Toniolo and Linder, Equation (7)

Average Error: 42.6 → 9.7

Time: 30.9s

Precision: binary64

Cost: 21080

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := 2 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-258}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_3 + \left(t_1 - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\frac{t_2}{\frac{\ell}{\sqrt{2}} \cdot \frac{-\ell}{t \cdot x} - t_2}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{\sqrt{\mathsf{fma}\left(2, t_1, t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* l (/ l x))) (t_2 (* t (sqrt 2.0))) (t_3 (* 2.0 (* t t))))
   (if (<= t -8e+45)
     (/ t_2 (* t_2 (- (sqrt (/ (+ x 1.0) (+ x -1.0))))))
     (if (<= t -3.1e-258)
       (/ t_2 (sqrt (+ (/ l (/ x l)) (+ t_3 (- t_1 (/ (* t t) (/ x -4.0)))))))
       (if (<= t 2e-286)
         (/ t_2 (- (* (/ l (sqrt 2.0)) (/ (- l) (* t x))) t_2))
         (if (<= t 2.65e-236)
           (/ (sqrt x) (/ l t))
           (if (<= t 4e-161)
             1.0
             (if (<= t 4.259322231528669e+74)
               (/ (sqrt (* t (* t 2.0))) (sqrt (fma 2.0 t_1 t_3)))
               1.0))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = l * (l / x);
	double t_2 = t * sqrt(2.0);
	double t_3 = 2.0 * (t * t);
	double tmp;
	if (t <= -8e+45) {
		tmp = t_2 / (t_2 * -sqrt(((x + 1.0) / (x + -1.0))));
	} else if (t <= -3.1e-258) {
		tmp = t_2 / sqrt(((l / (x / l)) + (t_3 + (t_1 - ((t * t) / (x / -4.0))))));
	} else if (t <= 2e-286) {
		tmp = t_2 / (((l / sqrt(2.0)) * (-l / (t * x))) - t_2);
	} else if (t <= 2.65e-236) {
		tmp = sqrt(x) / (l / t);
	} else if (t <= 4e-161) {
		tmp = 1.0;
	} else if (t <= 4.259322231528669e+74) {
		tmp = sqrt((t * (t * 2.0))) / sqrt(fma(2.0, t_1, t_3));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(l * Float64(l / x))
	t_2 = Float64(t * sqrt(2.0))
	t_3 = Float64(2.0 * Float64(t * t))
	tmp = 0.0
	if (t <= -8e+45)
		tmp = Float64(t_2 / Float64(t_2 * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))));
	elseif (t <= -3.1e-258)
		tmp = Float64(t_2 / sqrt(Float64(Float64(l / Float64(x / l)) + Float64(t_3 + Float64(t_1 - Float64(Float64(t * t) / Float64(x / -4.0)))))));
	elseif (t <= 2e-286)
		tmp = Float64(t_2 / Float64(Float64(Float64(l / sqrt(2.0)) * Float64(Float64(-l) / Float64(t * x))) - t_2));
	elseif (t <= 2.65e-236)
		tmp = Float64(sqrt(x) / Float64(l / t));
	elseif (t <= 4e-161)
		tmp = 1.0;
	elseif (t <= 4.259322231528669e+74)
		tmp = Float64(sqrt(Float64(t * Float64(t * 2.0))) / sqrt(fma(2.0, t_1, t_3)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8e+45], N[(t$95$2 / N[(t$95$2 * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.1e-258], N[(t$95$2 / N[Sqrt[N[(N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 + N[(t$95$1 - N[(N[(t * t), $MachinePrecision] / N[(x / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e-286], N[(t$95$2 / N[(N[(N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[((-l) / N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.65e-236], N[(N[Sqrt[x], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-161], 1.0, If[LessEqual[t, 4.259322231528669e+74], N[(N[Sqrt[N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(2.0 * t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.0]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -7.9999999999999994e45

   1. Initial program 43.4

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

   2. Taylor expanded in t around -inf 3.6

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

   3. Simplified 3.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{x + 1}{x + -1}} \cdot \left(-t \cdot \sqrt{2}\right)}} \]
      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 -7.9999999999999994e45 < t < -3.09999999999999999e-258

   1. Initial program 39.1

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

   2. Taylor expanded in x around inf 16.9

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

   3. Simplified 16.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 2 points increase in error, 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 16.9

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

   5. Simplified 16.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(2 \cdot \left(t \cdot t\right) - \frac{\left(t \cdot t\right) \cdot -4 - \ell \cdot \ell}{x}\right)}}} \]
      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. Applied egg-rr 12.9

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

   if -3.09999999999999999e-258 < t < 2.0000000000000001e-286

   1. Initial program 62.7

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

   2. Taylor expanded in x around inf 31.7

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

   3. Simplified 31.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 2 points increase in error, 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 31.7

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

   5. Simplified 31.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\left(2 \cdot \left(t \cdot t\right) - \frac{\left(t \cdot t\right) \cdot -4 - \ell \cdot \ell}{x}\right)}}} \]
      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 31.7

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

   7. Simplified 31.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) - \color{blue}{\frac{-\ell \cdot \ell}{x}}\right)}} \]
      Proof
      (/.f64 (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. Taylor expanded in t around -inf 30.3

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

   9. Simplified 30.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\ell}{\sqrt{2}} \cdot \left(-\frac{\ell}{t \cdot x}\right) - \sqrt{2} \cdot t}} \]
      Proof
      (-.f64 (*.f64 (/.f64 l (sqrt.f64 2)) (neg.f64 (/.f64 l (*.f64 t x)))) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 l (sqrt.f64 2)) (/.f64 l (*.f64 t x))))) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (neg.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 (sqrt.f64 2) (*.f64 t x))))) (*.f64 (sqrt.f64 2) t)): 37 points increase in error, 13 points decrease in error
      (-.f64 (neg.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (sqrt.f64 2) (*.f64 t x)))) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (neg.f64 (/.f64 (pow.f64 l 2) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (sqrt.f64 2) t) x)))) (*.f64 (sqrt.f64 2) t)): 6 points increase in error, 2 points decrease in error
      (-.f64 (neg.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (pow.f64 l 2) x) (*.f64 (sqrt.f64 2) t)))) (*.f64 (sqrt.f64 2) t)): 7 points increase in error, 5 points decrease in error
      (-.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (/.f64 (pow.f64 l 2) x)) (*.f64 (sqrt.f64 2) t))) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 l 2) x))) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 (pow.f64 l 2) x) -1)) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= metadata-eval (+.f64 -1/2 -1/2))) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 l 2) x)) (*.f64 -1/2 (/.f64 (pow.f64 l 2) x)))) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 -1/2 (+.f64 (/.f64 (pow.f64 l 2) x) (/.f64 (pow.f64 l 2) x)))) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 -1/2 (+.f64 (/.f64 (pow.f64 l 2) x) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (pow.f64 l 2) x))))) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 -1/2 (+.f64 (/.f64 (pow.f64 l 2) x) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (/.f64 (pow.f64 l 2) x)))) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 -1/2 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))))) (*.f64 (sqrt.f64 2) t)) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))) (*.f64 (sqrt.f64 2) t)))) (*.f64 (sqrt.f64 2) t)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1/2 (/.f64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))) (*.f64 (sqrt.f64 2) t))) (neg.f64 (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (-.f64 (/.f64 (pow.f64 l 2) x) (*.f64 -1 (/.f64 (pow.f64 l 2) x))) (*.f64 (sqrt.f64 2) t))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error

   if 2.0000000000000001e-286 < t < 2.6500000000000001e-236

   1. Initial program 64.0

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

   2. Taylor expanded in l around -inf 63.4

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

   3. Simplified 63.4

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

   4. Applied egg-rr 53.6

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

   5. Taylor expanded in x around inf 36.0

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

   6. Applied egg-rr 36.2

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

   if 2.6500000000000001e-236 < t < 4.00000000000000011e-161 or 4.2593222315286691e74 < t

   1. Initial program 50.3

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

   2. Taylor expanded in x around inf 10.8

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   3. Applied egg-rr 10.0

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

   if 4.00000000000000011e-161 < t < 4.2593222315286691e74

   1. Initial program 26.5

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

   2. Applied egg-rr 35.2

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

   3. Taylor expanded in x around inf 20.3

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

   4. Simplified 4.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{\ell}{x} \cdot \ell, \mathsf{fma}\left(2, t \cdot t, 4 \cdot \left(\frac{t}{x} \cdot t\right)\right) + 0\right)}}} \]
      Proof
      (fma.f64 2 (*.f64 (/.f64 l x) l) (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 4 (*.f64 (/.f64 t x) t))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/r/_binary64 (/.f64 l (/.f64 x l))) (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 4 (*.f64 (/.f64 t x) t))) 0)): 9 points increase in error, 5 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 4 (*.f64 (/.f64 t x) t))) 0)): 19 points increase in error, 6 points decrease in error
      (fma.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (+.f64 (fma.f64 2 (*.f64 t t) (*.f64 4 (*.f64 (/.f64 t x) t))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 4 (*.f64 (/.f64 t x) t))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (fma.f64 2 (pow.f64 t 2) (*.f64 4 (Rewrite<= associate-/r/_binary64 (/.f64 t (/.f64 x t))))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (fma.f64 2 (pow.f64 t 2) (*.f64 4 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (fma.f64 2 (pow.f64 t 2) (*.f64 4 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 4 (/.f64 (pow.f64 t 2) x)))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) 0)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= mul0-lft_binary64 (*.f64 0 (pow.f64 l 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (+.f64 -1 1)) (pow.f64 l 2)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 -1 (pow.f64 l 2)) (pow.f64 l 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (pow.f64 l 2))) (pow.f64 l 2)))): 8 points increase in error, 9 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2))))) (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (pow.f64 l 2) x) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 l 2) x)) (+.f64 (pow.f64 l 2) (+.f64 (*.f64 4 (/.f64 (pow.f64 t 2) x)) (+.f64 (*.f64 2 (pow.f64 t 2)) (*.f64 -1 (pow.f64 l 2))))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around inf 4.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{\ell}{x} \cdot \ell, \color{blue}{2 \cdot {t}^{2}} + 0\right)}} \]

   6. Simplified 4.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\mathsf{fma}\left(2, \frac{\ell}{x} \cdot \ell, \color{blue}{2 \cdot \left(t \cdot t\right)} + 0\right)}} \]
      Proof
      (*.f64 2 (*.f64 t t)): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 4.6

      \[\leadsto \frac{\color{blue}{\sqrt{t \cdot \left(2 \cdot t\right)}}}{\sqrt{\mathsf{fma}\left(2, \frac{\ell}{x} \cdot \ell, 2 \cdot \left(t \cdot t\right) + 0\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-258}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \frac{\ell}{x} - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\ell}{\sqrt{2}} \cdot \frac{-\ell}{t \cdot x} - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{\sqrt{\mathsf{fma}\left(2, \ell \cdot \frac{\ell}{x}, 2 \cdot \left(t \cdot t\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	39496

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell}{\sqrt{x}}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(\mathsf{hypot}\left(t_1, t_2\right), t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	9.3
Cost	33868

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := 2 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-258}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_3 + \left(t_1 - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-161}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;\frac{\sqrt{t \cdot \left(t \cdot 2\right)}}{\sqrt{\mathsf{fma}\left(2, t_1, t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	10.2
Cost	20952

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := 2 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-258}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_3 + \left(t_1 - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-286}:\\ \;\;\;\;\frac{t_2}{\frac{\ell}{\sqrt{2}} \cdot \frac{-\ell}{t \cdot x} - t_2}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;\frac{t_2}{\sqrt{\mathsf{fma}\left(2, t_1, t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	10.0
Cost	20688

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := 2 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t_2}{t_2 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_3 + \left(t_1 - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;\frac{t_2}{\sqrt{\mathsf{fma}\left(2, t_1, t_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	10.0
Cost	20356

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \frac{\ell}{x} - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t_1}{t_1 \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	10.1
Cost	20100

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \frac{\ell}{x} - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{if}\;t \leq -1.0718741334502336 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \left(-\sqrt{\frac{x + -1}{x + 1}}\right)\right)\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	10.1
Cost	15184

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \frac{\ell}{x} - \frac{t \cdot t}{\frac{x}{-4}}\right)\right)}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	14.6
Cost	14420

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-139}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\left(x \cdot \left(t \cdot t\right)\right) \cdot \frac{2}{2 \cdot \left(\ell \cdot \ell\right)}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + 2 \cdot \left(t \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	10.2
Cost	14416

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{t_1}{\sqrt{2 \cdot \left(t \cdot t\right) + 2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+45}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-140}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.259322231528669 \cdot 10^{+74}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	14.8
Cost	13768

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	14.6
Cost	13768

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{t_1}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	15.2
Cost	13444

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-175}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \left(-\sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{1}{\frac{\ell}{t \cdot \sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13
Error	35.8
Cost	7248

\[\begin{array}{l} t_1 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{-t}}\\ \mathbf{elif}\;\ell \leq 9.67813334717305 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{+203}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	35.7
Cost	7248

\[\begin{array}{l} t_1 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{x} \cdot \frac{-t}{\ell}\\ \mathbf{elif}\;\ell \leq 9.67813334717305 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{+203}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	34.9
Cost	7248

\[\begin{array}{l} t_1 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{\ell} \cdot \frac{t}{{x}^{-0.5}}\\ \mathbf{elif}\;\ell \leq 9.67813334717305 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{+203}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	37.7
Cost	7116

\[\begin{array}{l} t_1 := \frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{if}\;\ell \leq 9.67813334717305 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.85 \cdot 10^{+203}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	35.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 2.65 \cdot 10^{-236}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 18
Error	39.4
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022325 
(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)))))