Toniolo and Linder, Equation (7)

Average Error: 67.1% → 12.57%

Time: 28.7s

Precision: binary64

Cost: 33548

Math

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

↓

\[\begin{array}{l} t_1 := t + \frac{t}{x}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_3 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t_1, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+89}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-300}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\ell}{\sqrt{x}}, \sqrt{t_1 \cdot \left(t \cdot 2\right)}\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

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 (/ t x)))
        (t_2 (+ 2.0 (+ (/ 2.0 x) (/ 2.0 x))))
        (t_3 (* t (sqrt (/ 1.0 (fma t t_1 (/ l (/ x l))))))))
   (if (<= t -6.6e+89)
     (- -1.0 (/ -1.0 x))
     (if (<= t -7.5e-155)
       t_3
       (if (<= t 4.2e-300)
         (*
          t
          (/
           (sqrt 2.0)
           (hypot (* (sqrt 2.0) (/ l (sqrt x))) (sqrt (* t_1 (* t 2.0))))))
         (if (<= t 8.5e-156)
           (*
            t
            (/
             (sqrt 2.0)
             (fma t (sqrt t_2) (* (sqrt (/ 1.0 t_2)) (* (/ l x) (/ l t))))))
           (if (<= t 8.2e+68)
             t_3
             (+
              1.0
              (+ (/ -1.0 x) (+ (/ 0.5 (* x x)) (/ -0.5 (pow x 3.0))))))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t + (t / x);
	double t_2 = 2.0 + ((2.0 / x) + (2.0 / x));
	double t_3 = t * sqrt((1.0 / fma(t, t_1, (l / (x / l)))));
	double tmp;
	if (t <= -6.6e+89) {
		tmp = -1.0 - (-1.0 / x);
	} else if (t <= -7.5e-155) {
		tmp = t_3;
	} else if (t <= 4.2e-300) {
		tmp = t * (sqrt(2.0) / hypot((sqrt(2.0) * (l / sqrt(x))), sqrt((t_1 * (t * 2.0)))));
	} else if (t <= 8.5e-156) {
		tmp = t * (sqrt(2.0) / fma(t, sqrt(t_2), (sqrt((1.0 / t_2)) * ((l / x) * (l / t)))));
	} else if (t <= 8.2e+68) {
		tmp = t_3;
	} else {
		tmp = 1.0 + ((-1.0 / x) + ((0.5 / (x * x)) + (-0.5 / pow(x, 3.0))));
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(t + Float64(t / x))
	t_2 = Float64(2.0 + Float64(Float64(2.0 / x) + Float64(2.0 / x)))
	t_3 = Float64(t * sqrt(Float64(1.0 / fma(t, t_1, Float64(l / Float64(x / l))))))
	tmp = 0.0
	if (t <= -6.6e+89)
		tmp = Float64(-1.0 - Float64(-1.0 / x));
	elseif (t <= -7.5e-155)
		tmp = t_3;
	elseif (t <= 4.2e-300)
		tmp = Float64(t * Float64(sqrt(2.0) / hypot(Float64(sqrt(2.0) * Float64(l / sqrt(x))), sqrt(Float64(t_1 * Float64(t * 2.0))))));
	elseif (t <= 8.5e-156)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(t, sqrt(t_2), Float64(sqrt(Float64(1.0 / t_2)) * Float64(Float64(l / x) * Float64(l / t))))));
	elseif (t <= 8.2e+68)
		tmp = t_3;
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) + Float64(Float64(0.5 / Float64(x * x)) + Float64(-0.5 / (x ^ 3.0)))));
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t + N[(t / x), $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[N[(1.0 / N[(t * t$95$1 + N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.6e+89], N[(-1.0 - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.5e-155], t$95$3, If[LessEqual[t, 4.2e-300], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[Sqrt[N[(t$95$1 * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-156], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * N[Sqrt[t$95$2], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(N[(l / x), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.2e+68], t$95$3, N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.59999999999999948e89

   1. Initial program 76.86

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

   2. Simplified 76.81

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

      [Start] 76.86	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 76.81	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 76.81	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 76.81	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around -inf 4.89

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

   4. Simplified 4.86

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

      [Start] 4.89	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right) \]
      mul-1-neg [=>] 4.89	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
      associate-*l* [=>] 4.86	\[ -\color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      sub-neg [=>] 4.86	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]
      metadata-eval [=>] 4.86	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]
      +-commutative [=>] 4.86	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]
      +-commutative [=>] 4.86	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   5. Applied egg-rr 3.41

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right)} \]

   6. Simplified 3.38

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

      [Start] 3.41	\[ -\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right) \]
      expm1-def [=>] 3.4	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]
      expm1-log1p [=>] 3.38	\[ -\color{blue}{\sqrt{\frac{-1 + x}{x + 1}}} \]

   7. Taylor expanded in x around inf 4.41

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

   if -6.59999999999999948e89 < t < -7.5000000000000006e-155 or 8.5e-156 < t < 8.1999999999999998e68

   1. Initial program 42.14

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

   2. Simplified 42.05

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

      [Start] 42.14	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 42.05	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 42.05	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 42.05	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 16.12

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

   4. Simplified 16.12

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

      [Start] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 16.12	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 16.63

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

   6. Simplified 16.63

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

      [Start] 16.63	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 16.63	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 8.31

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}}} \cdot t \]

   8. Simplified 8.02

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

      [Start] 8.31	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 8.02	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 8.02	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 8.03	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 8.03	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right) + \frac{\ell}{x} \cdot \ell}}} \cdot t \]
      fma-udef [=>] 8.03	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \frac{\ell}{x} \cdot \ell\right)} + \frac{\ell}{x} \cdot \ell}} \cdot t \]
      associate-+l+ [=>] 8.02	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      fma-udef [=>] 8.02	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      distribute-rgt-out [=>] 8.02	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      count-2 [=>] 8.02	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      *-commutative [=>] 8.02	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]

   9. Applied egg-rr 7.82

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

   if -7.5000000000000006e-155 < t < 4.20000000000000007e-300

   1. Initial program 97.64

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

   2. Simplified 97.64

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

      [Start] 97.64	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 97.64	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 97.64	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 97.64	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 54

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

   4. Simplified 54

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

      [Start] 54	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 54

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

   6. Simplified 54

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

      [Start] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 54	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   7. Applied egg-rr 51.03

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}}} \cdot t \]

   8. Simplified 51.01

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

      [Start] 51.03	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25} \cdot {\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{0.25}} \cdot t \]
      pow-sqr [=>] 51.01	\[ \frac{\sqrt{2}}{\color{blue}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\left(2 \cdot 0.25\right)}}} \cdot t \]
      metadata-eval [=>] 51.01	\[ \frac{\sqrt{2}}{{\left(\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)\right)}^{\color{blue}{0.5}}} \cdot t \]
      unpow1/2 [=>] 51.01	\[ \frac{\sqrt{2}}{\color{blue}{\sqrt{\frac{\ell}{x} \cdot \ell + \mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      +-commutative [=>] 51.01	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right), \frac{\ell}{x} \cdot \ell\right) + \frac{\ell}{x} \cdot \ell}}} \cdot t \]
      fma-udef [=>] 51.01	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \frac{\ell}{x} \cdot \ell\right)} + \frac{\ell}{x} \cdot \ell}} \cdot t \]
      associate-+l+ [=>] 51.01	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      fma-udef [=>] 51.01	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      distribute-rgt-out [=>] 51.01	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left(t \cdot \left(t + \frac{t}{x}\right)\right)} + \left(\frac{\ell}{x} \cdot \ell + \frac{\ell}{x} \cdot \ell\right)}} \cdot t \]
      count-2 [=>] 51.01	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \cdot t \]
      *-commutative [=>] 51.01	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{x}\right)}}} \cdot t \]

   9. Applied egg-rr 40.25

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

   if 4.20000000000000007e-300 < t < 8.5e-156

   1. Initial program 97.57

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

   2. Simplified 97.57

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

      [Start] 97.57	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 97.57	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 97.57	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 97.57	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 53.11

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

   4. Simplified 53.11

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

      [Start] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 53.11	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around inf 37.84

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

   6. Simplified 37.94

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

      [Start] 37.84	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      *-commutative [<=] 37.84	\[ \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      fma-def [=>] 37.84	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      distribute-lft-in [=>] 37.84	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 37.84	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 37.84	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 37.84	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 37.84	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 37.84	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 37.84	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]

   if 8.1999999999999998e68 < t

   1. Initial program 72.15

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

   2. Simplified 72.09

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

      [Start] 72.15	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 72.09	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 72.09	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 72.09	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in t around -inf 98.42

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

   4. Simplified 98.42

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

      [Start] 98.42	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}\right) \]
      mul-1-neg [=>] 98.42	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
      associate-*l* [=>] 98.42	\[ -\color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]
      sub-neg [=>] 98.42	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]
      metadata-eval [=>] 98.42	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]
      +-commutative [=>] 98.42	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]
      +-commutative [=>] 98.42	\[ -\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   5. Applied egg-rr 98.42

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right)} \]

   6. Simplified 98.42

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

      [Start] 98.42	\[ -\left(e^{\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)} - 1\right) \]
      expm1-def [=>] 98.42	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]
      expm1-log1p [=>] 98.42	\[ -\color{blue}{\sqrt{\frac{-1 + x}{x + 1}}} \]

   7. Taylor expanded in x around -inf 100

      \[\leadsto -\color{blue}{\left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + \left(-0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right)\right)} \]

   8. Simplified 4.92

      \[\leadsto -\color{blue}{\left(-1 + \left(\frac{1}{x} + \left(\frac{-0.5}{x \cdot x} + \frac{0.5}{{x}^{3}}\right)\right)\right)} \]
      Proof

      [Start] 100	\[ -\left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + \left(-0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right)\right) \]
      associate-+r+ [=>] 100	\[ -\color{blue}{\left(\left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right) + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right)} \]
      +-commutative [=>] 100	\[ -\color{blue}{\left(\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right) + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right)} \]
      unpow2 [=>] 100	\[ -\left(\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x}\right) + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right) \]
      rem-square-sqrt [=>] 100	\[ -\left(\left(\color{blue}{-1} + \frac{1}{x}\right) + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right) \]
      associate-+l+ [=>] 100	\[ -\color{blue}{\left(-1 + \left(\frac{1}{x} + \left(0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + -0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}}\right)\right)\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 12.57

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+89}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-300}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\ell}{\sqrt{x}}, \sqrt{\left(t + \frac{t}{x}\right) \cdot \left(t \cdot 2\right)}\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}, \sqrt{\frac{1}{2 + \left(\frac{2}{x} + \frac{2}{x}\right)}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	14.44%
Cost	28240

\[\begin{array}{l} t_1 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ t_2 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+90}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-303}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;t \leq 7.3 \cdot 10^{-156}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{t_1}, \sqrt{\frac{1}{t_1}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 2
Error	14.42%
Cost	27600

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+90}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-302}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \left(\ell \cdot \ell + t \cdot t\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 3
Error	15.16%
Cost	14672

\[\begin{array}{l} t_1 := t + \frac{t}{x}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+89}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-211}:\\ \;\;\;\;t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t_1, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+69}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(t \cdot t_1\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 4
Error	15.29%
Cost	14288

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{1}{\mathsf{fma}\left(t, t + \frac{t}{x}, \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+91}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-298}:\\ \;\;\;\;\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 5
Error	22.69%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-200}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-299}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \left(\frac{0.5}{x \cdot x} + \frac{-0.5}{{x}^{3}}\right)\right)\\ \end{array} \]

Alternative 6
Error	22.75%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-200}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-299}:\\ \;\;\;\;t \cdot \left(\sqrt{x} \cdot \frac{1}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 7
Error	23.05%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-200}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 8
Error	22.79%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -7.3 \cdot 10^{-200}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{-300}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 9
Error	23.36%
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-258}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq 3.75 \cdot 10^{-302}:\\ \;\;\;\;\frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5}{x \cdot x}\right)\\ \end{array} \]

Alternative 10
Error	23.84%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-300}:\\ \;\;\;\;\frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 11
Error	23.48%
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-259}:\\ \;\;\;\;-1 - \frac{-1}{x}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-303}:\\ \;\;\;\;\frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Error	24.18%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.04 \cdot 10^{-299}:\\ \;\;\;\;-\frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	24.14%
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-258}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-301}:\\ \;\;\;\;\frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14
Error	24.15%
Cost	196

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

Alternative 15
Error	60.86%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023115 
(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)))))