Toniolo and Linder, Equation (7)

Average Error: 42.6 → 9.0

Time: 32.4s

Precision: binary64

Cost: 54288

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := t \cdot \left(t \cdot 2\right)\\ t_4 := \sqrt[3]{\mathsf{fma}\left(\ell, \ell, t_3\right)}\\ t_5 := 1 + \frac{2}{x}\\ t_6 := \sqrt{2 \cdot t_5}\\ t_7 := \sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + t_1\right)}\\ \mathbf{if}\;t \leq -1.1292640349786462 \cdot 10^{+46}:\\ \;\;\;\;\frac{t_2}{-\mathsf{fma}\left(t, t_6, \sqrt{\frac{0.5}{t_5}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{t_2}{t_7}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-184}:\\ \;\;\;\;\frac{t_2}{t_6 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-190}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, {t_4}^{2} \cdot \left(t_4 \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-272}:\\ \;\;\;\;\frac{t_2}{\sqrt{2 \cdot t_1}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 2.2960935021905734 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sqrt{t_3}}{t_7}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

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 (* l (/ l x)))
        (t_2 (* t (sqrt 2.0)))
        (t_3 (* t (* t 2.0)))
        (t_4 (cbrt (fma l l t_3)))
        (t_5 (+ 1.0 (/ 2.0 x)))
        (t_6 (sqrt (* 2.0 t_5)))
        (t_7 (sqrt (* 2.0 (+ (* t (+ t (/ t x))) t_1)))))
   (if (<= t -1.1292640349786462e+46)
     (/ t_2 (- (fma t t_6 (* (sqrt (/ 0.5 t_5)) (/ (* l (/ l t)) x)))))
     (if (<= t -8.6e-151)
       (/ t_2 t_7)
       (if (<= t -2.1e-184)
         (/ t_2 (* t_6 (- t)))
         (if (<= t -1e-190)
           (/
            t_2
            (sqrt
             (+
              (/ l (/ x l))
              (fma
               2.0
               (+ (/ t (/ x t)) (* t t))
               (* (pow t_4 2.0) (* t_4 (/ 1.0 x)))))))
           (if (<= t 5.4e-272)
             (/ t_2 (sqrt (* 2.0 t_1)))
             (if (<= t 4.5e-150)
               (*
                (sqrt 2.0)
                (/
                 t
                 (fma
                  t
                  (sqrt 2.0)
                  (*
                   (/ 0.5 (sqrt 2.0))
                   (/ (* 2.0 (fma 2.0 (* t t) (* l l))) (* t x))))))
               (if (<= t 2.2960935021905734e+108)
                 (/ (sqrt t_3) t_7)
                 (sqrt (/ (+ x -1.0) (+ 1.0 x))))))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = l * (l / x);
	double t_2 = t * sqrt(2.0);
	double t_3 = t * (t * 2.0);
	double t_4 = cbrt(fma(l, l, t_3));
	double t_5 = 1.0 + (2.0 / x);
	double t_6 = sqrt((2.0 * t_5));
	double t_7 = sqrt((2.0 * ((t * (t + (t / x))) + t_1)));
	double tmp;
	if (t <= -1.1292640349786462e+46) {
		tmp = t_2 / -fma(t, t_6, (sqrt((0.5 / t_5)) * ((l * (l / t)) / x)));
	} else if (t <= -8.6e-151) {
		tmp = t_2 / t_7;
	} else if (t <= -2.1e-184) {
		tmp = t_2 / (t_6 * -t);
	} else if (t <= -1e-190) {
		tmp = t_2 / sqrt(((l / (x / l)) + fma(2.0, ((t / (x / t)) + (t * t)), (pow(t_4, 2.0) * (t_4 * (1.0 / x))))));
	} else if (t <= 5.4e-272) {
		tmp = t_2 / sqrt((2.0 * t_1));
	} else if (t <= 4.5e-150) {
		tmp = sqrt(2.0) * (t / fma(t, sqrt(2.0), ((0.5 / sqrt(2.0)) * ((2.0 * fma(2.0, (t * t), (l * l))) / (t * x)))));
	} else if (t <= 2.2960935021905734e+108) {
		tmp = sqrt(t_3) / t_7;
	} else {
		tmp = sqrt(((x + -1.0) / (1.0 + x)));
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(l * Float64(l / x))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(t * Float64(t * 2.0))
	t_4 = cbrt(fma(l, l, t_3))
	t_5 = Float64(1.0 + Float64(2.0 / x))
	t_6 = sqrt(Float64(2.0 * t_5))
	t_7 = sqrt(Float64(2.0 * Float64(Float64(t * Float64(t + Float64(t / x))) + t_1)))
	tmp = 0.0
	if (t <= -1.1292640349786462e+46)
		tmp = Float64(t_2 / Float64(-fma(t, t_6, Float64(sqrt(Float64(0.5 / t_5)) * Float64(Float64(l * Float64(l / t)) / x)))));
	elseif (t <= -8.6e-151)
		tmp = Float64(t_2 / t_7);
	elseif (t <= -2.1e-184)
		tmp = Float64(t_2 / Float64(t_6 * Float64(-t)));
	elseif (t <= -1e-190)
		tmp = Float64(t_2 / sqrt(Float64(Float64(l / Float64(x / l)) + fma(2.0, Float64(Float64(t / Float64(x / t)) + Float64(t * t)), Float64((t_4 ^ 2.0) * Float64(t_4 * Float64(1.0 / x)))))));
	elseif (t <= 5.4e-272)
		tmp = Float64(t_2 / sqrt(Float64(2.0 * t_1)));
	elseif (t <= 4.5e-150)
		tmp = Float64(sqrt(2.0) * Float64(t / fma(t, sqrt(2.0), Float64(Float64(0.5 / sqrt(2.0)) * Float64(Float64(2.0 * fma(2.0, Float64(t * t), Float64(l * l))) / Float64(t * x))))));
	elseif (t <= 2.2960935021905734e+108)
		tmp = Float64(sqrt(t_3) / t_7);
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(1.0 + x)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(l * l + t$95$3), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$5 = N[(1.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(2.0 * t$95$5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(2.0 * N[(N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.1292640349786462e+46], N[(t$95$2 / (-N[(t * t$95$6 + N[(N[Sqrt[N[(0.5 / t$95$5), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -8.6e-151], N[(t$95$2 / t$95$7), $MachinePrecision], If[LessEqual[t, -2.1e-184], N[(t$95$2 / N[(t$95$6 * (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1e-190], N[(t$95$2 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(N[Power[t$95$4, 2.0], $MachinePrecision] * N[(t$95$4 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e-272], N[(t$95$2 / N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.5e-150], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(t * N[Sqrt[2.0], $MachinePrecision] + N[(N[(0.5 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2960935021905734e+108], N[(N[Sqrt[t$95$3], $MachinePrecision] / t$95$7), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if t < -1.129264034978646e46

   1. Initial program 43.6

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

   2. Taylor expanded in x around inf 42.5

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

   3. Simplified 42.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      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))): 2 points increase in error, 3 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 13.0

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

   5. Simplified 5.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}, \sqrt{\frac{0.5}{1 + \frac{2}{x}}} \cdot \frac{\frac{\ell}{t} \cdot \ell}{x}\right)}} \]
      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)))): 0 points increase in error, 0 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)))): 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 (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)))))): 5 points increase in error, 13 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)))))): 34 points increase in error, 6 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 -1.129264034978646e46 < t < -8.60000000000000035e-151

   1. Initial program 27.9

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

   2. Taylor expanded in x around inf 9.4

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

   3. Simplified 9.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      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))): 2 points increase in error, 3 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 9.6

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

   5. Simplified 4.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      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))): 23 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 19 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 9.6

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

   7. Simplified 4.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell}{x} \cdot \ell\right)}}} \]
      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))): 1 points increase in error, 1 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))): 1 points increase in error, 2 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))))): 8 points increase in error, 5 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)))): 20 points increase in error, 4 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, 1 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

   if -8.60000000000000035e-151 < t < -2.0999999999999999e-184

   1. Initial program 56.4

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

   2. Taylor expanded in x around inf 32.7

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

   3. Simplified 32.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      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))): 2 points increase in error, 3 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 28.0

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

   5. Simplified 28.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \left(-\sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}\right)}} \]
      Proof
      (*.f64 t (neg.f64 (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 2 x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.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-rgt-neg-in_binary64 (neg.f64 (*.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 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 -2.0999999999999999e-184 < t < -1e-190

   1. Initial program 64.0

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

   2. Taylor expanded in x around inf 35.3

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

   3. Simplified 35.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      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))): 2 points increase in error, 3 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. Applied egg-rr 35.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\ell, \ell, \left(2 \cdot t\right) \cdot t\right)}\right)}^{2} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\ell, \ell, \left(2 \cdot t\right) \cdot t\right)} \cdot \frac{1}{x}\right)}\right)}} \]

   if -1e-190 < t < 5.39999999999999985e-272

   1. Initial program 63.1

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

   2. Taylor expanded in x around inf 32.3

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

   3. Simplified 32.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      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))): 2 points increase in error, 3 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 32.3

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

   5. Simplified 31.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      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))): 23 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 19 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 32.3

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

   7. Simplified 31.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(\frac{\ell}{x} \cdot \ell\right)}}} \]
      Proof
      (*.f64 2 (*.f64 (/.f64 l x) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l)))): 23 points increase in error, 12 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x))): 37 points increase in error, 19 points decrease in error
      (*.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error

   if 5.39999999999999985e-272 < t < 4.5000000000000002e-150

   1. Initial program 62.1

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

   2. Simplified 63.4

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 6 points increase in error, 17 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 1 points increase in error, 9 points decrease in error

   3. Taylor expanded in x around inf 23.7

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

   4. Simplified 23.7

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}} \]
      Proof
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2)))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (+.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1/2 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (sqrt.f64 2) (*.f64 t x))))): 9 points increase in error, 5 points decrease in error
      (fma.f64 t (sqrt.f64 2) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 2)) (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) t)) (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))) (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error

   if 4.5000000000000002e-150 < t < 2.29609350219057336e108

   1. Initial program 25.8

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

   2. Taylor expanded in x around inf 10.8

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

   3. Simplified 10.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      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))): 2 points increase in error, 3 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 5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      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))): 23 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 19 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 5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell}{x} \cdot \ell\right)}}} \]
      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))): 1 points increase in error, 1 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))): 1 points increase in error, 2 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))))): 8 points increase in error, 5 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)))): 20 points increase in error, 4 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, 1 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 5.0

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

   if 2.29609350219057336e108 < t

   1. Initial program 51.4

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

   2. Applied egg-rr 52.5

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

   3. Taylor expanded in t around inf 2.2

      \[\leadsto \sqrt{\color{blue}{\frac{x - 1}{1 + x}}} \]
3. Recombined 8 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1292640349786462 \cdot 10^{+46}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}, \sqrt{\frac{0.5}{1 + \frac{2}{x}}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-184}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(1 + \frac{2}{x}\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-190}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, {\left(\sqrt[3]{\mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right)}\right)}^{2} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\ell, \ell, t \cdot \left(t \cdot 2\right)\right)} \cdot \frac{1}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-272}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 2.2960935021905734 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.0
Cost	40208

\[\begin{array}{l} t_1 := t \cdot \left(t + \frac{t}{x}\right)\\ t_2 := \ell \cdot \frac{\ell}{x}\\ t_3 := t \cdot \sqrt{2}\\ t_4 := 1 + \frac{2}{x}\\ t_5 := \sqrt{2 \cdot t_4}\\ t_6 := \sqrt{2 \cdot \left(t_1 + t_2\right)}\\ \mathbf{if}\;t \leq -1.1292640349786462 \cdot 10^{+46}:\\ \;\;\;\;\frac{t_3}{-\mathsf{fma}\left(t, t_5, \sqrt{\frac{0.5}{t_4}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{t_3}{t_6}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-184}:\\ \;\;\;\;\frac{t_3}{t_5 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-195}:\\ \;\;\;\;\frac{t_3}{\sqrt{2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{\ell}\right)}^{2}, \frac{\ell}{x} \cdot \sqrt[3]{\ell}, t_1\right)}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-272}:\\ \;\;\;\;\frac{t_3}{\sqrt{2 \cdot t_2}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 2.2960935021905734 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{t_6}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 2
Error	9.0
Cost	33868

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 1 + \frac{2}{x}\\ t_3 := \sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}\\ \mathbf{if}\;t \leq -1.1292640349786462 \cdot 10^{+46}:\\ \;\;\;\;\frac{t_1}{-\mathsf{fma}\left(t, \sqrt{2 \cdot t_2}, \sqrt{\frac{0.5}{t_2}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-272}:\\ \;\;\;\;\frac{t_1}{t_3}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 2.2960935021905734 \cdot 10^{+108}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 3
Error	9.8
Cost	27524

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 1 + \frac{2}{x}\\ \mathbf{if}\;t \leq -1.1292640349786462 \cdot 10^{+46}:\\ \;\;\;\;\frac{t_1}{-\mathsf{fma}\left(t, \sqrt{2 \cdot t_2}, \sqrt{\frac{0.5}{t_2}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 2.2960935021905734 \cdot 10^{+108}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 4
Error	9.5
Cost	14280

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.1292640349786462 \cdot 10^{+46}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(1 + \frac{2}{x}\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.2960935021905734 \cdot 10^{+108}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 5
Error	15.1
Cost	14032

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-243}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 6
Error	14.5
Cost	14032

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-184}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(1 + \frac{2}{x}\right)} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-243}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 7
Error	15.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-243}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 8
Error	15.3
Cost	13188

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{x \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 9
Error	38.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;t \leq 2.75 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{x \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 10
Error	38.4
Cost	7044

\[\begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-275}:\\ \;\;\;\;\sqrt{\frac{t}{\ell} \cdot \frac{-t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{1 + x}}\\ \end{array} \]

Alternative 11
Error	39.2
Cost	6848

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

Alternative 12
Error	39.4
Cost	448

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

Alternative 13
Error	39.4
Cost	320

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

Alternative 14
Error	62.1
Cost	64

\[x \]

Alternative 15
Error	39.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022316 
(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)))))