Toniolo and Linder, Equation (7)

Average Error: 43.5 → 10.4

Time: 26.5s

Precision: binary64

Cost: 21200

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \frac{\ell}{\frac{x}{\ell}}\\ t_3 := t + \frac{t}{x}\\ \mathbf{if}\;t \leq -8.707137624821387 \cdot 10^{+78}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-206}:\\ \;\;\;\;\frac{t_1}{{\left(t \cdot \left(2 \cdot t_3\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-78}:\\ \;\;\;\;\frac{t_1}{\sqrt{t_2 + \mathsf{fma}\left(2, t \cdot t_3, t_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))) (t_2 (/ l (/ x l))) (t_3 (+ t (/ t x))))
   (if (<= t -8.707137624821387e+78)
     (+ -1.0 (+ (/ 1.0 x) (* (/ 1.0 (* x x)) (+ (/ 0.5 x) -0.5))))
     (if (<= t -5.8e-206)
       (/ t_1 (pow (+ (* t (* 2.0 t_3)) (* 2.0 (* l (/ l x)))) 0.5))
       (if (<= t -4.7e-232)
         -1.0
         (if (<= t 9e-78)
           (/ t_1 (sqrt (+ t_2 (fma 2.0 (* t t_3) t_2))))
           (sqrt (/ (+ -1.0 x) (+ 1.0 x)))))))))

C

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

↓

double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = l / (x / l);
	double t_3 = t + (t / x);
	double tmp;
	if (t <= -8.707137624821387e+78) {
		tmp = -1.0 + ((1.0 / x) + ((1.0 / (x * x)) * ((0.5 / x) + -0.5)));
	} else if (t <= -5.8e-206) {
		tmp = t_1 / pow(((t * (2.0 * t_3)) + (2.0 * (l * (l / x)))), 0.5);
	} else if (t <= -4.7e-232) {
		tmp = -1.0;
	} else if (t <= 9e-78) {
		tmp = t_1 / sqrt((t_2 + fma(2.0, (t * t_3), t_2)));
	} else {
		tmp = sqrt(((-1.0 + x) / (1.0 + x)));
	}
	return tmp;
}

Julia

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

↓

function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = Float64(l / Float64(x / l))
	t_3 = Float64(t + Float64(t / x))
	tmp = 0.0
	if (t <= -8.707137624821387e+78)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) + Float64(Float64(1.0 / Float64(x * x)) * Float64(Float64(0.5 / x) + -0.5))));
	elseif (t <= -5.8e-206)
		tmp = Float64(t_1 / (Float64(Float64(t * Float64(2.0 * t_3)) + Float64(2.0 * Float64(l * Float64(l / x)))) ^ 0.5));
	elseif (t <= -4.7e-232)
		tmp = -1.0;
	elseif (t <= 9e-78)
		tmp = Float64(t_1 / sqrt(Float64(t_2 + fma(2.0, Float64(t * t_3), t_2))));
	else
		tmp = sqrt(Float64(Float64(-1.0 + x) / Float64(1.0 + 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[(x / l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.707137624821387e+78], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] + N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.8e-206], N[(t$95$1 / N[Power[N[(N[(t * N[(2.0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.7e-232], -1.0, If[LessEqual[t, 9e-78], N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[(2.0 * N[(t * t$95$3), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8.70713762482138709e78

   1. Initial program 48.8

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

   2. Taylor expanded in t around inf 63.0

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

   3. Simplified 63.0

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + -1}{x + 1}}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x -1) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x))))): 13 points increase in error, 0 points decrease in error

   4. Applied egg-rr 63.0

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

   5. Applied egg-rr 63.5

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

   6. Taylor expanded in x around -inf 64.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{2 \cdot \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \left(2 + \frac{1}{{\left(\sqrt{-1}\right)}^{4}}\right)}{{x}^{3}} + \left(-0.5 \cdot \frac{2 + {\left(\frac{-1}{\sqrt{-1}}\right)}^{2}}{{x}^{2}} + \left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)\right)} \]

   7. Simplified 2.9

      \[\leadsto \color{blue}{-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)} \]
      Proof
      (+.f64 -1 (+.f64 (/.f64 1 x) (*.f64 (/.f64 1 (*.f64 x x)) (+.f64 (/.f64 1/2 x) -1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) (+.f64 (/.f64 1 x) (*.f64 (/.f64 1 (*.f64 x x)) (+.f64 (/.f64 1/2 x) -1/2)))): 256 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)) (+.f64 (/.f64 1 x) (*.f64 (/.f64 1 (*.f64 x x)) (+.f64 (/.f64 1/2 x) -1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (*.f64 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 x 2))) (+.f64 (/.f64 1/2 x) -1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 1 (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (Rewrite<= metadata-eval (+.f64 1 0)) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (Rewrite<= metadata-eval (/.f64 1 1)) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (Rewrite<= metadata-eval (*.f64 -1 -1))) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) -1)) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)) -1)) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))))) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)))) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (Rewrite=> pow-sqr_binary64 (pow.f64 (sqrt.f64 -1) (*.f64 2 2)))) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) (Rewrite=> metadata-eval 4))) 0) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (Rewrite<= metadata-eval (+.f64 2 -2))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (Rewrite<= metadata-eval (*.f64 2 -1)))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (*.f64 2 (Rewrite<= metadata-eval (/.f64 1 -1))))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (*.f64 2 (/.f64 1 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))))))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (*.f64 2 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)))))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) 2))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) 2) (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 (/.f64 1/2 x) (/.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))))) (pow.f64 x 2))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 1/2 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))))) (*.f64 x (pow.f64 x 2)))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (/.f64 (*.f64 1/2 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))))) (*.f64 x (Rewrite=> unpow2_binary64 (*.f64 x x)))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (/.f64 (*.f64 1/2 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4))))) (Rewrite<= cube-mult_binary64 (pow.f64 x 3))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3)))) (*.f64 -1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1/2 1) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (Rewrite<= metadata-eval (+.f64 1 0))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (Rewrite<= metadata-eval (/.f64 1 1)) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (Rewrite<= metadata-eval (*.f64 -1 -1))) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) -1)) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)) -1)) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))))) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (*.f64 (pow.f64 (sqrt.f64 -1) 2) (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)))) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (Rewrite=> pow-sqr_binary64 (pow.f64 (sqrt.f64 -1) (*.f64 2 2)))) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) (Rewrite=> metadata-eval 4))) 0)) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (Rewrite<= metadata-eval (+.f64 2 -2)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (Rewrite<= metadata-eval (*.f64 2 -1))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (*.f64 2 (Rewrite<= metadata-eval (/.f64 1 -1)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (*.f64 2 (/.f64 1 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) (+.f64 2 (*.f64 2 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2))))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)) 2) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.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))))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.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)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.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)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.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)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.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)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.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)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.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)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (/.f64 1 (Rewrite=> metadata-eval 1)) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 (Rewrite=> metadata-eval 1) (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (*.f64 2 (/.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (*.f64 2 (/.f64 1 (Rewrite=> rem-square-sqrt_binary64 -1)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (*.f64 2 (Rewrite=> metadata-eval -1))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (+.f64 1 (Rewrite=> metadata-eval -2)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite=> metadata-eval -1))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite<= metadata-eval (/.f64 1 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 1 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 (sqrt.f64 -1)) (sqrt.f64 -1))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (sqrt.f64 -1)) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (/.f64 (neg.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1)))) (sqrt.f64 -1)) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (/.f64 (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2))) (sqrt.f64 -1)) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (pow.f64 (sqrt.f64 -1) 2) (sqrt.f64 -1)))) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (neg.f64 (/.f64 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) (sqrt.f64 -1))) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (neg.f64 (/.f64 (Rewrite=> rem-square-sqrt_binary64 -1) (sqrt.f64 -1))) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 -1 (sqrt.f64 -1)))) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (*.f64 (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) (/.f64 -1 (sqrt.f64 -1))) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)) (/.f64 -1 (sqrt.f64 -1))) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 -1 (sqrt.f64 -1)) (pow.f64 (sqrt.f64 -1) 2))) (sqrt.f64 -1)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 -1 (sqrt.f64 -1)) (/.f64 (pow.f64 (sqrt.f64 -1) 2) (sqrt.f64 -1)))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (*.f64 (/.f64 -1 (sqrt.f64 -1)) (/.f64 (Rewrite=> unpow2_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))) (sqrt.f64 -1))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (*.f64 (/.f64 -1 (sqrt.f64 -1)) (/.f64 (Rewrite=> rem-square-sqrt_binary64 -1) (sqrt.f64 -1))))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (/.f64 (*.f64 -1/2 (+.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)))) (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (pow.f64 (sqrt.f64 -1) 2) (+.f64 (/.f64 1 x) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (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<= associate-+l+_binary64 (+.f64 (+.f64 (pow.f64 (sqrt.f64 -1) 2) (/.f64 1 x)) (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (*.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 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2)))) (+.f64 (pow.f64 (sqrt.f64 -1) 2) (/.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 1/2 (/.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 2))) (+.f64 2 (/.f64 1 (pow.f64 (sqrt.f64 -1) 4)))) (pow.f64 x 3))) (+.f64 (*.f64 -1/2 (/.f64 (+.f64 2 (pow.f64 (/.f64 -1 (sqrt.f64 -1)) 2)) (pow.f64 x 2))) (+.f64 (pow.f64 (sqrt.f64 -1) 2) (/.f64 1 x))))): 0 points increase in error, 0 points decrease in error

   if -8.70713762482138709e78 < t < -5.8000000000000004e-206

   1. Initial program 33.2

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

   2. Taylor expanded in x around inf 13.6

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

   3. Simplified 13.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 2 points increase in error, 2 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))): 1 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 13.9

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

   5. Simplified 8.9

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      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))): 21 points increase in error, 18 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 48 points increase in error, 12 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 8.9

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

   7. Applied egg-rr 8.9

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

   if -5.8000000000000004e-206 < t < -4.70000000000000035e-232

   1. Initial program 61.7

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

   2. Taylor expanded in t around inf 62.6

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

   3. Simplified 62.6

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + -1}{x + 1}}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x -1) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x))))): 13 points increase in error, 0 points decrease in error

   4. Applied egg-rr 62.6

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

   5. Taylor expanded in x around -inf 64.0

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]

   6. Simplified 36.9

      \[\leadsto \color{blue}{-1} \]
      Proof
      -1: 0 points increase in error, 0 points decrease in error
      (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 -1) (sqrt.f64 -1))): 256 points increase in error, 0 points decrease in error
      (Rewrite<= unpow2_binary64 (pow.f64 (sqrt.f64 -1) 2)): 0 points increase in error, 0 points decrease in error

   if -4.70000000000000035e-232 < t < 9e-78

   1. Initial program 54.2

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

   2. Taylor expanded in x around inf 25.4

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

   3. Simplified 25.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]
      Proof
      (+.f64 (/.f64 l (/.f64 x l)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)) (fma.f64 2 (+.f64 (/.f64 t (/.f64 x t)) (*.f64 t t)) (/.f64 (fma.f64 2 (*.f64 t t) (*.f64 l l)) x))): 2 points increase in error, 2 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))): 1 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 25.5

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

   5. Simplified 23.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, \frac{t}{\frac{x}{t}} + t \cdot t, \color{blue}{\frac{\ell}{x} \cdot \ell}\right)}} \]
      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))): 21 points increase in error, 18 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x)): 48 points increase in error, 12 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 t around 0 25.5

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

   7. Simplified 23.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \color{blue}{\mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)}}} \]
      Proof
      (fma.f64 2 (*.f64 t (+.f64 t (/.f64 t x))) (/.f64 l (/.f64 x l))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 t (+.f64 t (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) x))) (/.f64 l (/.f64 x l))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 t (+.f64 t (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 x) t)))) (/.f64 l (/.f64 x l))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 t (+.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 t)) (*.f64 (/.f64 1 x) t))) (/.f64 l (/.f64 x l))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 t (Rewrite<= distribute-rgt-in_binary64 (*.f64 t (+.f64 1 (/.f64 1 x))))) (/.f64 l (/.f64 x l))): 0 points increase in error, 1 points decrease in error
      (fma.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 t t) (+.f64 1 (/.f64 1 x)))) (/.f64 l (/.f64 x l))): 2 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (+.f64 1 (/.f64 1 x))) (/.f64 l (/.f64 x l))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 l l) x))): 23 points increase in error, 4 points decrease in error
      (fma.f64 2 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x))) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x)))) (/.f64 (pow.f64 l 2) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 l 2) x) (*.f64 2 (*.f64 (pow.f64 t 2) (+.f64 1 (/.f64 1 x)))))): 0 points increase in error, 0 points decrease in error

   if 9e-78 < t

   1. Initial program 39.7

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

   2. Taylor expanded in t around inf 8.4

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

   3. Simplified 8.4

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + -1}{x + 1}}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x -1) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (+.f64 x (Rewrite<= metadata-eval (neg.f64 1))) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 x 1)) (+.f64 x 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 2) (*.f64 (sqrt.f64 1/2) (sqrt.f64 (/.f64 (-.f64 x 1) (Rewrite<= +-commutative_binary64 (+.f64 1 x)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sqrt.f64 2) (sqrt.f64 1/2)) (sqrt.f64 (/.f64 (-.f64 x 1) (+.f64 1 x))))): 13 points increase in error, 0 points decrease in error

   4. Applied egg-rr 7.5

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{x + 1}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.707137624821387 \cdot 10^{+78}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-206}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{{\left(t \cdot \left(2 \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)\right)}^{0.5}}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-78}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	14928

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := t + \frac{t}{x}\\ \mathbf{if}\;t \leq -8.707137624821387 \cdot 10^{+78}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-206}:\\ \;\;\;\;\frac{t_2}{{\left(t \cdot \left(2 \cdot t_3\right) + 2 \cdot t_1\right)}^{0.5}}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-78}:\\ \;\;\;\;\frac{t_2}{\sqrt{\frac{\ell}{\frac{x}{\ell}} + \left(t_1 + 2 \cdot \left(t \cdot t_3\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 2
Error	10.4
Cost	14736

\[\begin{array}{l} t_1 := \frac{t \cdot \sqrt{2}}{{\left(t \cdot \left(2 \cdot \left(t + \frac{t}{x}\right)\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)\right)}^{0.5}}\\ \mathbf{if}\;t \leq -8.707137624821387 \cdot 10^{+78}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	14024

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{1 + x}}\\ t_2 := \frac{\ell}{\frac{x}{\ell}}\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{t_2 + t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.7
Cost	13768

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{-232}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	14.3
Cost	7176

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-232}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{t}{{x}^{-0.5}} \cdot \frac{1}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-232}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-165}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-1 + x}{1 + x}}\\ \end{array} \]

Alternative 7
Error	14.4
Cost	7112

\[\begin{array}{l} t_1 := \sqrt{\frac{-1 + x}{1 + x}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-232}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-165}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	14.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-232}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{t}{\frac{\ell}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5 + \frac{-0.5}{x}}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 9
Error	14.5
Cost	6984

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-232}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-165}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5 + \frac{-0.5}{x}}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10
Error	15.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-290}:\\ \;\;\;\;-1 + \left(\frac{1}{x} + \frac{1}{x \cdot x} \cdot \left(\frac{0.5}{x} + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5 + \frac{-0.5}{x}}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 11
Error	15.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-290}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5 + \frac{-0.5}{x}}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 12
Error	15.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-290}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 13
Error	15.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-290}:\\ \;\;\;\;\frac{1}{x} + \left(-1 + \frac{\frac{-0.5}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{\frac{0.5}{x}}{x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 14
Error	15.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-290}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 15
Error	15.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-290}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 16
Error	15.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-290}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	39.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022297 
(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)))))