t \cdot \frac{\sqrt{2}}{-t_3}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_3}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{x} + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

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
         (/
          t
          (sqrt
           (/
            (+ (/ l (/ x l)) (+ (* (* t t) (+ 2.0 (/ 4.0 x))) (* l (/ l x))))
            2.0))))
        (t_2 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x))))
        (t_3 (fma t (sqrt t_2) (* (sqrt (/ 1.0 t_2)) (* (/ l x) (/ l t)))))
        (t_4 (* t (/ (sqrt 2.0) (- t_3)))))
   (if (<= t -9.5e+153)
     t_4
     (if (<= t -1.25e-165)
       t_1
       (if (<= t -1e-310)
         t_4
         (if (<= t 1.05e-175)
           (* t (/ (sqrt 2.0) t_3))
           (if (<= t 1.25e+63)
             t_1
             (+ (/ (/ 0.5 x) x) (+ 1.0 (/ -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((((l / (x / l)) + (((t * t) * (2.0 + (4.0 / x))) + (l * (l / x)))) / 2.0));
	double t_2 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_3 = fma(t, sqrt(t_2), (sqrt((1.0 / t_2)) * ((l / x) * (l / t))));
	double t_4 = t * (sqrt(2.0) / -t_3);
	double tmp;
	if (t <= -9.5e+153) {
		tmp = t_4;
	} else if (t <= -1.25e-165) {
		tmp = t_1;
	} else if (t <= -1e-310) {
		tmp = t_4;
	} else if (t <= 1.05e-175) {
		tmp = t * (sqrt(2.0) / t_3);
	} else if (t <= 1.25e+63) {
		tmp = t_1;
	} else {
		tmp = ((0.5 / x) / x) + (1.0 + (-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(Float64(Float64(Float64(l / Float64(x / l)) + Float64(Float64(Float64(t * t) * Float64(2.0 + Float64(4.0 / x))) + Float64(l * Float64(l / x)))) / 2.0)))
	t_2 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_3 = fma(t, sqrt(t_2), Float64(sqrt(Float64(1.0 / t_2)) * Float64(Float64(l / x) * Float64(l / t))))
	t_4 = Float64(t * Float64(sqrt(2.0) / Float64(-t_3)))
	tmp = 0.0
	if (t <= -9.5e+153)
		tmp = t_4;
	elseif (t <= -1.25e-165)
		tmp = t_1;
	elseif (t <= -1e-310)
		tmp = t_4;
	elseif (t <= 1.05e-175)
		tmp = Float64(t * Float64(sqrt(2.0) / t_3));
	elseif (t <= 1.25e+63)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(0.5 / x) / x) + Float64(1.0 + 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[N[(N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * t), $MachinePrecision] * N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / (-t$95$3)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.5e+153], t$95$4, If[LessEqual[t, -1.25e-165], t$95$1, If[LessEqual[t, -1e-310], t$95$4, If[LessEqual[t, 1.05e-175], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e+63], t$95$1, N[(N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision] + N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -9.4999999999999995e153 or -1.24999999999999995e-165 < t < -9.999999999999969e-311

   1. Initial program 62.8

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

   2. Simplified 62.8

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

      [Start] 62.8	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 62.8	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 62.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 62.8	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 52.0

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

   4. Simplified 52.0

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

      [Start] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 52.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around -inf 19.9

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

   6. Simplified 11.3

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

      [Start] 19.9	\[ \frac{\sqrt{2}}{-1 \cdot \left(\frac{{\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}}\right) + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 19.9	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} + -1 \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}\right)} \cdot t \]
      *-commutative [=>] 19.9	\[ \frac{\sqrt{2}}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right) + -1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      distribute-lft-out [=>] 19.9	\[ \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x} + \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t\right)}} \cdot t \]
      +-commutative [<=] 19.9	\[ \frac{\sqrt{2}}{-1 \cdot \color{blue}{\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      mul-1-neg [=>] 19.9	\[ \frac{\sqrt{2}}{\color{blue}{-\left(\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      *-commutative [<=] 19.9	\[ \frac{\sqrt{2}}{-\left(\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      fma-def [=>] 19.9	\[ \frac{\sqrt{2}}{-\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      distribute-lft-in [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      distribute-lft-in [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      unpow2 [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t \cdot x}\right)} \cdot t \]
      *-commutative [=>] 19.9	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \frac{\ell \cdot \ell}{\color{blue}{x \cdot t}}\right)} \cdot t \]
      times-frac [=>] 11.3	\[ \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \color{blue}{\left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)}\right)} \cdot t \]

   if -9.4999999999999995e153 < t < -1.24999999999999995e-165 or 1.05e-175 < t < 1.25000000000000003e63

   1. Initial program 28.0

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

   2. Simplified 28.0

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

      [Start] 28.0	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 28.0	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 28.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 28.0	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 27.8

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

   4. Taylor expanded in x around inf 11.1

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

   5. Simplified 11.1

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

      [Start] 11.1	\[ \frac{t}{\sqrt{\frac{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}{2}}} \]
      cancel-sign-sub-inv [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{2}}} \]
      associate-+l+ [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{2}}} \]
      unpow2 [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      distribute-lft-out [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{1} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      fma-def [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}{2}}} \]
      associate-*r/ [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{\frac{1 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}\right)}{2}}} \]
      *-lft-identity [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}{2}}} \]

   6. Taylor expanded in t around 0 11.1

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

   7. Simplified 5.8

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

      [Start] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      *-commutative [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{{t}^{2} \cdot \left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)} + \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      fma-def [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\mathsf{fma}\left({t}^{2}, 2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}}{2}}} \]
      unpow2 [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(\color{blue}{t \cdot t}, 2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      distribute-lft-in [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      associate-*r/ [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \left(2 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \left(2 + \frac{\color{blue}{2}}{x}\right) + 2 \cdot \frac{1}{x}, \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      associate-*r/ [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \left(2 + \frac{2}{x}\right) + \color{blue}{\frac{2 \cdot 1}{x}}, \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \left(2 + \frac{2}{x}\right) + \frac{\color{blue}{2}}{x}, \frac{{\ell}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 11.1	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \left(2 + \frac{2}{x}\right) + \frac{2}{x}, \frac{\color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      associate-*r/ [<=] 5.8	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(t \cdot t, \left(2 + \frac{2}{x}\right) + \frac{2}{x}, \color{blue}{\ell \cdot \frac{\ell}{x}}\right)}{2}}} \]

   8. Applied egg-rr 5.8

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

   if -9.999999999999969e-311 < t < 1.05e-175

   1. Initial program 63.1

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

   2. Simplified 63.1

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

      [Start] 63.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 63.1	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 63.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 63.1	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 33.3

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

   4. Simplified 33.3

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

      [Start] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 33.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around inf 24.6

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

   6. Simplified 24.8

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

      [Start] 24.6	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      *-commutative [<=] 24.6	\[ \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      fma-def [=>] 24.6	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      distribute-lft-in [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      distribute-lft-in [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      unpow2 [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t \cdot x}\right)} \cdot t \]
      *-commutative [=>] 24.6	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \frac{\ell \cdot \ell}{\color{blue}{x \cdot t}}\right)} \cdot t \]
      times-frac [=>] 24.8	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \color{blue}{\left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)}\right)} \cdot t \]

   if 1.25000000000000003e63 < t

   1. Initial program 46.2

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

   2. Taylor expanded in l around 0 3.5

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

   3. Taylor expanded in x around inf 3.8

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

   4. Simplified 3.8

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

      [Start] 3.8	\[ \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x} \]
      sub-neg [=>] 3.8	\[ \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{1}{x}\right)} \]
      +-commutative [=>] 3.8	\[ \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + 1\right)} + \left(-\frac{1}{x}\right) \]
      associate-+l+ [=>] 3.8	\[ \color{blue}{0.5 \cdot \frac{1}{{x}^{2}} + \left(1 + \left(-\frac{1}{x}\right)\right)} \]
      associate-*r/ [=>] 3.8	\[ \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(1 + \left(-\frac{1}{x}\right)\right) \]
      metadata-eval [=>] 3.8	\[ \frac{\color{blue}{0.5}}{{x}^{2}} + \left(1 + \left(-\frac{1}{x}\right)\right) \]
      unpow2 [=>] 3.8	\[ \frac{0.5}{\color{blue}{x \cdot x}} + \left(1 + \left(-\frac{1}{x}\right)\right) \]
      associate-/r* [=>] 3.8	\[ \color{blue}{\frac{\frac{0.5}{x}}{x}} + \left(1 + \left(-\frac{1}{x}\right)\right) \]
      distribute-neg-frac [=>] 3.8	\[ \frac{\frac{0.5}{x}}{x} + \left(1 + \color{blue}{\frac{-1}{x}}\right) \]
      metadata-eval [=>] 3.8	\[ \frac{\frac{0.5}{x}}{x} + \left(1 + \frac{\color{blue}{-1}}{x}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+153}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-165}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}{2}}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+63}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{x} + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.9
Cost	28108

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}{2}}}\\ t_2 := \frac{t}{-t}\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+66}:\\ \;\;\;\;t_2 + \frac{t_2}{-x}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-263}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-173}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{t_3}, \sqrt{\frac{1}{t_3}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{x} + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 2
Error	9.5
Cost	8528

\[\begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}{2}}}\\ t_2 := 1 + \frac{-1}{x}\\ t_3 := \frac{t}{-t}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+66}:\\ \;\;\;\;t_3 + \frac{t_3}{-x}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{x} + t_2\\ \end{array} \]

Alternative 3
Error	15.2
Cost	7044

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

Alternative 4
Error	15.4
Cost	900

\[\begin{array}{l} t_1 := \frac{t}{-t}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_1 + \frac{t_1}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{x} + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 5
Error	15.7
Cost	836

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

Alternative 6
Error	15.7
Cost	452

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

Alternative 7
Error	16.0
Cost	196

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

Alternative 8
Error	39.5
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022364 
(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)))))