Toniolo and Linder, Equation (7)

Average Error: 66.2% → 16.41%

Time: 29.5s

Precision: binary64

Cost: 21460

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}{2}}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-160}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-211}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t_1} + t \cdot t_1}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 2.0 (/ 4.0 x))))
        (t_2 (/ (* l l) x))
        (t_3
         (/
          t
          (sqrt
           (/
            (+
             (/ l (/ x l))
             (+
              (/ (fma 2.0 (* t t) (* l l)) x)
              (* 2.0 (+ (* t t) (/ t (/ x t))))))
            2.0)))))
   (if (<= t -1.5e+79)
     (+ -1.0 (+ (/ 1.0 x) (* (/ 1.0 (* x x)) (+ (/ 0.5 x) -0.5))))
     (if (<= t -4.1e-160)
       t_3
       (if (<= t -1.4e-211)
         -1.0
         (if (<= t 1.55e-268)
           (*
            t
            (/
             (sqrt 2.0)
             (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x))))))))
           (if (<= t 2.6e-159)
             (* t (/ (sqrt 2.0) (+ (* (/ l t) (/ (/ l x) t_1)) (* t t_1))))
             (if (<= t 2.1e+23)
               t_3
               (+ 1.0 (+ (/ (/ 0.5 x) x) (/ -1.0 x)))))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = sqrt((2.0 + (4.0 / x)));
	double t_2 = (l * l) / x;
	double t_3 = t / sqrt((((l / (x / l)) + ((fma(2.0, (t * t), (l * l)) / x) + (2.0 * ((t * t) + (t / (x / t)))))) / 2.0));
	double tmp;
	if (t <= -1.5e+79) {
		tmp = -1.0 + ((1.0 / x) + ((1.0 / (x * x)) * ((0.5 / x) + -0.5)));
	} else if (t <= -4.1e-160) {
		tmp = t_3;
	} else if (t <= -1.4e-211) {
		tmp = -1.0;
	} else if (t <= 1.55e-268) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	} else if (t <= 2.6e-159) {
		tmp = t * (sqrt(2.0) / (((l / t) * ((l / x) / t_1)) + (t * t_1)));
	} else if (t <= 2.1e+23) {
		tmp = t_3;
	} else {
		tmp = 1.0 + (((0.5 / x) / x) + (-1.0 / x));
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = sqrt(Float64(2.0 + Float64(4.0 / x)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(t / sqrt(Float64(Float64(Float64(l / Float64(x / l)) + Float64(Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x) + Float64(2.0 * Float64(Float64(t * t) + Float64(t / Float64(x / t)))))) / 2.0)))
	tmp = 0.0
	if (t <= -1.5e+79)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) + Float64(Float64(1.0 / Float64(x * x)) * Float64(Float64(0.5 / x) + -0.5))));
	elseif (t <= -4.1e-160)
		tmp = t_3;
	elseif (t <= -1.4e-211)
		tmp = -1.0;
	elseif (t <= 1.55e-268)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))));
	elseif (t <= 2.6e-159)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(Float64(l / t) * Float64(Float64(l / x) / t_1)) + Float64(t * t_1))));
	elseif (t <= 2.1e+23)
		tmp = t_3;
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(0.5 / x) / x) + Float64(-1.0 / x)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(t / N[Sqrt[N[(N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+79], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] + N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.1e-160], t$95$3, If[LessEqual[t, -1.4e-211], -1.0, If[LessEqual[t, 1.55e-268], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e-159], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(l / t), $MachinePrecision] * N[(N[(l / x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+23], t$95$3, N[(1.0 + N[(N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.49999999999999987e79

   1. Initial program 74.06

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

   2. Simplified 74.01

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

      [Start] 74.06	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 74.01	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 74.01	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 74.01	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 73.88

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

   4. Taylor expanded in t around inf 98.42

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

   5. Taylor expanded in x around -inf 100

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

   6. Simplified 5.09

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

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

   if -1.49999999999999987e79 < t < -4.10000000000000002e-160 or 2.5999999999999998e-159 < t < 2.1000000000000001e23

   1. Initial program 42.97

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

   2. Simplified 42.86

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

      [Start] 42.97	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 42.86	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 42.86	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 42.86	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 42.58

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

   4. Taylor expanded in x around inf 13.88

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

   5. Simplified 13.89

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

      [Start] 13.88	\[ \frac{t}{\sqrt{\frac{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}{2}}} \]
      cancel-sign-sub-inv [=>] 13.88	\[ \frac{t}{\sqrt{\frac{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}{2}}} \]
      associate-+l+ [=>] 13.88	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}{2}}} \]
      unpow2 [=>] 13.88	\[ \frac{t}{\sqrt{\frac{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      distribute-lft-out [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      unpow2 [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      associate-/l* [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \color{blue}{\frac{t}{\frac{x}{t}}}\right) + \left(--1\right) \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      metadata-eval [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + \color{blue}{1} \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}{2}}} \]
      +-commutative [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)}{2}}} \]
      unpow2 [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)}{2}}} \]
      fma-def [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)}{2}}} \]
      unpow2 [=>] 13.89	\[ \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right) + 1 \cdot \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)}{2}}} \]

   if -4.10000000000000002e-160 < t < -1.3999999999999999e-211

   1. Initial program 98.18

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

   2. Simplified 98.18

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

      [Start] 98.18	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 98.18	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 98.18	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 98.18	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 98.18

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

   4. Taylor expanded in t around inf 97.98

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

   5. Taylor expanded in x around -inf 100

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

   6. Simplified 49.49

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

      [Start] 100	\[ {\left(\sqrt{-1}\right)}^{2} \]
      unpow2 [=>] 100	\[ \color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]
      rem-square-sqrt [=>] 49.49	\[ \color{blue}{-1} \]

   if -1.3999999999999999e-211 < t < 1.5499999999999999e-268

   1. Initial program 97.93

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

   2. Simplified 97.93

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

      [Start] 97.93	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 97.93	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 97.93	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 97.93	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 50.3

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

   4. Simplified 50.3

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

      [Start] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around 0 50.3

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

   6. Simplified 50.3

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

      [Start] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{{\ell}^{2}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 50.3	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]

   if 1.5499999999999999e-268 < t < 2.5999999999999998e-159

   1. Initial program 98.29

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

   2. Simplified 98.29

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

      [Start] 98.29	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 98.29	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 98.29	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 98.29	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Taylor expanded in x around inf 52.99

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

   4. Simplified 52.99

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

      [Start] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]
      associate--l+ [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]
      unpow2 [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      distribute-lft-out [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      unpow2 [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]
      mul-1-neg [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\left(-\frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}\right)}} \cdot t \]
      +-commutative [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{2 \cdot {t}^{2} + {\ell}^{2}}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}}{x}\right)\right)}} \cdot t \]
      fma-udef [<=] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}\right)\right)}} \cdot t \]
      unpow2 [=>] 52.99	\[ \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \left(-\frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}\right)\right)}} \cdot t \]

   5. Taylor expanded in t around inf 38.01

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

   6. Simplified 37.8

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

      [Start] 38.01	\[ \frac{\sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}} \cdot t + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      *-commutative [<=] 38.01	\[ \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} + \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}} \cdot t \]
      fma-def [=>] 38	\[ \frac{\sqrt{2}}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)}} \cdot t \]
      distribute-lft-in [=>] 38	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 38	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-+l+ [=>] 38	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{\color{blue}{2 + \left(2 \cdot \frac{1}{x} + 2 \cdot \frac{1}{x}\right)}}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 38	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\color{blue}{\frac{2 \cdot 1}{x}} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 38	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{\color{blue}{2}}{x} + 2 \cdot \frac{1}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      associate-*r/ [=>] 38	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \color{blue}{\frac{2 \cdot 1}{x}}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]
      metadata-eval [=>] 38	\[ \frac{\sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2 + \left(\frac{2}{x} + \frac{\color{blue}{2}}{x}\right)}, \sqrt{\frac{1}{2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}}} \cdot \frac{{\ell}^{2}}{t \cdot x}\right)} \cdot t \]

   7. Applied egg-rr 37.81

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

   if 2.1000000000000001e23 < t

   1. Initial program 64.77

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

   2. Simplified 64.72

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

      [Start] 64.77	\[ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
      associate-*l/ [<=] 64.72	\[ \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]
      +-commutative [=>] 64.72	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]
      fma-def [=>] 64.72	\[ \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \cdot t \]

   3. Applied egg-rr 64.55

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

   4. Taylor expanded in t around inf 7.18

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

   5. Taylor expanded in x around inf 7.78

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

   6. Simplified 7.78

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

      [Start] 7.78	\[ \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x} \]
      sub-neg [=>] 7.78	\[ \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) + \left(-\frac{1}{x}\right)} \]
      associate-+l+ [=>] 7.78	\[ \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\frac{1}{x}\right)\right)} \]
      sub-neg [<=] 7.78	\[ 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]
      associate-*r/ [=>] 7.78	\[ 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]
      metadata-eval [=>] 7.78	\[ 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]
      unpow2 [=>] 7.78	\[ 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]
      associate-/r* [=>] 7.78	\[ 1 + \left(\color{blue}{\frac{\frac{0.5}{x}}{x}} - \frac{1}{x}\right) \]

3. Recombined 6 regimes into one program.

4. Final simplification 16.41

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-211}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{\sqrt{2 + \frac{4}{x}}} + t \cdot \sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	17.38%
Cost	69324

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ t_2 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_3}, \sqrt{\frac{1}{t_3}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-157}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t_1} + t \cdot t_1}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \mathsf{fma}\left(-1, \frac{\left(-t_2\right) - t_2}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right)\right) + \left(\frac{t_2}{{x}^{3}} + \frac{t_2}{x}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 2
Error	17.45%
Cost	54540

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ t_2 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_3 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_3}, \sqrt{\frac{1}{t_3}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-159}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t_1} + t \cdot t_1}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, \mathsf{fma}\left(2, \frac{t_2}{x}, 2 \cdot \left(t \cdot t + \frac{t_2}{{x}^{3}}\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 3
Error	17.4%
Cost	34636

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-160}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t_1} + t \cdot t_1}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, 2 \cdot \left(t \cdot t + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 4
Error	17.24%
Cost	27908

\[\begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ t_2 := 2 + \left(\frac{2}{x} + \frac{2}{x}\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \left(\frac{\ell}{x} \cdot \frac{\ell}{t}\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \frac{\frac{\ell}{x}}{t_1} + t \cdot t_1}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 5
Error	17.64%
Cost	15576

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := \frac{t}{\sqrt{\frac{\frac{\ell}{\frac{x}{\ell}} + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + 2 \cdot \left(t \cdot t + \frac{t}{\frac{x}{t}}\right)\right)}{2}}}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+79}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.95 \cdot 10^{-209}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{t}{\mathsf{expm1}\left(\mathsf{log1p}\left(t\right)\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 6
Error	17.9%
Cost	15320

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(t_1 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -3700:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -1.26 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-210}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-268}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-175}:\\ \;\;\;\;\frac{t}{\mathsf{expm1}\left(\mathsf{log1p}\left(t\right)\right)}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 7
Error	22.45%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+225}:\\ \;\;\;\;\frac{\frac{\sqrt{2 \cdot x}}{\sqrt{2}}}{\ell} \cdot \left(-t\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\sqrt{-0.5 + x \cdot 0.5} \cdot \frac{\sqrt{2}}{\ell}\right)\\ \end{array} \]

Alternative 8
Error	22.41%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+225}:\\ \;\;\;\;\frac{\frac{\sqrt{2 \cdot x}}{\sqrt{2}}}{\ell} \cdot \left(-t\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+97}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 9
Error	22.49%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+225}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(t, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \end{array} \]

Alternative 10
Error	22.43%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-209}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 11
Error	21.79%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-211}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 12
Error	21.57%
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-210}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-268}:\\ \;\;\;\;t \cdot \sqrt{\frac{\frac{x}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	22.61%
Cost	6984

\[\begin{array}{l} t_1 := \frac{1}{x \cdot x}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-212}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + t_1 \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 + \frac{-0.5}{x}\right) + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 14
Error	22.63%
Cost	6984

\[\begin{array}{l} t_1 := \frac{1}{x \cdot x}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-212}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + t_1 \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 + \frac{-0.5}{x}\right) + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 15
Error	23.37%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;t \leq 1.85 \cdot 10^{-305}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 16
Error	23.32%
Cost	1220

\[\begin{array}{l} t_1 := \frac{1}{x \cdot x}\\ \mathbf{if}\;t \leq 1.85 \cdot 10^{-305}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + t_1 \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(0.5 + \frac{-0.5}{x}\right) + \left(1 + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 17
Error	23.64%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-299}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 18
Error	23.53%
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 19
Error	24.09%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-299}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	23.75%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-299}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 21
Error	24.44%
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 22
Error	61.49%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023089 
(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)))))