Toniolo and Linder, Equation (7)

Average Error: 43.1 → 9.9

Time: 31.3s

Precision: binary64

Cost: 21328

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \ell \cdot \frac{\ell}{x}\\ \mathbf{if}\;t \leq -4.337537014097745 \cdot 10^{+116}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq -4.25 \cdot 10^{-153}:\\ \;\;\;\;\frac{t_2}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + t_3\right)}}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;t_1 \cdot \frac{-t}{t}\\ \mathbf{elif}\;t \leq 8.866609018474782 \cdot 10^{+47}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t}{t}\\ \end{array} \]

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 (sqrt (/ 2.0 (+ 2.0 (/ 4.0 x)))))
        (t_2 (* t (sqrt 2.0)))
        (t_3 (* l (/ l x))))
   (if (<= t -4.337537014097745e+116)
     (/ t_2 (* (sqrt (/ (+ x 1.0) (+ x -1.0))) (* t (- (sqrt 2.0)))))
     (if (<= t -4.25e-153)
       (/ t_2 (sqrt (* 2.0 (+ (* t (+ t (/ t x))) t_3))))
       (if (<= t -8.5e-255)
         (* t_1 (/ (- t) t))
         (if (<= t 8.866609018474782e+47)
           (/
            t_2
            (sqrt (+ (/ l (/ x l)) (fma 2.0 (+ (/ t (/ x t)) (* t t)) t_3))))
           (* t_1 (/ t t))))))))

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 = sqrt((2.0 / (2.0 + (4.0 / x))));
	double t_2 = t * sqrt(2.0);
	double t_3 = l * (l / x);
	double tmp;
	if (t <= -4.337537014097745e+116) {
		tmp = t_2 / (sqrt(((x + 1.0) / (x + -1.0))) * (t * -sqrt(2.0)));
	} else if (t <= -4.25e-153) {
		tmp = t_2 / sqrt((2.0 * ((t * (t + (t / x))) + t_3)));
	} else if (t <= -8.5e-255) {
		tmp = t_1 * (-t / t);
	} else if (t <= 8.866609018474782e+47) {
		tmp = t_2 / sqrt(((l / (x / l)) + fma(2.0, ((t / (x / t)) + (t * t)), t_3)));
	} else {
		tmp = t_1 * (t / t);
	}
	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 = sqrt(Float64(2.0 / Float64(2.0 + Float64(4.0 / x))))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(l * Float64(l / x))
	tmp = 0.0
	if (t <= -4.337537014097745e+116)
		tmp = Float64(t_2 / Float64(sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))) * Float64(t * Float64(-sqrt(2.0)))));
	elseif (t <= -4.25e-153)
		tmp = Float64(t_2 / sqrt(Float64(2.0 * Float64(Float64(t * Float64(t + Float64(t / x))) + t_3))));
	elseif (t <= -8.5e-255)
		tmp = Float64(t_1 * Float64(Float64(-t) / t));
	elseif (t <= 8.866609018474782e+47)
		tmp = Float64(t_2 / sqrt(Float64(Float64(l / Float64(x / l)) + fma(2.0, Float64(Float64(t / Float64(x / t)) + Float64(t * t)), t_3))));
	else
		tmp = Float64(t_1 * Float64(t / t));
	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[Sqrt[N[(2.0 / N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.337537014097745e+116], 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, -4.25e-153], N[(t$95$2 / N[Sqrt[N[(2.0 * N[(N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -8.5e-255], N[(t$95$1 * N[((-t) / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.866609018474782e+47], N[(t$95$2 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(t / t), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4.3375370140977447e116

   1. Initial program 53.3

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

   2. Taylor expanded in t around -inf 1.5

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

   3. Simplified 1.5

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

   if -4.3375370140977447e116 < t < -4.2499999999999998e-153

   1. Initial program 25.8

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

   2. Taylor expanded in x around inf 10.5

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

   3. Simplified 10.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 3 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 10.9

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

   5. Simplified 5.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      Proof
      (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 32 points increase in error, 17 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 21 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in l around 0 10.9

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

   7. Simplified 5.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell}{x} \cdot \ell\right)}}} \]
      Proof
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (/.f64 t x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 t (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 x) t)))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) (*.f64 (/.f64 1 x) t))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 t (Rewrite<= distribute-rgt-in_binary64 (*.f64 t (+.f64 1 (/.f64 1 x))))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t t) (+.f64 1 (/.f64 1 x)))) (*.f64 (/.f64 l x) l))): 1 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 1 (/.f64 1 x))) (*.f64 (/.f64 l x) l))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))))): 11 points increase in error, 3 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)))): 18 points increase in error, 12 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (*.f64 (pow.f64 t 2) (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 1 x) 1))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (pow.f64 t 2) (/.f64 1 x)) (*.f64 (pow.f64 t 2) 1))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (pow.f64 t 2) 1) x)) (*.f64 (pow.f64 t 2) 1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (/.f64 (Rewrite=> *-rgt-identity_binary64 (pow.f64 t 2)) x) (*.f64 (pow.f64 t 2) 1)) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (+.f64 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite=> *-rgt-identity_binary64 (pow.f64 t 2))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 l 2) x)) (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))))): 0 points increase in error, 0 points decrease in error

   if -4.2499999999999998e-153 < t < -8.49999999999999982e-255

   1. Initial program 60.4

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

   2. Taylor expanded in x around inf 34.0

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

   3. Simplified 34.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      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))): 3 points increase in error, 7 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 34.0

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

   5. Simplified 34.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(2 \cdot \left(t \cdot t\right) - \frac{\left(t \cdot t\right) \cdot -4 - \ell \cdot \ell}{x}\right)}}} \]
      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 -inf 34.1

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

   7. Simplified 34.1

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

   8. Applied egg-rr 34.1

      \[\leadsto \color{blue}{-\sqrt{\frac{2}{2 + \frac{4}{x}}} \cdot \frac{t}{t}} \]

   if -8.49999999999999982e-255 < t < 8.866609018474782e47

   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 x around inf 20.5

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

   3. Simplified 20.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 3 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 (pow.f64 l 2) x) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (/.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) x)))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 20.7

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

   5. Simplified 17.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      Proof
      (*.f64 (/.f64 l x) l): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))): 32 points increase in error, 17 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 37 points increase in error, 21 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   if 8.866609018474782e47 < t

   1. Initial program 45.0

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

   2. Taylor expanded in x around inf 43.5

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

   3. Simplified 43.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 3 points increase in error, 7 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 43.5

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

   5. Simplified 43.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(2 \cdot \left(t \cdot t\right) - \frac{\left(t \cdot t\right) \cdot -4 - \ell \cdot \ell}{x}\right)}}} \]
      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 -inf 63.0

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

   7. Simplified 63.0

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

   8. Applied egg-rr 4.4

      \[\leadsto \color{blue}{\frac{t}{t} \cdot \sqrt{\frac{2}{2 + \frac{4}{x}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.337537014097745 \cdot 10^{+116}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq -4.25 \cdot 10^{-153}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}} \cdot \frac{-t}{t}\\ \mathbf{elif}\;t \leq 8.866609018474782 \cdot 10^{+47}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}} \cdot \frac{t}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.9
Cost	20356

\[\begin{array}{l} t_1 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \frac{t_2}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{if}\;t \leq -4.337537014097745 \cdot 10^{+116}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(t \cdot \left(-\sqrt{2}\right)\right)}\\ \mathbf{elif}\;t \leq -4.25 \cdot 10^{-153}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;t_1 \cdot \frac{-t}{t}\\ \mathbf{elif}\;t \leq 8.866609018474782 \cdot 10^{+47}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t}{t}\\ \end{array} \]

Alternative 2
Error	11.8
Cost	14544

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell \cdot \ell}{x}\right)}}\\ t_2 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ t_3 := t_2 \cdot \frac{-t}{t}\\ \mathbf{if}\;t \leq -1.9431531786930383 \cdot 10^{+70}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.25 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 8.866609018474782 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t}{t}\\ \end{array} \]

Alternative 3
Error	10.0
Cost	14544

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ t_3 := t_2 \cdot \frac{-t}{t}\\ \mathbf{if}\;t \leq -4.337537014097745 \cdot 10^{+116}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.25 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 8.866609018474782 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t}{t}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	13768

\[\begin{array}{l} t_1 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;t_1 \cdot \frac{-t}{t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t}{t}\\ \end{array} \]

Alternative 5
Error	14.3
Cost	13768

\[\begin{array}{l} t_1 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-255}:\\ \;\;\;\;t_1 \cdot \frac{-t}{t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t}{t}\\ \end{array} \]

Alternative 6
Error	34.4
Cost	7368

\[\begin{array}{l} t_1 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{elif}\;\ell \leq 0.052:\\ \;\;\;\;\sqrt{\frac{2}{2 + \frac{4}{x}}} \cdot \frac{t}{t}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+185}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	14.4
Cost	7368

\[\begin{array}{l} t_1 := \sqrt{\frac{2}{2 + \frac{4}{x}}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-252}:\\ \;\;\;\;t_1 \cdot \frac{-t}{t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t}{t}\\ \end{array} \]

Alternative 8
Error	34.6
Cost	7248

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{elif}\;\ell \leq 0.052:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+185}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \end{array} \]

Alternative 9
Error	34.6
Cost	7248

\[\begin{array}{l} t_1 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{t \cdot \left(-\sqrt{x}\right)}{\ell}\\ \mathbf{elif}\;\ell \leq 0.052:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{+134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+185}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	35.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	34.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	38.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022315 
(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)))))