Toniolo and Linder, Equation (7)

Average Error: 42.7 → 6.7

Time: 25.3s

Precision: binary64

Cost: 33420

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := 2 + \frac{2}{x}\\ \mathbf{if}\;t \leq -2.113404131062392 \cdot 10^{+33}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot t_3\right) + t_1 \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 10^{-165}:\\ \;\;\;\;\frac{t_2}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\ell}{\sqrt{x}}, t \cdot \sqrt{t_3}\right)}\\ \mathbf{elif}\;t \leq 1.8896406000205735 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t \cdot t\right)}}{\sqrt{\left(t \cdot t\right) \cdot t_3 + 2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

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 (* l (/ l x))) (t_2 (* t (sqrt 2.0))) (t_3 (+ 2.0 (/ 2.0 x))))
   (if (<= t -2.113404131062392e+33)
     (- (sqrt (* 2.0 (/ (* 0.5 (+ x -1.0)) (+ x 1.0)))))
     (if (<= t -1e-220)
       (/
        t_2
        (sqrt
         (+
          (/ l (/ x l))
          (+
           (+ (fma 2.0 (* t t) (* (/ l x) (/ l x))) (* (/ t (/ x t)) t_3))
           (* t_1 (+ 1.0 (/ 1.0 x)))))))
       (if (<= t 1e-165)
         (/ t_2 (hypot (* (sqrt 2.0) (/ l (sqrt x))) (* t (sqrt t_3))))
         (if (<= t 1.8896406000205735e+52)
           (/ (sqrt (* 2.0 (* t t))) (sqrt (+ (* (* t t) t_3) (* 2.0 t_1))))
           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 = l * (l / x);
	double t_2 = t * sqrt(2.0);
	double t_3 = 2.0 + (2.0 / x);
	double tmp;
	if (t <= -2.113404131062392e+33) {
		tmp = -sqrt((2.0 * ((0.5 * (x + -1.0)) / (x + 1.0))));
	} else if (t <= -1e-220) {
		tmp = t_2 / sqrt(((l / (x / l)) + ((fma(2.0, (t * t), ((l / x) * (l / x))) + ((t / (x / t)) * t_3)) + (t_1 * (1.0 + (1.0 / x))))));
	} else if (t <= 1e-165) {
		tmp = t_2 / hypot((sqrt(2.0) * (l / sqrt(x))), (t * sqrt(t_3)));
	} else if (t <= 1.8896406000205735e+52) {
		tmp = sqrt((2.0 * (t * t))) / sqrt((((t * t) * t_3) + (2.0 * t_1)));
	} else {
		tmp = 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(l * Float64(l / x))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(2.0 + Float64(2.0 / x))
	tmp = 0.0
	if (t <= -2.113404131062392e+33)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(Float64(0.5 * Float64(x + -1.0)) / Float64(x + 1.0)))));
	elseif (t <= -1e-220)
		tmp = Float64(t_2 / sqrt(Float64(Float64(l / Float64(x / l)) + Float64(Float64(fma(2.0, Float64(t * t), Float64(Float64(l / x) * Float64(l / x))) + Float64(Float64(t / Float64(x / t)) * t_3)) + Float64(t_1 * Float64(1.0 + Float64(1.0 / x)))))));
	elseif (t <= 1e-165)
		tmp = Float64(t_2 / hypot(Float64(sqrt(2.0) * Float64(l / sqrt(x))), Float64(t * sqrt(t_3))));
	elseif (t <= 1.8896406000205735e+52)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t * t))) / sqrt(Float64(Float64(Float64(t * t) * t_3) + Float64(2.0 * t_1))));
	else
		tmp = 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[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.113404131062392e+33], (-N[Sqrt[N[(2.0 * N[(N[(0.5 * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t, -1e-220], N[(t$95$2 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(N[(l / x), $MachinePrecision] * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-165], N[(t$95$2 / N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[(l / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(t * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8896406000205735e+52], N[(N[Sqrt[N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(t * t), $MachinePrecision] * t$95$3), $MachinePrecision] + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.1134041310623919e33

   1. Initial program 42.6

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

   2. Simplified 42.6

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      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))))): 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<= *-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))))): 1 points increase in error, 12 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))))): 2 points increase in error, 17 points decrease in error

   3. Applied egg-rr 55.3

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

   4. Taylor expanded in t around -inf 4.7

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

   5. Simplified 4.7

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\sqrt{0.5} \cdot \sqrt{\frac{x + -1}{x + 1}}\right)} \]
      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)))))): 6 points increase in error, 0 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

   6. Applied egg-rr 3.8

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

   if -2.1134041310623919e33 < t < -9.99999999999999992e-221

   1. Initial program 35.8

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

   2. Taylor expanded in x around inf 16.4

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

   3. Simplified 15.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} \cdot \left(\frac{-1}{x} - 1\right)\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 6 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 x x)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 x x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (pow.f64 l 2) (pow.f64 x 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 (/.f64 2 x) (/.f64 (pow.f64 t 2) x))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 x x))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (/.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2)))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (/.f64 (pow.f64 t 2) x))) (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (/.f64 -1 x) (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (*.f64 x x))) (*.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (/.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= unpow2_binary64 (pow.f64 x 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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 2))) (*.f64 (neg.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 2))) (*.f64 (Rewrite=> metadata-eval -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) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))))) (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 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 15.7

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

   5. Simplified 11.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)} \cdot \left(\frac{-1}{x} - 1\right)\right)}} \]
      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))): 21 points increase in error, 26 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 50 points increase in error, 15 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   if -9.99999999999999992e-221 < t < 1e-165

   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 38.1

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

   3. Simplified 33.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} \cdot \left(\frac{-1}{x} - 1\right)\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 6 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 x x)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 x x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (pow.f64 l 2) (pow.f64 x 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 (/.f64 2 x) (/.f64 (pow.f64 t 2) x))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 x x))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (/.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2)))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (/.f64 (pow.f64 t 2) x))) (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (/.f64 -1 x) (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (*.f64 x x))) (*.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (/.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= unpow2_binary64 (pow.f64 x 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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 2))) (*.f64 (neg.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 2))) (*.f64 (Rewrite=> metadata-eval -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) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))))) (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 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 33.2

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

   5. Simplified 32.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)} \cdot \left(\frac{-1}{x} - 1\right)\right)}} \]
      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))): 21 points increase in error, 26 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 50 points increase in error, 15 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 x around inf 33.3

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

   7. Simplified 32.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 + \frac{2}{x}\right) \cdot \left(t \cdot t\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}} \]
      Proof
      (+.f64 (*.f64 (+.f64 2 (/.f64 2 x)) (*.f64 t t)) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)) (*.f64 t t)) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))) (*.f64 t t)) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 2 (*.f64 2 (/.f64 1 x))) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 t 2) (+.f64 2 (*.f64 2 (/.f64 1 x))))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 1 x)) 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (*.f64 2 (/.f64 1 x)) (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 1) x)) (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 (/.f64 (Rewrite=> metadata-eval 2) x) (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 2 (pow.f64 t 2)) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 t 2) x))) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 l 1)) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (Rewrite<= associate-*r/_binary64 (*.f64 l (/.f64 1 x)))))): 10 points increase in error, 3 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l l) (/.f64 1 x))))): 20 points increase in error, 3 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 l 2) 1) x)))): 4 points increase in error, 6 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (/.f64 (Rewrite=> *-rgt-identity_binary64 (pow.f64 l 2)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= count-2_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (/.f64 (pow.f64 l 2) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (pow.f64 l 2) x)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (pow.f64 l 2) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (pow.f64 l 2) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_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 (pow.f64 l 2) x)))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 12.4

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

   if 1e-165 < t < 1.8896406000205735e52

   1. Initial program 29.1

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

   2. Taylor expanded in x around inf 10.0

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

   3. Simplified 9.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} \cdot \left(\frac{-1}{x} - 1\right)\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 6 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 (/.f64 l x) (/.f64 l x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 x x)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 x x))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (fma.f64 2 (pow.f64 t 2) (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (pow.f64 l 2) (pow.f64 x 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 (/.f64 t (/.f64 x t)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (+.f64 2 (/.f64 2 x)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 (/.f64 2 x) (/.f64 (pow.f64 t 2) x))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 x x))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (/.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2)))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (/.f64 (pow.f64 t 2) x))) (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2)))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (*.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x) (-.f64 (/.f64 -1 x) 1)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (/.f64 -1 x) (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)) (*.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (*.f64 x x))) (*.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (/.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= unpow2_binary64 (pow.f64 x 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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 2))) (*.f64 (neg.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 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2))))) (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 2))) (*.f64 (Rewrite=> metadata-eval -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) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (+.f64 (/.f64 (pow.f64 l 2) (pow.f64 x 2)) (*.f64 2 (pow.f64 t 2)))))) (+.f64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (pow.f64 x 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 9.8

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

   5. Simplified 5.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)} \cdot \left(\frac{-1}{x} - 1\right)\right)}} \]
      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))): 21 points increase in error, 26 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 50 points increase in error, 15 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 x around inf 9.9

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

   7. Simplified 5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 + \frac{2}{x}\right) \cdot \left(t \cdot t\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}} \]
      Proof
      (+.f64 (*.f64 (+.f64 2 (/.f64 2 x)) (*.f64 t t)) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)) (*.f64 t t)) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))) (*.f64 t t)) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (+.f64 2 (*.f64 2 (/.f64 1 x))) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 t 2) (+.f64 2 (*.f64 2 (/.f64 1 x))))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 1 x)) 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (*.f64 2 (/.f64 1 x)) (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2)))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 1 points decrease in error
      (+.f64 (+.f64 (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 1) x)) (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 (/.f64 (Rewrite=> metadata-eval 2) x) (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 2 (pow.f64 t 2)) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 t 2) x))) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 l x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 l 1)) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 l (Rewrite<= associate-*r/_binary64 (*.f64 l (/.f64 1 x)))))): 10 points increase in error, 3 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 l l) (/.f64 1 x))))): 20 points increase in error, 3 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 l 2) 1) x)))): 4 points increase in error, 6 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 2 (/.f64 (Rewrite=> *-rgt-identity_binary64 (pow.f64 l 2)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= count-2_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (/.f64 (pow.f64 l 2) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (pow.f64 l 2) x)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))))) (/.f64 (pow.f64 l 2) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (pow.f64 l 2) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= cancel-sign-sub-inv_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 (pow.f64 l 2) x)))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 5.1

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

   if 1.8896406000205735e52 < t

   1. Initial program 45.0

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

   2. Simplified 45.1

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      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))))): 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<= *-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))))): 1 points increase in error, 12 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))))): 2 points increase in error, 17 points decrease in error

   3. Taylor expanded in x around inf 4.6

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

   4. Applied egg-rr 4.8

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

   5. Applied egg-rr 5.8

      \[\leadsto \sqrt{2} \cdot \color{blue}{\log \left(e^{\frac{\frac{t}{t}}{\sqrt{2}}}\right)} \]

   6. Taylor expanded in t around 0 4.4

      \[\leadsto \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.113404131062392 \cdot 10^{+33}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 10^{-165}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\ell}{\sqrt{x}}, t \cdot \sqrt{2 + \frac{2}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.8896406000205735 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t \cdot t\right)}}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{2}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	9.5
Cost	22344

\[\begin{array}{l} \mathbf{if}\;t \leq -2.113404131062392 \cdot 10^{+33}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{elif}\;t \leq 1.8896406000205735 \cdot 10^{+52}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	9.6
Cost	14408

\[\begin{array}{l} \mathbf{if}\;t \leq -2.113404131062392 \cdot 10^{+33}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{elif}\;t \leq 1.8896406000205735 \cdot 10^{+52}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{2}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	14.0
Cost	13772

\[\begin{array}{l} t_1 := \sqrt{\frac{2}{x}}\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{-219}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{elif}\;t \leq 10^{-240}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{t_1 \cdot \left(-\ell\right)}\\ \mathbf{elif}\;t \leq 10^{-168}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5 + \frac{-0.5}{x}}{x \cdot x}\right)\\ \end{array} \]

Alternative 4
Error	13.4
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-219}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{elif}\;t \leq 10^{-168}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5 + \frac{-0.5}{x}}{x \cdot x}\right)\\ \end{array} \]

Alternative 5
Error	14.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-221}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{elif}\;t \leq 10^{-168}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5 + \frac{-0.5}{x}}{x \cdot x}\right)\\ \end{array} \]

Alternative 6
Error	14.4
Cost	7300

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{0.5 \cdot \left(x + -1\right)}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5 + \frac{-0.5}{x}}{x \cdot x}\right)\\ \end{array} \]

Alternative 7
Error	14.6
Cost	1092

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{0.5 + \frac{-0.5}{x}}{x \cdot x}\right)\\ \end{array} \]

Alternative 8
Error	14.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 9
Error	14.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 10
Error	14.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	14.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 12
Error	15.1
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-304}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	39.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022308 
(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)))))