Toniolo and Linder, Equation (7)

Average Error: 43.3 → 7.5

Time: 42.8s

Precision: binary64

Cost: 33944

Math

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

↓

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ t_2 := \mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)\\ t_3 := t \cdot \sqrt{2}\\ t_4 := 1 + \frac{2}{x}\\ \mathbf{if}\;t \leq -1.8141869063142166 \cdot 10^{+21}:\\ \;\;\;\;\frac{t_3}{t_3 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_3}{\sqrt{2} \cdot \sqrt{t_2}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;\frac{t_3}{-\mathsf{fma}\left(t, \sqrt{2 \cdot t_4}, \sqrt{\frac{0.5}{t_4}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_3}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_3}{\left(\ell \cdot \frac{\frac{\ell}{t}}{x}\right) \cdot \sqrt{\frac{1}{t_1}} + t \cdot \sqrt{t_1}}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;{\left(\sqrt{t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot t_2}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

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 (+ 2.0 (/ 4.0 x)))
        (t_2 (fma t (+ t (/ t x)) (* l (/ l x))))
        (t_3 (* t (sqrt 2.0)))
        (t_4 (+ 1.0 (/ 2.0 x))))
   (if (<= t -1.8141869063142166e+21)
     (/ t_3 (* t_3 (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t -1.6e-164)
       (/ t_3 (* (sqrt 2.0) (sqrt t_2)))
       (if (<= t -3.2e-271)
         (/
          t_3
          (-
           (fma
            t
            (sqrt (* 2.0 t_4))
            (* (sqrt (/ 0.5 t_4)) (/ (* l (/ l t)) x)))))
         (if (<= t 3.6e-266)
           (/ t_3 (fabs (* l (sqrt (/ 2.0 x)))))
           (if (<= t 4.2e-163)
             (/
              t_3
              (+ (* (* l (/ (/ l t) x)) (sqrt (/ 1.0 t_1))) (* t (sqrt t_1))))
             (if (<= t 4.299023181944863e+110)
               (pow (sqrt (* t (/ (sqrt 2.0) (sqrt (* 2.0 t_2))))) 2.0)
               (sqrt (* 2.0 (/ (* 0.5 (+ x -1.0)) (+ x 1.0))))))))))))

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 = 2.0 + (4.0 / x);
	double t_2 = fma(t, (t + (t / x)), (l * (l / x)));
	double t_3 = t * sqrt(2.0);
	double t_4 = 1.0 + (2.0 / x);
	double tmp;
	if (t <= -1.8141869063142166e+21) {
		tmp = t_3 / (t_3 * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= -1.6e-164) {
		tmp = t_3 / (sqrt(2.0) * sqrt(t_2));
	} else if (t <= -3.2e-271) {
		tmp = t_3 / -fma(t, sqrt((2.0 * t_4)), (sqrt((0.5 / t_4)) * ((l * (l / t)) / x)));
	} else if (t <= 3.6e-266) {
		tmp = t_3 / fabs((l * sqrt((2.0 / x))));
	} else if (t <= 4.2e-163) {
		tmp = t_3 / (((l * ((l / t) / x)) * sqrt((1.0 / t_1))) + (t * sqrt(t_1)));
	} else if (t <= 4.299023181944863e+110) {
		tmp = pow(sqrt((t * (sqrt(2.0) / sqrt((2.0 * t_2))))), 2.0);
	} else {
		tmp = sqrt((2.0 * ((0.5 * (x + -1.0)) / (x + 1.0))));
	}
	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(2.0 + Float64(4.0 / x))
	t_2 = fma(t, Float64(t + Float64(t / x)), Float64(l * Float64(l / x)))
	t_3 = Float64(t * sqrt(2.0))
	t_4 = Float64(1.0 + Float64(2.0 / x))
	tmp = 0.0
	if (t <= -1.8141869063142166e+21)
		tmp = Float64(t_3 / Float64(t_3 * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= -1.6e-164)
		tmp = Float64(t_3 / Float64(sqrt(2.0) * sqrt(t_2)));
	elseif (t <= -3.2e-271)
		tmp = Float64(t_3 / Float64(-fma(t, sqrt(Float64(2.0 * t_4)), Float64(sqrt(Float64(0.5 / t_4)) * Float64(Float64(l * Float64(l / t)) / x)))));
	elseif (t <= 3.6e-266)
		tmp = Float64(t_3 / abs(Float64(l * sqrt(Float64(2.0 / x)))));
	elseif (t <= 4.2e-163)
		tmp = Float64(t_3 / Float64(Float64(Float64(l * Float64(Float64(l / t) / x)) * sqrt(Float64(1.0 / t_1))) + Float64(t * sqrt(t_1))));
	elseif (t <= 4.299023181944863e+110)
		tmp = sqrt(Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * t_2))))) ^ 2.0;
	else
		tmp = sqrt(Float64(2.0 * Float64(Float64(0.5 * Float64(x + -1.0)) / Float64(x + 1.0))));
	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[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.8141869063142166e+21], N[(t$95$3 / N[(t$95$3 * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.6e-164], N[(t$95$3 / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.2e-271], N[(t$95$3 / (-N[(t * N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(0.5 / t$95$4), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 3.6e-266], N[(t$95$3 / N[Abs[N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-163], N[(t$95$3 / N[(N[(N[(l * N[(N[(l / t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.299023181944863e+110], N[Power[N[Sqrt[N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], N[Sqrt[N[(2.0 * N[(N[(0.5 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -1.8141869063142166e21

   1. Initial program 42.8

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

   2. Taylor expanded in t around -inf 4.5

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

   3. Simplified 4.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-t \cdot \sqrt{2}\right)}} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 (+.f64 x 1) (+.f64 x -1))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 x)) (+.f64 x -1))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (Rewrite<= sub-neg_binary64 (-.f64 x 1)))) (neg.f64 (*.f64 t (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1))) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1))) (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (/.f64 (+.f64 1 x) (-.f64 x 1)))))): 0 points increase in error, 0 points decrease in error

   if -1.8141869063142166e21 < t < -1.6e-164

   1. Initial program 32.0

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

   2. Taylor expanded in x around inf 10.9

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

   3. Simplified 10.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 11.2

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

   5. Simplified 5.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      Proof
      (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 20 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 33 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in l around 0 11.2

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

   7. Simplified 5.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell}{x} \cdot \ell\right)}}} \]
      Proof
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (/.f64 t x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 x) t)))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) (*.f64 (/.f64 1 x) t))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (Rewrite<= distribute-rgt-in_binary64 (*.f64 t (+.f64 1 (/.f64 1 x))))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 2 points decrease in error
      (*.f64 2 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t t) (+.f64 1 (/.f64 1 x)))) (*.f64 (/.f64 l x) l))): 2 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 1 (/.f64 1 x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))))): 9 points increase in error, 3 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)))): 17 points increase in error, 2 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 1 x) 1))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (pow.f64 t 2) (/.f64 1 x)) (*.f64 (pow.f64 t 2) 1))) (/.f64 (pow.f64 l 2) x))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 t 2) 1) x)) (*.f64 (pow.f64 t 2) 1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (pow.f64 t 2)) x) (*.f64 (pow.f64 t 2) 1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite=> *-rgt-identity_binary64 (pow.f64 t 2))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 l 2) x)) (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 5.2

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

   if -1.6e-164 < t < -3.19999999999999978e-271

   1. Initial program 62.7

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

   2. Taylor expanded in x around inf 36.1

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

   3. Simplified 36.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around -inf 23.0

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

   5. Simplified 22.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}, \sqrt{\frac{0.5}{1 + \frac{2}{x}}} \cdot \frac{\frac{\ell}{t} \cdot \ell}{x}\right)}} \]
      Proof
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 2 x)))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 1 points increase in error, 1 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 1 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 2 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 l t) (/.f64 l x)))))): 3 points increase in error, 17 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 t x)))))): 33 points increase in error, 12 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t)) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x))) (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))) (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (+.f64 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))) (Rewrite=> *-commutative_binary64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))) (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))) (*.f64 -1 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error

   if -3.19999999999999978e-271 < t < 3.5999999999999999e-266

   1. Initial program 63.0

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

   2. Taylor expanded in x around inf 30.6

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

   3. Simplified 30.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 30.6

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

   5. Simplified 30.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      Proof
      (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 20 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 33 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in l around inf 30.6

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

   7. Simplified 30.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{2}{x} \cdot \left(\ell \cdot \ell\right)}}} \]
      Proof
      (*.f64 (/.f64 2 x) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 x) (Rewrite<= unpow2_binary64 (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 x (pow.f64 l 2)))): 16 points increase in error, 14 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 (pow.f64 l 2)) x)): 12 points increase in error, 11 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 l 2) x))): 1 points increase in error, 1 points decrease in error

   8. Applied egg-rr 18.8

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

   if 3.5999999999999999e-266 < t < 4.19999999999999996e-163

   1. Initial program 63.4

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

   2. Taylor expanded in x around inf 34.5

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

   3. Simplified 34.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around inf 23.2

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

   5. Simplified 23.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}, \sqrt{\frac{0.5}{1 + \frac{2}{x}}} \cdot \frac{\frac{\ell}{t} \cdot \ell}{x}\right)}} \]
      Proof
      (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 2 x)))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 1 points increase in error, 1 points decrease in error
      (fma.f64 t (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))) (/.f64 (*.f64 (/.f64 l t) l) x))): 1 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 2 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 l t) (/.f64 l x))))): 0 points increase in error, 20 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 t x))))): 35 points increase in error, 10 points decrease in error
      (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t)) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 23.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\frac{\frac{\ell}{t}}{x} \cdot \ell\right) \cdot \sqrt{\frac{1}{2 + \frac{4}{x}}} + t \cdot \sqrt{2 + \frac{4}{x}}}} \]

   if 4.19999999999999996e-163 < t < 4.2990231819448629e110

   1. Initial program 26.2

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

   2. Taylor expanded in x around inf 9.8

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

   3. Simplified 9.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 10.2

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

   5. Simplified 5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      Proof
      (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 20 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 33 points increase in error, 10 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in l around 0 10.2

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

   7. Simplified 5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell}{x} \cdot \ell\right)}}} \]
      Proof
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (/.f64 t x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 x) t)))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) (*.f64 (/.f64 1 x) t))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (Rewrite<= distribute-rgt-in_binary64 (*.f64 t (+.f64 1 (/.f64 1 x))))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 2 points decrease in error
      (*.f64 2 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t t) (+.f64 1 (/.f64 1 x)))) (*.f64 (/.f64 l x) l))): 2 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 1 (/.f64 1 x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))))): 9 points increase in error, 3 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)))): 17 points increase in error, 2 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 1 x) 1))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (pow.f64 t 2) (/.f64 1 x)) (*.f64 (pow.f64 t 2) 1))) (/.f64 (pow.f64 l 2) x))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 t 2) 1) x)) (*.f64 (pow.f64 t 2) 1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (pow.f64 t 2)) x) (*.f64 (pow.f64 t 2) 1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite=> *-rgt-identity_binary64 (pow.f64 t 2))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 l 2) x)) (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 5.2

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

   if 4.2990231819448629e110 < t

   1. Initial program 53.2

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

   2. Taylor expanded in t around -inf 63.0

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

   3. Simplified 63.0

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{x + -1}{x + 1}}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (neg.f64 (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x -1) (+.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (neg.f64 (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (neg.f64 (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 x 1)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (neg.f64 (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 1 x))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x))))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x)))))): 12 points increase in error, 1 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 2.3

      \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{0.5 \cdot \left(x - 1\right)}{x + 1}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8141869063142166 \cdot 10^{+21}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}, \sqrt{\frac{0.5}{1 + \frac{2}{x}}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(\ell \cdot \frac{\frac{\ell}{t}}{x}\right) \cdot \sqrt{\frac{1}{2 + \frac{4}{x}}} + t \cdot \sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;{\left(\sqrt{t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	27788

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{t_2}{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ t_4 := 1 + \frac{2}{x}\\ \mathbf{if}\;t \leq -1.8141869063142166 \cdot 10^{+21}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;\frac{t_2}{-\mathsf{fma}\left(t, \sqrt{2 \cdot t_4}, \sqrt{\frac{0.5}{t_4}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_2}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_2}{\left(\ell \cdot \frac{\frac{\ell}{t}}{x}\right) \cdot \sqrt{\frac{1}{t_1}} + t \cdot \sqrt{t_1}}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 2
Error	8.2
Cost	27480

\[\begin{array}{l} t_1 := 2 + \frac{4}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{t_2}{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(t, t + \frac{t}{x}, \ell \cdot \frac{\ell}{x}\right)}}\\ t_4 := \sqrt{t_1}\\ \mathbf{if}\;t \leq -1.8141869063142166 \cdot 10^{+21}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;\frac{t_2}{t \cdot \left(-{\left(\sqrt[3]{t_4}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_2}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_2}{\left(\ell \cdot \frac{\frac{\ell}{t}}{x}\right) \cdot \sqrt{\frac{1}{t_1}} + t \cdot t_4}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 3
Error	8.4
Cost	26828

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := 2 + \frac{4}{x}\\ t_4 := \sqrt{t_3}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, t_1\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;\frac{t_2}{t \cdot \left(-{\left(\sqrt[3]{t_4}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_2}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_2}{\left(\ell \cdot \frac{\frac{\ell}{t}}{x}\right) \cdot \sqrt{\frac{1}{t_3}} + t \cdot t_4}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_2 \cdot {\left(2 \cdot \mathsf{fma}\left(t, t + \frac{t}{x}, t_1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 4
Error	8.4
Cost	21588

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, t_1\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_2}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{t_2}{\left(\ell \cdot \frac{\frac{\ell}{t}}{x}\right) \cdot \sqrt{\frac{1}{t_3}} + t \cdot \sqrt{t_3}}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_2 \cdot {\left(2 \cdot \mathsf{fma}\left(t, t + \frac{t}{x}, t_1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 5
Error	9.3
Cost	21144

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ t_3 := t + \frac{t}{x}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t_2 + t \cdot t_3\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_1}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, \frac{\ell}{t} \cdot \frac{\ell}{t}, 1\right)}{x}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_1 \cdot {\left(2 \cdot \mathsf{fma}\left(t, t_3, t_2\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 6
Error	9.3
Cost	21144

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, t_1\right)}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_2}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, \frac{\ell}{t} \cdot \frac{\ell}{t}, 1\right)}{x}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_2 \cdot {\left(2 \cdot \mathsf{fma}\left(t, t + \frac{t}{x}, t_1\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 7
Error	9.3
Cost	20820

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_3 := \sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_1}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, \frac{\ell}{t} \cdot \frac{\ell}{t}, 1\right)}{x}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Error	9.3
Cost	20820

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_1}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;1 - \frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, \frac{\ell}{t} \cdot \frac{\ell}{t}, 1\right)}{x}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 9
Error	8.7
Cost	20304

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_3 := \sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-189}:\\ \;\;\;\;\frac{t_1}{\left|\ell \cdot \sqrt{\frac{2}{x}}\right|}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t_1}\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	9.8
Cost	14808

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_3 := \sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{if}\;t \leq -84532766400.66748:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-275}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{2}{\frac{x}{\ell \cdot \ell}}}}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	14.6
Cost	14684

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{2}{\frac{x}{\ell \cdot \ell}}}}\\ t_3 := \sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{if}\;t \leq -7.3 \cdot 10^{-75}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-275}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{-150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.299023181944863 \cdot 10^{+110}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	14.7
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{if}\;t \leq -7.3 \cdot 10^{-75}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-110}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-275}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2}{\frac{x}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	14.1
Cost	13768

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{-275}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2}{\frac{x}{\ell \cdot \ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	14.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \end{array} \]

Alternative 15
Error	14.6
Cost	7368

\[\begin{array}{l} t_1 := \sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	14.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{1 + \frac{2}{x}}}\\ \end{array} \]

Alternative 17
Error	15.2
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-271}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	14.9
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-294}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-266}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	15.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-295}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 20
Error	15.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-295}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 21
Error	39.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022300 
(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)))))