Toniolo and Linder, Equation (7)

Average Error: 43.3 → 7.5

Time: 32.1s

Precision: binary64

Cost: 39628

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\sqrt{x}}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t \cdot \left(t \cdot 2\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -11911421963951122:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-243}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 10^{-125}:\\ \;\;\;\;\frac{t_2}{\mathsf{hypot}\left(\mathsf{hypot}\left(t_2, t_1\right), t_1\right)}\\ \mathbf{elif}\;t \leq 1.5955510907711053 \cdot 10^{+73}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ l (sqrt x)))
        (t_2 (* t (sqrt 2.0)))
        (t_3
         (/ t_2 (sqrt (+ (/ l (/ x l)) (+ (* t (* t 2.0)) (* l (/ l x))))))))
   (if (<= t -11911421963951122.0)
     (/ t_2 (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (* t (- (sqrt 2.0)))))
     (if (<= t -1e-243)
       t_3
       (if (<= t 1e-125)
         (/ t_2 (hypot (hypot t_2 t_1) t_1))
         (if (<= t 1.5955510907711053e+73) t_3 (sqrt (+ 1.0 (/ -2.0 x)))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = l / sqrt(x);
	double t_2 = t * sqrt(2.0);
	double t_3 = t_2 / sqrt(((l / (x / l)) + ((t * (t * 2.0)) + (l * (l / x)))));
	double tmp;
	if (t <= -11911421963951122.0) {
		tmp = t_2 / (sqrt(((x + 1.0) / (x + -1.0))) * (t * -sqrt(2.0)));
	} else if (t <= -1e-243) {
		tmp = t_3;
	} else if (t <= 1e-125) {
		tmp = t_2 / hypot(hypot(t_2, t_1), t_1);
	} else if (t <= 1.5955510907711053e+73) {
		tmp = t_3;
	} else {
		tmp = sqrt((1.0 + (-2.0 / x)));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double l, double t) {
	double t_1 = l / Math.sqrt(x);
	double t_2 = t * Math.sqrt(2.0);
	double t_3 = t_2 / Math.sqrt(((l / (x / l)) + ((t * (t * 2.0)) + (l * (l / x)))));
	double tmp;
	if (t <= -11911421963951122.0) {
		tmp = t_2 / (Math.sqrt(((x + 1.0) / (x + -1.0))) * (t * -Math.sqrt(2.0)));
	} else if (t <= -1e-243) {
		tmp = t_3;
	} else if (t <= 1e-125) {
		tmp = t_2 / Math.hypot(Math.hypot(t_2, t_1), t_1);
	} else if (t <= 1.5955510907711053e+73) {
		tmp = t_3;
	} else {
		tmp = Math.sqrt((1.0 + (-2.0 / x)));
	}
	return tmp;
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

↓

def code(x, l, t):
	t_1 = l / math.sqrt(x)
	t_2 = t * math.sqrt(2.0)
	t_3 = t_2 / math.sqrt(((l / (x / l)) + ((t * (t * 2.0)) + (l * (l / x)))))
	tmp = 0
	if t <= -11911421963951122.0:
		tmp = t_2 / (math.sqrt(((x + 1.0) / (x + -1.0))) * (t * -math.sqrt(2.0)))
	elif t <= -1e-243:
		tmp = t_3
	elif t <= 1e-125:
		tmp = t_2 / math.hypot(math.hypot(t_2, t_1), t_1)
	elif t <= 1.5955510907711053e+73:
		tmp = t_3
	else:
		tmp = math.sqrt((1.0 + (-2.0 / x)))
	return tmp

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(l / sqrt(x))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(t_2 / sqrt(Float64(Float64(l / Float64(x / l)) + Float64(Float64(t * Float64(t * 2.0)) + Float64(l * Float64(l / x))))))
	tmp = 0.0
	if (t <= -11911421963951122.0)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(t * Float64(-sqrt(2.0)))));
	elseif (t <= -1e-243)
		tmp = t_3;
	elseif (t <= 1e-125)
		tmp = Float64(t_2 / hypot(hypot(t_2, t_1), t_1));
	elseif (t <= 1.5955510907711053e+73)
		tmp = t_3;
	else
		tmp = sqrt(Float64(1.0 + Float64(-2.0 / x)));
	end
	return tmp
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

↓

function tmp_2 = code(x, l, t)
	t_1 = l / sqrt(x);
	t_2 = t * sqrt(2.0);
	t_3 = t_2 / sqrt(((l / (x / l)) + ((t * (t * 2.0)) + (l * (l / x)))));
	tmp = 0.0;
	if (t <= -11911421963951122.0)
		tmp = t_2 / (sqrt(((x + 1.0) / (x + -1.0))) * (t * -sqrt(2.0)));
	elseif (t <= -1e-243)
		tmp = t_3;
	elseif (t <= 1e-125)
		tmp = t_2 / hypot(hypot(t_2, t_1), t_1);
	elseif (t <= 1.5955510907711053e+73)
		tmp = t_3;
	else
		tmp = sqrt((1.0 + (-2.0 / x)));
	end
	tmp_2 = 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[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -11911421963951122.0], N[(t$95$2 / N[(N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1e-243], t$95$3, If[LessEqual[t, 1e-125], N[(t$95$2 / N[Sqrt[N[Sqrt[t$95$2 ^ 2 + t$95$1 ^ 2], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5955510907711053e+73], t$95$3, N[Sqrt[N[(1.0 + N[(-2.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -11911421963951122

   1. Initial program 43.7

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

   2. Taylor expanded in t around -inf 4.8

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

   3. Simplified 4.8

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

   if -11911421963951122 < t < -9.99999999999999995e-244 or 1.00000000000000001e-125 < t < 1.5955510907711053e73

   1. Initial program 33.1

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

   2. Taylor expanded in x around inf 13.6

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

   3. Simplified 13.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around -inf 13.6

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

   5. Simplified 13.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(2 \cdot \left(t \cdot t\right) - \frac{\left(t \cdot t\right) \cdot -4 - \ell \cdot \ell}{x}\right)}}} \]
      Proof
      (-.f64 (*.f64 2 (*.f64 t t)) (/.f64 (-.f64 (*.f64 (*.f64 t t) -4) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (-.f64 (*.f64 (*.f64 t t) -4) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) -4) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (*.f64 (pow.f64 t 2) (Rewrite<= metadata-eval (+.f64 -2 -2))) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -2 (pow.f64 t 2)))) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -2 (pow.f64 t 2))) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) (pow.f64 t 2))) (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (pow.f64 t 2))))) (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (Rewrite=> unsub-neg_binary64 (-.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2)))) (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 -2 (pow.f64 t 2)) (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (*.f64 -2 (pow.f64 t 2)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (neg.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 2 (pow.f64 t 2)) (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (neg.f64 (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)) (*.f64 2 (pow.f64 t 2)))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in t around 0 13.9

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

   7. Simplified 13.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) - \color{blue}{\frac{\ell \cdot \left(-\ell\right)}{x}}\right)}} \]
      Proof
      (/.f64 (*.f64 l (neg.f64 l)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 l l))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 l 2))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 9.3

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

   if -9.99999999999999995e-244 < t < 1.00000000000000001e-125

   1. Initial program 58.5

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

   2. Taylor expanded in x around inf 27.7

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

   3. Simplified 27.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x around -inf 27.7

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

   5. Simplified 27.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(2 \cdot \left(t \cdot t\right) - \frac{\left(t \cdot t\right) \cdot -4 - \ell \cdot \ell}{x}\right)}}} \]
      Proof
      (-.f64 (*.f64 2 (*.f64 t t)) (/.f64 (-.f64 (*.f64 (*.f64 t t) -4) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (-.f64 (*.f64 (*.f64 t t) -4) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) -4) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (*.f64 (pow.f64 t 2) (Rewrite<= metadata-eval (+.f64 -2 -2))) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -2 (pow.f64 t 2)))) (*.f64 l l)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -2 (pow.f64 t 2))) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) (pow.f64 t 2))) (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (pow.f64 t 2))))) (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (Rewrite=> unsub-neg_binary64 (-.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 2 (pow.f64 t 2)))) (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 -2 (pow.f64 t 2)) (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2)))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (-.f64 (*.f64 -2 (pow.f64 t 2)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 2 (pow.f64 t 2)) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (neg.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 2 (pow.f64 t 2)) (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (neg.f64 (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (pow.f64 t 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (+.f64 (*.f64 -2 (pow.f64 t 2)) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) x)) (*.f64 2 (pow.f64 t 2)))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in t around 0 27.8

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

   7. Simplified 27.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) - \color{blue}{\frac{\ell \cdot \left(-\ell\right)}{x}}\right)}} \]
      Proof
      (/.f64 (*.f64 l (neg.f64 l)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 l l))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 l 2))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 13.9

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

   if 1.5955510907711053e73 < t

   1. Initial program 47.3

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

   2. Taylor expanded in x around inf 46.8

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

   3. Simplified 46.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around -inf 63.0

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

   5. Simplified 63.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \left(-\sqrt{2 + 2 \cdot \frac{2}{x}}\right)}} \]
      Proof
      (*.f64 t (neg.f64 (sqrt.f64 (+.f64 2 (*.f64 2 (/.f64 2 x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (*.f64 2 (/.f64 2 x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (+.f64 (*.f64 2 1) (*.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (+.f64 (*.f64 2 1) (*.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (+.f64 (*.f64 2 1) (*.f64 2 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 1 (+.f64 (/.f64 1 x) (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))))): 2 points increase in error, 0 points decrease in error
      (*.f64 t (neg.f64 (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 46.2

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

   7. Taylor expanded in x around inf 3.5

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

   8. Simplified 3.5

      \[\leadsto \sqrt{\color{blue}{1 + \frac{-2}{x}}} \]
      Proof
      (+.f64 1 (/.f64 -2 x)): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (/.f64 (Rewrite<= metadata-eval (neg.f64 2)) x)): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 2 x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 1 (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 2 (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -11911421963951122:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-243}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t \cdot \left(t \cdot 2\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 10^{-125}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\mathsf{hypot}\left(t \cdot \sqrt{2}, \frac{\ell}{\sqrt{x}}\right), \frac{\ell}{\sqrt{x}}\right)}\\ \mathbf{elif}\;t \leq 1.5955510907711053 \cdot 10^{+73}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t \cdot \left(t \cdot 2\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.3
Cost	33868

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t \cdot \left(t \cdot 2\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -11911421963951122:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 1.5955510907711053 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	20356

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t \cdot \left(t \cdot 2\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -11911421963951122:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.5955510907711053 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 3
Error	9.9
Cost	14672

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t \cdot \left(t \cdot 2\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -11911421963951122:\\ \;\;\;\;\frac{t_1}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.5955510907711053 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 4
Error	15.0
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -4.55 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{1}{2 + \frac{4}{x}}}\right)\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 5
Error	14.8
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -4.55 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{2}}{t} \cdot \frac{-t}{\sqrt{2 + \frac{4}{x}}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 6
Error	14.7
Cost	14032

\[\begin{array}{l} t_1 := \sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -4.55 \cdot 10^{-243}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 7
Error	14.7
Cost	14032

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -4.55 \cdot 10^{-243}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-264}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + t_2}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-130}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2 \cdot t_2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 8
Error	15.9
Cost	13572

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{1}{2 + \frac{4}{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 9
Error	15.7
Cost	13444

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(-\sqrt{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 10
Error	16.0
Cost	13188

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 11
Error	27.4
Cost	7364

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{-141}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 12
Error	25.9
Cost	7364

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{t \cdot t}{2 + \frac{4}{x}}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \frac{-2}{x}}\\ \end{array} \]

Alternative 13
Error	38.8
Cost	6720

\[\sqrt{1 + \frac{-2}{x}} \]

Alternative 14
Error	39.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022306 
(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)))))