Toniolo and Linder, Equation (7)

Average Error: 67.75% → 14.3%

Time: 27.4s

Precision: binary64

Cost: 40916

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ t_4 := -t_3\\ t_5 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+42}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_2}{\sqrt{2 \cdot \left(t_5 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-268}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-295}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-157}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(0.5, \frac{t_1 + t_1}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_2\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_5 + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (fma 2.0 (* t t) (* l l)))
        (t_2 (* t (sqrt 2.0)))
        (t_3 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_4 (- t_3))
        (t_5 (* l (/ l x))))
   (if (<= t -1.35e+42)
     t_4
     (if (<= t -2e-163)
       (/ t_2 (sqrt (* 2.0 (+ t_5 (* t (+ t (/ t x)))))))
       (if (<= t -3.9e-268)
         t_4
         (if (<= t 1.15e-295)
           (* t (sqrt (/ x (* l l))))
           (if (<= t 2.3e-157)
             (/ t_2 (fma 0.5 (/ (+ t_1 t_1) (* (sqrt 2.0) (* t x))) t_2))
             (if (<= t 1.35e+69)
               (/
                t_2
                (sqrt
                 (+ (/ l (/ x l)) (+ t_5 (* 2.0 (+ (* t t) (/ t (/ x t))))))))
               t_3))))))))

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 = fma(2.0, (t * t), (l * l));
	double t_2 = t * sqrt(2.0);
	double t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_4 = -t_3;
	double t_5 = l * (l / x);
	double tmp;
	if (t <= -1.35e+42) {
		tmp = t_4;
	} else if (t <= -2e-163) {
		tmp = t_2 / sqrt((2.0 * (t_5 + (t * (t + (t / x))))));
	} else if (t <= -3.9e-268) {
		tmp = t_4;
	} else if (t <= 1.15e-295) {
		tmp = t * sqrt((x / (l * l)));
	} else if (t <= 2.3e-157) {
		tmp = t_2 / fma(0.5, ((t_1 + t_1) / (sqrt(2.0) * (t * x))), t_2);
	} else if (t <= 1.35e+69) {
		tmp = t_2 / sqrt(((l / (x / l)) + (t_5 + (2.0 * ((t * t) + (t / (x / t)))))));
	} else {
		tmp = t_3;
	}
	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 = fma(2.0, Float64(t * t), Float64(l * l))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_4 = Float64(-t_3)
	t_5 = Float64(l * Float64(l / x))
	tmp = 0.0
	if (t <= -1.35e+42)
		tmp = t_4;
	elseif (t <= -2e-163)
		tmp = Float64(t_2 / sqrt(Float64(2.0 * Float64(t_5 + Float64(t * Float64(t + Float64(t / x)))))));
	elseif (t <= -3.9e-268)
		tmp = t_4;
	elseif (t <= 1.15e-295)
		tmp = Float64(t * sqrt(Float64(x / Float64(l * l))));
	elseif (t <= 2.3e-157)
		tmp = Float64(t_2 / fma(0.5, Float64(Float64(t_1 + t_1) / Float64(sqrt(2.0) * Float64(t * x))), t_2));
	elseif (t <= 1.35e+69)
		tmp = Float64(t_2 / sqrt(Float64(Float64(l / Float64(x / l)) + Float64(t_5 + Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t))))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = (-t$95$3)}, Block[{t$95$5 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.35e+42], t$95$4, If[LessEqual[t, -2e-163], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(t$95$5 + N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.9e-268], t$95$4, If[LessEqual[t, 1.15e-295], N[(t * N[Sqrt[N[(x / N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e-157], N[(t$95$2 / N[(0.5 * N[(N[(t$95$1 + t$95$1), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+69], N[(t$95$2 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.35e42 or -1.99999999999999985e-163 < t < -3.8999999999999998e-268

   1. Initial program 76.73

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

   2. Simplified 76.26

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

      [Start] 76.73	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 76.78	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 90.27	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 76.72	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 76.72	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 76.72	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 76.72	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 76.72	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 76.72	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 76.26	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around inf 89.85

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

   4. Simplified 89.85

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

      [Start] 89.85	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \]
      *-commutative [=>] 89.85	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \]
      unpow2 [=>] 89.85	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + x\right)}{x - 1}}} \]
      +-commutative [=>] 89.85	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \]
      sub-neg [=>] 89.85	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 89.85	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 89.85	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}}}} \]

   5. Taylor expanded in t around -inf 17.38

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

   6. Simplified 17.38

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

      [Start] 17.38	\[ -1 \cdot \sqrt{\frac{x - 1}{1 + x}} \]
      mul-1-neg [=>] 17.38	\[ \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]
      sub-neg [=>] 17.38	\[ -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]
      metadata-eval [=>] 17.38	\[ -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   if -1.35e42 < t < -1.99999999999999985e-163

   1. Initial program 46.49

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

   2. Taylor expanded in x around inf 15.06

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

   3. Simplified 15.06

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

      [Start] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 15.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 15.5

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

   5. Simplified 8.13

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

      [Start] 15.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      mul-1-neg [=>] 15.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]
      unpow2 [=>] 15.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \]
      associate-*r/ [<=] 8.13	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\color{blue}{\ell \cdot \frac{\ell}{x}}\right)\right)}} \]
      distribute-lft-neg-in [=>] 8.13	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\ell\right) \cdot \frac{\ell}{x}}\right)}} \]
      *-commutative [=>] 8.13	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \]

   6. Taylor expanded in l around 0 15.5

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

   7. Simplified 8.12

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

      [Start] 15.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)}} \]
      distribute-lft-out [=>] 15.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{{\ell}^{2}}{x} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}}} \]
      unpow2 [=>] 15.5	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}} \]
      associate-*r/ [<=] 8.12	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\color{blue}{\ell \cdot \frac{\ell}{x}} + \left(\frac{{t}^{2}}{x} + {t}^{2}\right)\right)}} \]
      +-commutative [=>] 8.12	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)}\right)}} \]
      unpow2 [=>] 8.12	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right)\right)}} \]
      unpow2 [=>] 8.12	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right)\right)}} \]
      associate-*r/ [<=] 8.12	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \left(t \cdot t + \color{blue}{t \cdot \frac{t}{x}}\right)\right)}} \]
      distribute-lft-in [<=] 8.12	\[ \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + \color{blue}{t \cdot \left(t + \frac{t}{x}\right)}\right)}} \]

   if -3.8999999999999998e-268 < t < 1.15e-295

   1. Initial program 99.1

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

   2. Simplified 95.94

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

      [Start] 99.1	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 99.1	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 95.3	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 98.53	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 98.53	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 98.53	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 98.53	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 98.53	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 98.53	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 95.94	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around 0 95.94

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

   4. Simplified 95.94

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

      [Start] 95.94	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{{\ell}^{2}}{x - 1}, -\ell \cdot \ell\right)}} \]
      unpow2 [=>] 95.94	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\color{blue}{\ell \cdot \ell}}{x - 1}, -\ell \cdot \ell\right)}} \]
      sub-neg [=>] 95.94	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\ell \cdot \ell}{\color{blue}{x + \left(-1\right)}}, -\ell \cdot \ell\right)}} \]
      metadata-eval [=>] 95.94	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\ell \cdot \ell}{x + \color{blue}{-1}}, -\ell \cdot \ell\right)}} \]
      +-commutative [=>] 95.94	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(x + 1, \frac{\ell \cdot \ell}{\color{blue}{-1 + x}}, -\ell \cdot \ell\right)}} \]

   5. Taylor expanded in x around inf 44.15

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

   6. Simplified 44.15

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

      [Start] 44.15	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x}}} \]
      unpow2 [=>] 44.15	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{x}}} \]

   7. Applied egg-rr 72.89

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

   8. Simplified 72.84

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

      [Start] 72.89	\[ \sqrt{\frac{2 \cdot \left(t \cdot t\right)}{2} \cdot \frac{x}{\ell \cdot \ell}} \]
      associate-*r/ [=>] 72.48	\[ \sqrt{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot t\right)}{2} \cdot x}{\ell \cdot \ell}}} \]
      associate-/l* [=>] 72.84	\[ \sqrt{\color{blue}{\frac{\frac{2 \cdot \left(t \cdot t\right)}{2}}{\frac{\ell \cdot \ell}{x}}}} \]
      *-commutative [=>] 72.84	\[ \sqrt{\frac{\frac{\color{blue}{\left(t \cdot t\right) \cdot 2}}{2}}{\frac{\ell \cdot \ell}{x}}} \]
      associate-/l* [=>] 72.84	\[ \sqrt{\frac{\color{blue}{\frac{t \cdot t}{\frac{2}{2}}}}{\frac{\ell \cdot \ell}{x}}} \]
      metadata-eval [=>] 72.84	\[ \sqrt{\frac{\frac{t \cdot t}{\color{blue}{1}}}{\frac{\ell \cdot \ell}{x}}} \]
      associate-*r/ [<=] 72.84	\[ \sqrt{\frac{\color{blue}{t \cdot \frac{t}{1}}}{\frac{\ell \cdot \ell}{x}}} \]
      /-rgt-identity [=>] 72.84	\[ \sqrt{\frac{t \cdot \color{blue}{t}}{\frac{\ell \cdot \ell}{x}}} \]

   9. Applied egg-rr 45

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

   if 1.15e-295 < t < 2.29999999999999989e-157

   1. Initial program 98.35

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

   2. Taylor expanded in x around inf 39.56

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

   3. Simplified 39.56

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

      [Start] 39.56	\[ \frac{\sqrt{2} \cdot t}{0.5 \cdot \frac{\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)} + \sqrt{2} \cdot t} \]
      fma-def [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{\left({\ell}^{2} + 2 \cdot {t}^{2}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)}} \]
      +-commutative [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)} - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      unpow2 [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      fma-udef [<=] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)} - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      unpow2 [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right) - -1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      mul-1-neg [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \color{blue}{\left(-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)\right)}}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      +-commutative [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \left(-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      unpow2 [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \left(-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      fma-udef [<=] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \left(-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      unpow2 [=>] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \left(-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \sqrt{2} \cdot t\right)} \]
      *-commutative [<=] 39.56	\[ \frac{\sqrt{2} \cdot t}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \left(-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, \color{blue}{t \cdot \sqrt{2}}\right)} \]

   if 2.29999999999999989e-157 < t < 1.3499999999999999e69

   1. Initial program 41.67

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

   2. Taylor expanded in x around inf 13.99

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

   3. Simplified 13.99

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

      [Start] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \]
      associate--l+ [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \]
      unpow2 [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      distribute-lft-out [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      +-commutative [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      unpow2 [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-/l* [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \]
      associate-*r/ [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \]
      mul-1-neg [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \]
      +-commutative [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}\right)}} \]
      unpow2 [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}\right)}} \]
      fma-udef [<=] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}} \]
      unpow2 [=>] 13.99	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}} \]

   4. Taylor expanded in t around 0 14.68

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

   5. Simplified 8.27

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

      [Start] 14.68	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - -1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]
      mul-1-neg [=>] 14.68	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]
      unpow2 [=>] 14.68	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \]
      associate-*r/ [<=] 8.27	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \left(-\color{blue}{\ell \cdot \frac{\ell}{x}}\right)\right)}} \]
      distribute-lft-neg-in [=>] 8.27	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\left(-\ell\right) \cdot \frac{\ell}{x}}\right)}} \]
      *-commutative [=>] 8.27	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{\ell}{x} \cdot \left(-\ell\right)}\right)}} \]

   if 1.3499999999999999e69 < t

   1. Initial program 74.18

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

   2. Simplified 74.15

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

      [Start] 74.18	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*r/ [<=] 74.23	\[ \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]
      associate-*l/ [=>] 91.25	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(x + 1\right) \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)}{x - 1}} - \ell \cdot \ell}} \]
      associate-*r/ [<=] 74.22	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}} - \ell \cdot \ell}} \]
      *-lft-identity [<=] 74.22	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \left(x + 1\right)\right)} \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} - \ell \cdot \ell}} \]
      associate-*r* [<=] 74.22	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{1 \cdot \left(\left(x + 1\right) \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 74.22	\[ \sqrt{2} \cdot \frac{t}{\sqrt{1 \cdot \color{blue}{\left(\frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1} \cdot \left(x + 1\right)\right)} - \ell \cdot \ell}} \]
      associate-*r* [=>] 74.22	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right) \cdot \left(x + 1\right)} - \ell \cdot \ell}} \]
      *-commutative [<=] 74.22	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(x + 1\right) \cdot \left(1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}\right)} - \ell \cdot \ell}} \]
      fma-neg [=>] 74.15	\[ \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(x + 1, 1 \cdot \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x - 1}, -\ell \cdot \ell\right)}}} \]

   3. Taylor expanded in t around inf 90.1

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

   4. Simplified 90.1

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

      [Start] 90.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}} \]
      *-commutative [=>] 90.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{{t}^{2} \cdot \left(1 + x\right)}}{x - 1}}} \]
      unpow2 [=>] 90.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\color{blue}{\left(t \cdot t\right)} \cdot \left(1 + x\right)}{x - 1}}} \]
      +-commutative [=>] 90.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \]
      sub-neg [=>] 90.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{x + \left(-1\right)}}}} \]
      metadata-eval [=>] 90.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{x + \color{blue}{-1}}}} \]
      +-commutative [=>] 90.1	\[ \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \frac{\left(t \cdot t\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}}}} \]

   5. Taylor expanded in t around 0 5.61

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+42}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-268}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-295}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-157}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\ell \cdot \frac{\ell}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.59%
Cost	40844

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := \ell \cdot \frac{\ell}{x}\\ t_4 := 1 + \frac{1}{x}\\ \mathbf{if}\;t \leq -9.8 \cdot 10^{+41}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t_3 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-272}:\\ \;\;\;\;\frac{t_1}{-0.5 \cdot \left(\frac{\sqrt{2} \cdot {\ell}^{2}}{t \cdot x} \cdot \sqrt{\frac{1}{t_4}}\right) - t_1 \cdot \sqrt{t_4}}\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{+69}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_3 + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	15.89%
Cost	15320

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := -t_2\\ t_4 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-160}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t_4 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-270}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-269}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+69}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_4 + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	15.88%
Cost	14808

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -1.26 \cdot 10^{+42}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-270}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-273}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	18.57%
Cost	8920

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := -t_1\\ t_3 := \frac{\ell \cdot \ell}{x}\\ t_4 := \frac{t}{\sqrt{\frac{2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \left(t_3 + t_3\right)}{2}}}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-160}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-268}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-269}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-157}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-71}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	54.36%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{x}}{\ell} \cdot \left(-t\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+170}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 6
Error	22.4%
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-268}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-273}:\\ \;\;\;\;t \cdot \sqrt{\frac{x}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	54.59%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{x}}{\ell} \cdot \left(-t\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{0.5}{x \cdot x} + \left(1 + \frac{-1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \end{array} \]

Alternative 8
Error	52.11%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq 3.9 \cdot 10^{-269}:\\ \;\;\;\;\frac{t}{\sqrt{\ell \cdot \frac{\ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 9
Error	54.8%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-303}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x \cdot x} + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Error	54.78%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x \cdot x} + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 11
Error	61.55%
Cost	704

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

Alternative 12
Error	61.68%
Cost	320

\[1 + \frac{-1}{x} \]

Alternative 13
Error	62.08%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023088 
(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)))))