Toniolo and Linder, Equation (7)

Average Error: 42.7 → 9.7

Time: 34.0s

Precision: binary64

Cost: 35544

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_2 := 2 \cdot t_1\\ t_3 := t \cdot \sqrt{2}\\ t_4 := 1 + \frac{2}{x}\\ t_5 := \frac{t}{\frac{x}{t}} + t \cdot t\\ t_6 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_7 := \frac{\ell}{\frac{x}{\ell}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;-t_6\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-253}:\\ \;\;\;\;\frac{t_3}{\sqrt{t_7 + \mathsf{fma}\left(2, t_5, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-280}:\\ \;\;\;\;\frac{t_3}{-\mathsf{fma}\left(t, \sqrt{2 \cdot t_4}, \sqrt{\frac{0.5}{t_4}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{t_3}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t_3}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{t_2}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+34}:\\ \;\;\;\;\frac{t_3}{\sqrt{t_7 + \left(\mathsf{fma}\left(2, t_5, \frac{t_2}{x \cdot x}\right) + \frac{t_1}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

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 (fma 2.0 (* t t) (* l l)))
        (t_2 (* 2.0 t_1))
        (t_3 (* t (sqrt 2.0)))
        (t_4 (+ 1.0 (/ 2.0 x)))
        (t_5 (+ (/ t (/ x t)) (* t t)))
        (t_6 (sqrt (/ (+ -1.0 x) (+ x 1.0))))
        (t_7 (/ l (/ x l))))
   (if (<= t -1.5e+39)
     (- t_6)
     (if (<= t -9e-253)
       (/ t_3 (sqrt (+ t_7 (fma 2.0 t_5 (* l (/ l x))))))
       (if (<= t -4e-280)
         (/
          t_3
          (-
           (fma
            t
            (sqrt (* 2.0 t_4))
            (* (sqrt (/ 0.5 t_4)) (/ (* l (/ l t)) x)))))
         (if (<= t 7.6e-306)
           (/ t_3 (* l (sqrt (/ 2.0 x))))
           (if (<= t 2.4e-155)
             (/ t_3 (fma t (sqrt 2.0) (* (/ 0.5 (sqrt 2.0)) (/ t_2 (* t x)))))
             (if (<= t 9e+34)
               (/
                t_3
                (sqrt (+ t_7 (+ (fma 2.0 t_5 (/ t_2 (* x x))) (/ t_1 x)))))
               t_6))))))))

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 = fma(2.0, (t * t), (l * l));
	double t_2 = 2.0 * t_1;
	double t_3 = t * sqrt(2.0);
	double t_4 = 1.0 + (2.0 / x);
	double t_5 = (t / (x / t)) + (t * t);
	double t_6 = sqrt(((-1.0 + x) / (x + 1.0)));
	double t_7 = l / (x / l);
	double tmp;
	if (t <= -1.5e+39) {
		tmp = -t_6;
	} else if (t <= -9e-253) {
		tmp = t_3 / sqrt((t_7 + fma(2.0, t_5, (l * (l / x)))));
	} else if (t <= -4e-280) {
		tmp = t_3 / -fma(t, sqrt((2.0 * t_4)), (sqrt((0.5 / t_4)) * ((l * (l / t)) / x)));
	} else if (t <= 7.6e-306) {
		tmp = t_3 / (l * sqrt((2.0 / x)));
	} else if (t <= 2.4e-155) {
		tmp = t_3 / fma(t, sqrt(2.0), ((0.5 / sqrt(2.0)) * (t_2 / (t * x))));
	} else if (t <= 9e+34) {
		tmp = t_3 / sqrt((t_7 + (fma(2.0, t_5, (t_2 / (x * x))) + (t_1 / x))));
	} else {
		tmp = t_6;
	}
	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 = fma(2.0, Float64(t * t), Float64(l * l))
	t_2 = Float64(2.0 * t_1)
	t_3 = Float64(t * sqrt(2.0))
	t_4 = Float64(1.0 + Float64(2.0 / x))
	t_5 = Float64(Float64(t / Float64(x / t)) + Float64(t * t))
	t_6 = sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0)))
	t_7 = Float64(l / Float64(x / l))
	tmp = 0.0
	if (t <= -1.5e+39)
		tmp = Float64(-t_6);
	elseif (t <= -9e-253)
		tmp = Float64(t_3 / sqrt(Float64(t_7 + fma(2.0, t_5, Float64(l * Float64(l / x))))));
	elseif (t <= -4e-280)
		tmp = Float64(t_3 / Float64(-fma(t, sqrt(Float64(2.0 * t_4)), Float64(sqrt(Float64(0.5 / t_4)) * Float64(Float64(l * Float64(l / t)) / x)))));
	elseif (t <= 7.6e-306)
		tmp = Float64(t_3 / Float64(l * sqrt(Float64(2.0 / x))));
	elseif (t <= 2.4e-155)
		tmp = Float64(t_3 / fma(t, sqrt(2.0), Float64(Float64(0.5 / sqrt(2.0)) * Float64(t_2 / Float64(t * x)))));
	elseif (t <= 9e+34)
		tmp = Float64(t_3 / sqrt(Float64(t_7 + Float64(fma(2.0, t_5, Float64(t_2 / Float64(x * x))) + Float64(t_1 / x)))));
	else
		tmp = t_6;
	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[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t / N[(x / t), $MachinePrecision]), $MachinePrecision] + N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(l / N[(x / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+39], (-t$95$6), If[LessEqual[t, -9e-253], N[(t$95$3 / N[Sqrt[N[(t$95$7 + N[(2.0 * t$95$5 + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4e-280], N[(t$95$3 / (-N[(t * N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(0.5 / t$95$4), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 7.6e-306], N[(t$95$3 / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e-155], N[(t$95$3 / N[(t * N[Sqrt[2.0], $MachinePrecision] + N[(N[(0.5 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e+34], N[(t$95$3 / N[Sqrt[N[(t$95$7 + N[(N[(2.0 * t$95$5 + N[(t$95$2 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$6]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -1.5e39

   1. Initial program 43.1

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

   2. Simplified 43.2

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 4 points increase in error, 15 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 4 points increase in error, 9 points decrease in error

   3. Applied egg-rr 43.1

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

   4. Taylor expanded in t around inf 54.5

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

   5. Simplified 39.6

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(1 + x\right) \cdot 2}{\frac{x + -1}{t \cdot t}}}}} \]
      Proof
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (+.f64 x -1) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (-.f64 x 1) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 x) (/.f64 (-.f64 x 1) (pow.f64 t 2))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (-.f64 x 1))) 2): 59 points increase in error, 23 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in t around -inf 4.2

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

   7. Simplified 4.2

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

   if -1.5e39 < t < -9.00000000000000057e-253

   1. Initial program 40.0

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

   2. Taylor expanded in x around inf 17.2

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

   3. Simplified 17.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 1 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 17.5

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

   5. Simplified 13.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      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))): 29 points increase in error, 16 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 38 points increase in error, 20 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   if -9.00000000000000057e-253 < t < -3.9999999999999998e-280

   1. Initial program 61.5

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

   2. Taylor expanded in x around inf 35.6

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

   3. Simplified 35.5

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

   4. Taylor expanded in t around -inf 22.1

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

   5. Simplified 22.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}, \sqrt{\frac{0.5}{1 + \frac{2}{x}}} \cdot \frac{\frac{\ell}{t} \cdot \ell}{x}\right)}} \]
      Proof
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 2 x)))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (+.f64 1 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 3 points increase in error, 2 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1/2 (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (+.f64 1 (/.f64 2 x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x)))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (+.f64 1 (Rewrite<= count-2_binary64 (+.f64 (/.f64 1 x) (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 (/.f64 1 2) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 2 (+.f64 (+.f64 1 (/.f64 1 x)) (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))) (/.f64 (*.f64 (/.f64 l t) l) x)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 l t) (/.f64 l x)))))): 2 points increase in error, 24 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 t x)))))): 32 points increase in error, 8 points decrease in error
      (neg.f64 (fma.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x)))))): 0 points increase in error, 1 points decrease in error
      (neg.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t)) (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))) (/.f64 (pow.f64 l 2) (*.f64 t x))) (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))) (*.f64 (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) t))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (+.f64 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))) (Rewrite=> *-commutative_binary64 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))) (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 -1 (*.f64 (/.f64 (pow.f64 l 2) (*.f64 t x)) (sqrt.f64 (/.f64 1 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))))))) (*.f64 -1 (*.f64 t (sqrt.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))))))): 0 points increase in error, 0 points decrease in error

   if -3.9999999999999998e-280 < t < 7.6e-306

   1. Initial program 63.3

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

   2. Taylor expanded in x around inf 29.1

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

   3. Simplified 29.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 1 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 29.2

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

   5. Simplified 28.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      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))): 29 points increase in error, 16 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 38 points increase in error, 20 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 inf 29.1

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

   7. Simplified 29.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{2}{x} \cdot \left(\ell \cdot \ell\right)}}} \]
      Proof
      (*.f64 (/.f64 2 x) (*.f64 l l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 2 x) (Rewrite<= unpow2_binary64 (pow.f64 l 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 2 (/.f64 x (pow.f64 l 2)))): 21 points increase in error, 20 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 2 (pow.f64 l 2)) x)): 15 points increase in error, 10 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 2 points decrease in error

   8. Applied egg-rr 29.8

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

   if 7.6e-306 < t < 2.4e-155

   1. Initial program 61.5

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

   2. Taylor expanded in x around inf 25.9

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

   3. Simplified 25.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}} \]
      Proof
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2)))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (+.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (*.f64 (/.f64 1/2 (sqrt.f64 2)) (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 t x)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 t (sqrt.f64 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1/2 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (sqrt.f64 2) (*.f64 t x))))): 9 points increase in error, 4 points decrease in error
      (fma.f64 t (sqrt.f64 2) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (sqrt.f64 2)) (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))))): 1 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) t)) (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (-.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (sqrt.f64 2) (*.f64 t x)))) (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error

   if 2.4e-155 < t < 9.0000000000000001e34

   1. Initial program 30.3

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

   2. Taylor expanded in x around -inf 10.5

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

   3. Simplified 10.5

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \left(\mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (+.f64 (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 x x))) (/.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)) (+.f64 (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 x x))) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 1 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (+.f64 (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) x)) (*.f64 t t)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t)) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (/.f64 (*.f64 2 (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (*.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (fma.f64 2 (*.f64 t t) (*.f64 l l))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l l))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (*.f64 (+.f64 1 1) (fma.f64 2 (pow.f64 t 2) (Rewrite<= unpow2_binary64 (pow.f64 l 2)))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (pow.f64 t 2)) (pow.f64 l 2)))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (*.f64 (+.f64 1 1) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))) (*.f64 1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (+.f64 (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (+.f64 (neg.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (+.f64 (neg.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (neg.f64 (+.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (Rewrite=> mul-1-neg_binary64 (neg.f64 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))))) (*.f64 x x))) (/.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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (/.f64 (*.f64 -1 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2))))) (Rewrite<= unpow2_binary64 (pow.f64 x 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) (+.f64 (fma.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2)) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 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) (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (+.f64 (/.f64 (pow.f64 t 2) x) (pow.f64 t 2))) (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 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) (+.f64 (+.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 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 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) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (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) (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (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) (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (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) (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (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) (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (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) (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (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) (+.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (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<= sub-neg_binary64 (-.f64 (+.f64 (*.f64 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.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 -1 (/.f64 (-.f64 (*.f64 -1 (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (+.f64 (pow.f64 l 2) (*.f64 2 (pow.f64 t 2)))) (pow.f64 x 2))) (+.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

   if 9.0000000000000001e34 < t

   1. Initial program 42.5

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

   2. Simplified 42.5

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\frac{x + -1}{\mathsf{fma}\left(t, 2 \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \]
      Proof
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x -1) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (fma.f64 t (*.f64 2 t) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 t (*.f64 2 t)) (*.f64 l l))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 t) t)) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l)))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (/.f64 (+.f64 x 1) (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))))) (*.f64 l l))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (/.f64 t (sqrt.f64 (-.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l))))): 4 points increase in error, 15 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))): 4 points increase in error, 9 points decrease in error

   3. Applied egg-rr 42.4

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

   4. Taylor expanded in t around inf 53.6

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

   5. Simplified 38.4

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\left(1 + x\right) \cdot 2}{\frac{x + -1}{t \cdot t}}}}} \]
      Proof
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (+.f64 x -1) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (*.f64 t t))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 1 x) 2) (/.f64 (-.f64 x 1) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 1 x) (/.f64 (-.f64 x 1) (pow.f64 t 2))) 2)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (-.f64 x 1))) 2): 59 points increase in error, 23 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 (+.f64 1 x) (pow.f64 t 2)) (-.f64 x 1)))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in t around 0 4.1

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;-\sqrt{\frac{-1 + x}{x + 1}}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-253}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-280}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-\mathsf{fma}\left(t, \sqrt{2 \cdot \left(1 + \frac{2}{x}\right)}, \sqrt{\frac{0.5}{1 + \frac{2}{x}}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+34}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(\mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.7
Cost	34132

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := 2 \cdot \left(\ell \cdot \ell\right)\\ t_3 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_4 := 1 + \frac{2}{x}\\ t_5 := 2 \cdot \left(t \cdot t\right)\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-253}:\\ \;\;\;\;\frac{t_1}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-280}:\\ \;\;\;\;\frac{t_1}{-\mathsf{fma}\left(t, \sqrt{2 \cdot t_4}, \sqrt{\frac{0.5}{t_4}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{t_1}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(t, \sqrt{2}, \frac{0.5}{\sqrt{2}} \cdot \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot x}\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+34}:\\ \;\;\;\;\frac{t_1}{\sqrt{\left(2 \cdot \left(\frac{t_5}{x} + \left(t \cdot t + \frac{t_5}{x \cdot x}\right)\right) + \frac{t_2}{x}\right) + \frac{t_2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	9.1
Cost	27788

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_4 := 1 + \frac{2}{x}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-253}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, t_1\right)}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{t_2}{-\mathsf{fma}\left(t, \sqrt{2 \cdot t_4}, \sqrt{\frac{0.5}{t_4}} \cdot \frac{\ell \cdot \frac{\ell}{t}}{x}\right)}\\ \mathbf{elif}\;t \leq 7.4996005065882996 \cdot 10^{+109}:\\ \;\;\;\;\frac{t_2}{\sqrt{2 \cdot \left(t_1 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	9.4
Cost	21064

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_4 := -t_3\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-253}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, t_1\right)}}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-280}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 7.4996005065882996 \cdot 10^{+109}:\\ \;\;\;\;\frac{t_2}{\sqrt{2 \cdot \left(t_1 + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	9.4
Cost	14544

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x} + t \cdot \left(t + \frac{t}{x}\right)\right)}}\\ t_2 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+39}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7.4996005065882996 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	14.5
Cost	13904

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ t_2 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -2.85 \cdot 10^{-191}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	14.2
Cost	13904

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_3 := -t_2\\ \mathbf{if}\;t \leq -2.85 \cdot 10^{-191}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-253}:\\ \;\;\;\;\frac{t_1}{\sqrt{\ell \cdot \left(2 \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-280}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{t_1}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	14.6
Cost	13640

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_2 := -t_1\\ \mathbf{if}\;t \leq -2.85 \cdot 10^{-191}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-230}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-280}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	14.8
Cost	7308

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{x + 1}}\\ t_2 := -t_1\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-200}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-230}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-304}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	34.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 10
Error	35.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;t \leq 7.6 \cdot 10^{-306}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 11
Error	38.8
Cost	704

\[1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right) \]

Alternative 12
Error	38.9
Cost	320

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

Alternative 13
Error	39.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022295 
(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)))))