Toniolo and Linder, Equation (7)

Average Error: 43.2 → 9.3

Time: 30.9s

Precision: binary64

Cost: 28108

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \ell \cdot \frac{\ell}{x}\\ t_3 := 2 \cdot \left(t \cdot t\right)\\ t_4 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-253}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(t_2 + \left(\left(t \cdot t\right) \cdot t_4 + t_2\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-194}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(t_3 + \left(\ell \cdot \ell + t_3\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t_1\right)}\\ \mathbf{elif}\;t \leq 10^{+143}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(t_4, t \cdot t, t_2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (* l (/ l x)))
        (t_3 (* 2.0 (* t t)))
        (t_4 (+ 2.0 (/ 4.0 x))))
   (if (<= t -1.05e+108)
     (+ (/ 1.0 x) -1.0)
     (if (<= t 5e-253)
       (*
        t
        (/ (sqrt 2.0) (pow (pow (+ t_2 (+ (* (* t t) t_4) t_2)) 0.25) 2.0)))
       (if (<= t 7.8e-194)
         (/
          t_1
          (fma
           0.5
           (/ (+ (* l l) (+ t_3 (+ (* l l) t_3))) (* (sqrt 2.0) (* t x)))
           t_1))
         (if (<= t 1e+143)
           (* t (sqrt (/ 2.0 (fma l (/ l x) (fma t_4 (* t t) t_2)))))
           (+ 1.0 (+ (/ -1.0 x) (/ (/ 0.5 x) x)))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = l * (l / x);
	double t_3 = 2.0 * (t * t);
	double t_4 = 2.0 + (4.0 / x);
	double tmp;
	if (t <= -1.05e+108) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 5e-253) {
		tmp = t * (sqrt(2.0) / pow(pow((t_2 + (((t * t) * t_4) + t_2)), 0.25), 2.0));
	} else if (t <= 7.8e-194) {
		tmp = t_1 / fma(0.5, (((l * l) + (t_3 + ((l * l) + t_3))) / (sqrt(2.0) * (t * x))), t_1);
	} else if (t <= 1e+143) {
		tmp = t * sqrt((2.0 / fma(l, (l / x), fma(t_4, (t * t), t_2))));
	} else {
		tmp = 1.0 + ((-1.0 / x) + ((0.5 / x) / x));
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(l * Float64(l / x))
	t_3 = Float64(2.0 * Float64(t * t))
	t_4 = Float64(2.0 + Float64(4.0 / x))
	tmp = 0.0
	if (t <= -1.05e+108)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 5e-253)
		tmp = Float64(t * Float64(sqrt(2.0) / ((Float64(t_2 + Float64(Float64(Float64(t * t) * t_4) + t_2)) ^ 0.25) ^ 2.0)));
	elseif (t <= 7.8e-194)
		tmp = Float64(t_1 / fma(0.5, Float64(Float64(Float64(l * l) + Float64(t_3 + Float64(Float64(l * l) + t_3))) / Float64(sqrt(2.0) * Float64(t * x))), t_1));
	elseif (t <= 1e+143)
		tmp = Float64(t * sqrt(Float64(2.0 / fma(l, Float64(l / x), fma(t_4, Float64(t * t), t_2)))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / x) + Float64(Float64(0.5 / x) / x)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.05e+108], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 5e-253], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[N[Power[N[(t$95$2 + N[(N[(N[(t * t), $MachinePrecision] * t$95$4), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.8e-194], N[(t$95$1 / N[(0.5 * N[(N[(N[(l * l), $MachinePrecision] + N[(t$95$3 + N[(N[(l * l), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+143], N[(t * N[Sqrt[N[(2.0 / N[(l * N[(l / x), $MachinePrecision] + N[(t$95$4 * N[(t * t), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] + N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.05000000000000005e108

   1. Initial program 52.7

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

   2. Taylor expanded in t around -inf 3.5

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

   3. Simplified 3.5

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{x + 1}} \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)} \]
      Proof
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (+.f64 -1 x) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 1 x)))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 2.6

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

   5. Taylor expanded in x around inf 3.0

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

   if -1.05000000000000005e108 < t < 4.99999999999999971e-253

   1. Initial program 39.8

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

   2. Simplified 43.6

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}} \cdot t} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (fma.f64 (*.f64 2 t) t (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 2 t) t) (*.f64 l l)))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l)))) t): 12 points increase in error, 32 points decrease in error
      (Rewrite=> associate-*l/_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))))): 20 points increase in error, 14 points decrease in error

   3. Taylor expanded in x around inf 18.6

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

   4. Simplified 18.6

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]
      Proof
      (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.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 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))) (/.f64 (neg.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 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (/.f64 (neg.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<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (*.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (*.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (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
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= associate-*r/_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
      (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

   5. Applied egg-rr 18.7

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

   6. Taylor expanded in t around 0 18.7

      \[\leadsto \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\left(\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]

   7. Simplified 15.6

      \[\leadsto \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \ell \cdot \frac{\ell}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]
      Proof
      (fma.f64 (*.f64 t t) (+.f64 2 (+.f64 (/.f64 2 x) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 2 (+.f64 (/.f64 2 x) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (*.f64 2 (/.f64 1 x)) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (*.f64 2 (/.f64 1 x)) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 2 (*.f64 2 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 l (/.f64 l x))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (*.f64 2 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 1 (/.f64 1 x)))) (*.f64 2 (/.f64 1 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 l l) x))): 17 points increase in error, 2 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (pow.f64 t 2))) (/.f64 (pow.f64 l 2) x)): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 15.6

      \[\leadsto \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\left(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \ell \cdot \frac{\ell}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]

   if 4.99999999999999971e-253 < t < 7.7999999999999997e-194

   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 24.8

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

   3. Simplified 24.8

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(2 \cdot \left(t \cdot t\right) - \left(-\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}} \]
      Proof
      (fma.f64 1/2 (/.f64 (+.f64 (*.f64 l l) (-.f64 (*.f64 2 (*.f64 t t)) (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))))) (*.f64 (sqrt.f64 2) (*.f64 t x))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (+.f64 (*.f64 l l) (-.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (neg.f64 (+.f64 (*.f64 2 (*.f64 t t)) (*.f64 l l))))) (*.f64 (sqrt.f64 2) (*.f64 t x))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (+.f64 (*.f64 l l) (-.f64 (*.f64 2 (pow.f64 t 2)) (neg.f64 (+.f64 (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 t 2))) (*.f64 l l))))) (*.f64 (sqrt.f64 2) (*.f64 t x))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (+.f64 (*.f64 l l) (-.f64 (*.f64 2 (pow.f64 t 2)) (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (pow.f64 t 2))))))) (*.f64 (sqrt.f64 2) (*.f64 t x))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (+.f64 (*.f64 l l) (-.f64 (*.f64 2 (pow.f64 t 2)) (neg.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 2 (pow.f64 t 2)))))) (*.f64 (sqrt.f64 2) (*.f64 t x))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (+.f64 (*.f64 l l) (-.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))))))) (*.f64 (sqrt.f64 2) (*.f64 t x))) (*.f64 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 l l) (*.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 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 1/2 (/.f64 (-.f64 (+.f64 (Rewrite<= unpow2_binary64 (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 t (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (fma.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))) (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_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 7.7999999999999997e-194 < t < 1e143

   1. Initial program 27.9

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

   2. Simplified 38.0

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2 \cdot t, t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}} \cdot t} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (fma.f64 (*.f64 2 t) t (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 2 t) t) (*.f64 l l)))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 t t))) (*.f64 l l))) (+.f64 x -1)) (*.f64 l l)))) t): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (+.f64 x -1)) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (+.f64 x (Rewrite<= metadata-eval (neg.f64 1)))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (/.f64 (*.f64 (+.f64 x 1) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (Rewrite<= sub-neg_binary64 (-.f64 x 1))) (*.f64 l l)))) t): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) (sqrt.f64 (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t))))) (*.f64 l l)))) t): 12 points increase in error, 32 points decrease in error
      (Rewrite=> associate-*l/_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))))): 20 points increase in error, 14 points decrease in error

   3. Taylor expanded in x around inf 13.2

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

   4. Simplified 13.2

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]
      Proof
      (+.f64 (/.f64 (*.f64 l l) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l))) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x) (-.f64 (*.f64 2 (+.f64 (/.f64 (*.f64 t t) x) (*.f64 t t))) (/.f64 (neg.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 2 (+.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) x) (*.f64 t t))) (/.f64 (neg.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 2 (+.f64 (/.f64 (pow.f64 t 2) x) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))) (/.f64 (neg.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<= distribute-lft-out_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2)))) (/.f64 (neg.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (*.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.f64 (+.f64 (*.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (neg.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 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (/.f64 (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
      (+.f64 (/.f64 (pow.f64 l 2) x) (-.f64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) x)) (*.f64 2 (pow.f64 t 2))) (Rewrite<= associate-*r/_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
      (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

   5. Applied egg-rr 13.3

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

   6. Taylor expanded in t around 0 13.3

      \[\leadsto \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\left(\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]

   7. Simplified 8.0

      \[\leadsto \frac{\sqrt{2}}{{\left({\left(\frac{\ell}{x} \cdot \ell + \color{blue}{\mathsf{fma}\left(t \cdot t, 2 + \left(\frac{2}{x} + \frac{2}{x}\right), \ell \cdot \frac{\ell}{x}\right)}\right)}^{0.25}\right)}^{2}} \cdot t \]
      Proof
      (fma.f64 (*.f64 t t) (+.f64 2 (+.f64 (/.f64 2 x) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 2 (+.f64 (/.f64 2 x) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))) (/.f64 2 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (*.f64 2 (/.f64 1 x)) (/.f64 (Rewrite<= metadata-eval (*.f64 2 1)) x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 2 (+.f64 (*.f64 2 (/.f64 1 x)) (Rewrite<= associate-*r/_binary64 (*.f64 2 (/.f64 1 x))))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 2 (*.f64 2 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (*.f64 l (/.f64 l x))): 0 points increase in error, 1 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (+.f64 (Rewrite<= metadata-eval (*.f64 2 1)) (*.f64 2 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 1 (/.f64 1 x)))) (*.f64 2 (/.f64 1 x))) (*.f64 l (/.f64 l x))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 l l) x))): 17 points increase in error, 2 points decrease in error
      (fma.f64 (pow.f64 t 2) (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 t 2) (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x)))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (*.f64 2 (+.f64 1 (/.f64 1 x))) (*.f64 2 (/.f64 1 x))) (pow.f64 t 2))) (/.f64 (pow.f64 l 2) x)): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 39.4

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(\ell, \frac{\ell}{x}, \left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right)\right)\right)}}\right)} - 1\right)} \cdot t \]

   9. Simplified 8.3

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(2 + \frac{4}{x}, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}} \cdot t \]
      Proof
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (fma.f64 (+.f64 2 (/.f64 4 x)) (*.f64 t t) (*.f64 l (/.f64 l x)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (fma.f64 (+.f64 2 (/.f64 (Rewrite<= metadata-eval (*.f64 4 1)) x)) (*.f64 t t) (*.f64 l (/.f64 l x)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (fma.f64 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 4 (/.f64 1 x)))) (*.f64 t t) (*.f64 l (/.f64 l x)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (fma.f64 (+.f64 2 (*.f64 4 (/.f64 1 x))) (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (*.f64 l (/.f64 l x)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (+.f64 2 (*.f64 4 (/.f64 1 x))) (pow.f64 t 2)) (*.f64 l (/.f64 l x))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 t 2) (+.f64 2 (*.f64 4 (/.f64 1 x))))) (*.f64 l (/.f64 l x)))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 l (/.f64 l x)) (*.f64 (pow.f64 t 2) (+.f64 2 (*.f64 4 (/.f64 1 x))))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (+.f64 (*.f64 l (/.f64 l x)) (*.f64 (pow.f64 t 2) (+.f64 2 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 4 1) x)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (+.f64 (*.f64 l (/.f64 l x)) (*.f64 (pow.f64 t 2) (+.f64 2 (/.f64 (Rewrite=> metadata-eval 4) x))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (+.f64 (*.f64 l (/.f64 l x)) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 t t)) (+.f64 2 (/.f64 4 x))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (Rewrite<= fma-udef_binary64 (fma.f64 l (/.f64 l x) (*.f64 (*.f64 t t) (+.f64 2 (/.f64 4 x)))))))): 0 points increase in error, 2 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (fma.f64 l (/.f64 l x) (*.f64 (*.f64 t t) (+.f64 2 (/.f64 4 x)))))))))): 32 points increase in error, 23 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (sqrt.f64 (/.f64 2 (fma.f64 l (/.f64 l x) (fma.f64 l (/.f64 l x) (*.f64 (*.f64 t t) (+.f64 2 (/.f64 4 x))))))))) 1)): 49 points increase in error, 47 points decrease in error

   if 1e143 < t

   1. Initial program 59.5

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

   2. Taylor expanded in t around -inf 63.0

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

   3. Simplified 63.0

      \[\leadsto \color{blue}{-\sqrt{\frac{-1 + x}{x + 1}} \cdot \left(\sqrt{2} \cdot \sqrt{0.5}\right)} \]
      Proof
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (+.f64 -1 x) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x -1)) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 x 1))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (*.f64 (sqrt.f64 (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 1 x)))) (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 63.0

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

   5. Taylor expanded in x around -inf 64.0

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

   6. Simplified 2.2

      \[\leadsto -\color{blue}{\left(-1 + \left(\frac{1}{x} + \frac{\frac{-0.5}{x}}{x}\right)\right)} \cdot 1 \]
      Proof
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (/.f64 -1/2 x) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (Rewrite=> associate-/l/_binary64 (/.f64 -1/2 (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (Rewrite<= metadata-eval (*.f64 -1/2 1)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (Rewrite<= metadata-eval (+.f64 3 -2))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (Rewrite<= metadata-eval (+.f64 2 1)) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (Rewrite<= metadata-eval (/.f64 1 1))) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (Rewrite<= metadata-eval (*.f64 -1 -1)))) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) -1))) -2)) (*.f64 x x)))): 256 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)) -1))) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))))) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2))))) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (Rewrite=> pow-sqr_binary64 (pow.f64 (sqrt.f64 -1) (*.f64 2 2))))) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) (Rewrite=> metadata-eval 4)))) -2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))) (Rewrite<= metadata-eval (*.f64 2 -1)))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))) (*.f64 2 (Rewrite<= metadata-eval (/.f64 1 -1))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))) (*.f64 2 (/.f64 1 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))) (*.f64 2 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (Rewrite=> associate-+l+_binary64 (+.f64 2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) (Rewrite<= metadata-eval (*.f64 2 2)))) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (pow.f64 (sqrt.f64 -1) 2)))) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) (pow.f64 (sqrt.f64 -1) 2))) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (*.f64 (Rewrite=> rem-square-sqrt_binary64 -1) (pow.f64 (sqrt.f64 -1) 2))) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (*.f64 -1 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))))) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (*.f64 -1 (Rewrite=> rem-square-sqrt_binary64 -1))) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (Rewrite=> metadata-eval 1)) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (Rewrite=> metadata-eval 1) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (*.f64 2 (/.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (*.f64 2 (/.f64 1 (Rewrite=> rem-square-sqrt_binary64 -1)))))) (*.f64 x x)))): 0 points increase in error, 256 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (*.f64 2 (Rewrite=> metadata-eval -1))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (Rewrite=> metadata-eval -2)))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite=> metadata-eval -1))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite<= metadata-eval (/.f64 1 -1)))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 1 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))))) (*.f64 x x)))): 256 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 (sqrt.f64 -1)) (sqrt.f64 -1))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 -1 -1)) (sqrt.f64 -1)) (sqrt.f64 -1)))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 -1 (sqrt.f64 -1)) -1)) (sqrt.f64 -1)))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 -1 (sqrt.f64 -1)) (/.f64 -1 (sqrt.f64 -1)))))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)))) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (/.f64 (*.f64 -1/2 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2))) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (/.f64 1 x) (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (/.f64 1 x) (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2)))) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2))) (/.f64 1 x))) -1): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2))) (+.f64 (/.f64 1 x) -1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2))) (+.f64 (/.f64 1 x) (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2))) (+.f64 (/.f64 1 x) (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (sqrt.f64 -1) 2) (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-253}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(\ell \cdot \frac{\ell}{x} + \left(\left(t \cdot t\right) \cdot \left(2 + \frac{4}{x}\right) + \ell \cdot \frac{\ell}{x}\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-194}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\ell \cdot \ell + \left(2 \cdot \left(t \cdot t\right) + \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 10^{+143}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(2 + \frac{4}{x}, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	9.3
Cost	27596

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+106}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-255}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(t_1 + \left(\left(t \cdot t\right) \cdot t_2 + t_1\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-205}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(\frac{\ell \cdot \ell}{t \cdot x}, \sqrt{\frac{1}{t_2}}, t \cdot \sqrt{t_2}\right)}\\ \mathbf{elif}\;t \leq 1.56 \cdot 10^{+143}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(t_2, t \cdot t, t_1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 2
Error	9.7
Cost	21192

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := 2 + \frac{4}{x}\\ \mathbf{if}\;t \leq -7 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-252}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{{\left({\left(t_1 + \left(\left(t \cdot t\right) \cdot t_2 + t_1\right)\right)}^{0.25}\right)}^{2}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-207}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.72 \cdot 10^{+143}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(t_2, t \cdot t, t_1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 3
Error	9.9
Cost	21072

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\ell, \frac{\ell}{x}, \mathsf{fma}\left(2 + \frac{4}{x}, t \cdot t, \ell \cdot \frac{\ell}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-194}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 4
Error	12.1
Cost	15440

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{-32}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(\ell \cdot \frac{\ell}{x} + t_2\right)}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7500000:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{t_1 + \left(t_2 + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{{x}^{3}} + \left(\frac{-1}{x} + \left(1 + \frac{\frac{0.5}{x}}{x}\right)\right)\\ \end{array} \]

Alternative 5
Error	12.0
Cost	15440

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-31}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-253}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_1 + \left(\ell \cdot \frac{\ell}{x} + t_2\right)}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-203}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 80000000:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{t_1 + \left(t_2 + \frac{\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{{x}^{3}} + \left(\frac{-1}{x} + \left(1 + \frac{\frac{0.5}{x}}{x}\right)\right)\\ \end{array} \]

Alternative 6
Error	12.0
Cost	15056

\[\begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\ell \cdot \frac{\ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{-30}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-250}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{{x}^{3}} + \left(\frac{-1}{x} + \left(1 + \frac{\frac{0.5}{x}}{x}\right)\right)\\ \end{array} \]

Alternative 7
Error	13.8
Cost	13768

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-179}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-254}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{{x}^{3}} + \left(\frac{-1}{x} + \left(1 + \frac{\frac{0.5}{x}}{x}\right)\right)\\ \end{array} \]

Alternative 8
Error	14.4
Cost	7688

\[\begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-184}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-299}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{{x}^{3}} + \left(\frac{-1}{x} + \left(1 + \frac{\frac{0.5}{x}}{x}\right)\right)\\ \end{array} \]

Alternative 9
Error	14.5
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-187}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) + \frac{\frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-298}:\\ \;\;\;\;t \cdot \frac{-\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 10
Error	14.5
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-186}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) + \frac{\frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-297}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{-\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 11
Error	14.6
Cost	7048

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-176}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{1 + x}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-298}:\\ \;\;\;\;\frac{t \cdot \sqrt{x}}{-\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 12
Error	14.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{-239}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) + \frac{\frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-250}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 13
Error	14.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-239}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) + \frac{\frac{-0.5}{x}}{x}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-249}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 14
Error	14.7
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 15
Error	14.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) + \frac{\frac{-0.5}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{x} + \frac{\frac{0.5}{x}}{x}\right)\\ \end{array} \]

Alternative 16
Error	15.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 17
Error	14.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 18
Error	15.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19
Error	38.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022321 
(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)))))