Toniolo and Linder, Equation (7)

Average Error: 42.7 → 8.9

Time: 28.1s

Precision: binary64

Cost: 27660

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 2 + \frac{2}{x}\\ t_3 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot t_2\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ t_4 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.1074029748262555 \cdot 10^{+85}:\\ \;\;\;\;-t_4\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-268}:\\ \;\;\;\;\frac{t_1}{-\mathsf{fma}\left(t, \sqrt{t_2}, \sqrt{\frac{1}{t_2}} \cdot \frac{\ell}{t \cdot \frac{x}{\ell}}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_1}{{\left({\left(\mathsf{fma}\left(t_2, t \cdot t, \frac{\ell}{x} \cdot \left(2 \cdot \ell\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.69451875391852 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (+ 2.0 (/ 2.0 x)))
        (t_3
         (/
          t_1
          (sqrt
           (+
            (/ l (/ x l))
            (+
             (+ (fma 2.0 (* t t) (* (/ l x) (/ l x))) (* (/ t (/ x t)) t_2))
             (* (* l (/ l x)) (+ 1.0 (/ 1.0 x))))))))
        (t_4 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.1074029748262555e+85)
     (- t_4)
     (if (<= t -1e-121)
       t_3
       (if (<= t -1.35e-268)
         (/
          t_1
          (- (fma t (sqrt t_2) (* (sqrt (/ 1.0 t_2)) (/ l (* t (/ x l)))))))
         (if (<= t 2.9e-224)
           (/ t_1 (pow (pow (fma t_2 (* t t) (* (/ l x) (* 2.0 l))) 0.25) 2.0))
           (if (<= t 1.05e-164)
             1.0
             (if (<= t 2.69451875391852e+71) t_3 t_4))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = 2.0 + (2.0 / x);
	double t_3 = t_1 / sqrt(((l / (x / l)) + ((fma(2.0, (t * t), ((l / x) * (l / x))) + ((t / (x / t)) * t_2)) + ((l * (l / x)) * (1.0 + (1.0 / x))))));
	double t_4 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.1074029748262555e+85) {
		tmp = -t_4;
	} else if (t <= -1e-121) {
		tmp = t_3;
	} else if (t <= -1.35e-268) {
		tmp = t_1 / -fma(t, sqrt(t_2), (sqrt((1.0 / t_2)) * (l / (t * (x / l)))));
	} else if (t <= 2.9e-224) {
		tmp = t_1 / pow(pow(fma(t_2, (t * t), ((l / x) * (2.0 * l))), 0.25), 2.0);
	} else if (t <= 1.05e-164) {
		tmp = 1.0;
	} else if (t <= 2.69451875391852e+71) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(2.0 + Float64(2.0 / x))
	t_3 = Float64(t_1 / 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_2)) + Float64(Float64(l * Float64(l / x)) * Float64(1.0 + Float64(1.0 / x)))))))
	t_4 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.1074029748262555e+85)
		tmp = Float64(-t_4);
	elseif (t <= -1e-121)
		tmp = t_3;
	elseif (t <= -1.35e-268)
		tmp = Float64(t_1 / Float64(-fma(t, sqrt(t_2), Float64(sqrt(Float64(1.0 / t_2)) * Float64(l / Float64(t * Float64(x / l)))))));
	elseif (t <= 2.9e-224)
		tmp = Float64(t_1 / ((fma(t_2, Float64(t * t), Float64(Float64(l / x) * Float64(2.0 * l))) ^ 0.25) ^ 2.0));
	elseif (t <= 1.05e-164)
		tmp = 1.0;
	elseif (t <= 2.69451875391852e+71)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / 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$2), $MachinePrecision]), $MachinePrecision] + N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.1074029748262555e+85], (-t$95$4), If[LessEqual[t, -1e-121], t$95$3, If[LessEqual[t, -1.35e-268], N[(t$95$1 / (-N[(t * N[Sqrt[t$95$2], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision] * N[(l / N[(t * N[(x / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 2.9e-224], N[(t$95$1 / N[Power[N[Power[N[(t$95$2 * N[(t * t), $MachinePrecision] + N[(N[(l / x), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-164], 1.0, If[LessEqual[t, 2.69451875391852e+71], t$95$3, t$95$4]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.10740297482625545e85

   1. Initial program 48.8

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

   2. Taylor expanded in t around -inf 2.7

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

   3. Simplified 2.7

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

   4. Applied egg-rr 2.6

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

   if -1.10740297482625545e85 < t < -9.9999999999999998e-122 or 1.04999999999999995e-164 < t < 2.6945187539185201e71

   1. Initial program 25.7

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

   2. Taylor expanded in x around inf 9.8

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

   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)))): 2 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 9.9

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

   5. Simplified 5.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)} \cdot \left(\frac{-1}{x} - 1\right)\right)}} \]
      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))): 19 points increase in error, 28 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 42 points increase in error, 14 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   if -9.9999999999999998e-122 < t < -1.35e-268

   1. Initial program 56.0

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

   2. Taylor expanded in x around inf 31.6

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

   3. Simplified 28.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} \cdot \left(\frac{-1}{x} - 1\right)\right)}}} \]
      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)))): 2 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 28.1

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

   5. Simplified 24.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)} \cdot \left(\frac{-1}{x} - 1\right)\right)}} \]
      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))): 19 points increase in error, 28 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 42 points increase in error, 14 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 28.3

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

   7. Simplified 24.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 + \frac{2}{x}\right) \cdot \left(t \cdot t\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}} \]
      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)))))): 4 points increase in error, 5 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))))): 16 points increase in error, 4 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, 2 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))): 1 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. Taylor expanded in t around -inf 23.3

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

   9. Simplified 23.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{2}{x}}, \sqrt{\frac{1}{2 + \frac{2}{x}}} \cdot \frac{\ell}{t \cdot \frac{x}{\ell}}\right)}} \]
      Proof
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (/.f64 2 x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (/.f64 2 x)))) (/.f64 l (*.f64 t (/.f64 x l)))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (/.f64 2 x)))) (/.f64 l (*.f64 t (/.f64 x l)))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (/.f64 2 x)))) (/.f64 l (*.f64 t (/.f64 x l)))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (/.f64 l (*.f64 t (/.f64 x l)))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (/.f64 l (*.f64 t (/.f64 x l)))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 l (/.f64 x l)) t))))): 14 points increase in error, 7 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) t)))): 21 points increase in error, 1 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (/.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (Rewrite=> associate-/l/_binary64 (/.f64 (pow.f64 l 2) (*.f64 t x)))))): 3 points increase in error, 3 points decrease in error
      (neg.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.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 2 (*.f64 2 (/.f64 1 x)))) t)) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (*.f64 (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) t) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (*.f64 (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 1 x)))) t)) (*.f64 -1 (*.f64 (sqrt.f64 (/.f64 1 (+.f64 2 (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error

   if -1.35e-268 < t < 2.9e-224

   1. Initial program 63.2

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

   2. Taylor expanded in x around inf 36.8

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

   3. Simplified 30.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} \cdot \left(\frac{-1}{x} - 1\right)\right)}}} \]
      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)))): 2 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 30.9

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

   5. Simplified 30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) - \color{blue}{\left(\frac{\ell}{x} \cdot \ell\right)} \cdot \left(\frac{-1}{x} - 1\right)\right)}} \]
      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))): 19 points increase in error, 28 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 42 points increase in error, 14 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 30.9

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

   7. Simplified 30.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\left(2 + \frac{2}{x}\right) \cdot \left(t \cdot t\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}} \]
      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)))))): 4 points increase in error, 5 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))))): 16 points increase in error, 4 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, 2 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))): 1 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 30.3

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

   if 2.9e-224 < t < 1.04999999999999995e-164

   1. Initial program 63.0

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

   2. Simplified 62.7

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      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, 2 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))))): 0 points increase in error, 15 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))))): 3 points increase in error, 10 points decrease in error

   3. Taylor expanded in x around inf 35.2

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

   4. Applied egg-rr 35.3

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

   5. Applied egg-rr 35.3

      \[\leadsto \sqrt{2} \cdot \color{blue}{\frac{t}{\frac{\sqrt{2}}{\frac{1}{t}}}} \]

   6. Taylor expanded in t around 0 35.2

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

   if 2.6945187539185201e71 < t

   1. Initial program 46.8

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

   2. Taylor expanded in t around -inf 63.0

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

   3. Simplified 63.0

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

   4. Applied egg-rr 3.1

      \[\leadsto \color{blue}{\sqrt{\frac{x + -1}{x - -1}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1074029748262555 \cdot 10^{+85}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-121}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-268}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 + \frac{2}{x}}, \sqrt{\frac{1}{2 + \frac{2}{x}}} \cdot \frac{\ell}{t \cdot \frac{x}{\ell}}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left({\left(\mathsf{fma}\left(2 + \frac{2}{x}, t \cdot t, \frac{\ell}{x} \cdot \left(2 \cdot \ell\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.69451875391852 \cdot 10^{+71}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.8
Cost	33420

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 2 + \frac{2}{x}\\ t_3 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot t_2\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ t_4 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.1074029748262555 \cdot 10^{+85}:\\ \;\;\;\;-t_4\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-140}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 10^{-165}:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(\sqrt{2} \cdot \frac{\ell}{\sqrt{x}}, t \cdot \sqrt{t_2}\right)}\\ \mathbf{elif}\;t \leq 2.69451875391852 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 2
Error	9.4
Cost	27472

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 2 + \frac{2}{x}\\ t_3 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot t_2\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ t_4 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.1074029748262555 \cdot 10^{+85}:\\ \;\;\;\;-t_4\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-147}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-268}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_1}{{\left({\left(\mathsf{fma}\left(t_2, t \cdot t, \frac{\ell}{x} \cdot \left(2 \cdot \ell\right)\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.69451875391852 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 3
Error	9.4
Cost	22872

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := \frac{\ell}{\frac{x}{\ell}}\\ t_4 := \frac{t_1}{\sqrt{t_3 + \left(\left(\mathsf{fma}\left(2, t \cdot t, \frac{\ell}{x} \cdot \frac{\ell}{x}\right) + \frac{t}{\frac{x}{t}} \cdot \left(2 + \frac{2}{x}\right)\right) + \left(\ell \cdot \frac{\ell}{x}\right) \cdot \left(1 + \frac{1}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.1074029748262555 \cdot 10^{+85}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-147}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-268}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot t_3}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.69451875391852 \cdot 10^{+71}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	9.4
Cost	14936

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\left(t \cdot t\right) \cdot \left(2 + \frac{2}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.1074029748262555 \cdot 10^{+85}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-268}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-164}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.69451875391852 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	13.6
Cost	13768

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{-268}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	13.8
Cost	13704

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-270}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-224}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{2}{x}} \cdot \left(-\ell\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	14.5
Cost	7044

\[\begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-299}:\\ \;\;\;\;-t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	14.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 9
Error	14.8
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{-0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array} \]

Alternative 10
Error	15.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array} \]

Alternative 11
Error	14.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-299}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array} \]

Alternative 12
Error	15.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-299}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	38.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022294 
(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)))))