jeff quadratic root 1

Average Error: 20.2 → 6.3

Time: 21.3s

Precision: binary64

Cost: 26896

Math

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{if}\;b \leq -7.804026923526817 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-188}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - t_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - {\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{-4}\right)}^{1.5}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0)
   (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
   (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma b b (* -4.0 (* a c))))))
   (if (<= b -7.804026923526817e+154)
     (if (>= b 0.0)
       (/ (- (- b) b) (* 2.0 a))
       (/ (* 2.0 c) (- (- (* a (/ (* 2.0 c) b)) b) b)))
     (if (<= b -2.1e-188)
       (if (>= b 0.0)
         (* -0.5 (/ (+ b (pow (pow (* a (* c -4.0)) 0.25) 2.0)) a))
         (* -2.0 (/ c (- b t_0))))
       (if (<= b 2.7e+39)
         (if (>= b 0.0)
           (* -0.5 (/ (+ b t_0) a))
           (*
            -2.0
            (/ c (- b (pow (* (* (cbrt c) (cbrt a)) (cbrt -4.0)) 1.5)))))
         (if (>= b 0.0) (* -0.5 (/ (* b 2.0) a)) (/ b a)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + sqrt(((b * b) - ((4.0 * a) * c))));
	}
	return tmp;
}

↓

double code(double a, double b, double c) {
	double t_0 = sqrt(fma(b, b, (-4.0 * (a * c))));
	double tmp_1;
	if (b <= -7.804026923526817e+154) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = (-b - b) / (2.0 * a);
		} else {
			tmp_2 = (2.0 * c) / (((a * ((2.0 * c) / b)) - b) - b);
		}
		tmp_1 = tmp_2;
	} else if (b <= -2.1e-188) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -0.5 * ((b + pow(pow((a * (c * -4.0)), 0.25), 2.0)) / a);
		} else {
			tmp_3 = -2.0 * (c / (b - t_0));
		}
		tmp_1 = tmp_3;
	} else if (b <= 2.7e+39) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = -0.5 * ((b + t_0) / a);
		} else {
			tmp_4 = -2.0 * (c / (b - pow(((cbrt(c) * cbrt(a)) * cbrt(-4.0)), 1.5)));
		}
		tmp_1 = tmp_4;
	} else if (b >= 0.0) {
		tmp_1 = -0.5 * ((b * 2.0) / a);
	} else {
		tmp_1 = b / a;
	}
	return tmp_1;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))));
	end
	return tmp
end

↓

function code(a, b, c)
	t_0 = sqrt(fma(b, b, Float64(-4.0 * Float64(a * c))))
	tmp_1 = 0.0
	if (b <= -7.804026923526817e+154)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(Float64(Float64(-b) - b) / Float64(2.0 * a));
		else
			tmp_2 = Float64(Float64(2.0 * c) / Float64(Float64(Float64(a * Float64(Float64(2.0 * c) / b)) - b) - b));
		end
		tmp_1 = tmp_2;
	elseif (b <= -2.1e-188)
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(-0.5 * Float64(Float64(b + ((Float64(a * Float64(c * -4.0)) ^ 0.25) ^ 2.0)) / a));
		else
			tmp_3 = Float64(-2.0 * Float64(c / Float64(b - t_0)));
		end
		tmp_1 = tmp_3;
	elseif (b <= 2.7e+39)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(-0.5 * Float64(Float64(b + t_0) / a));
		else
			tmp_4 = Float64(-2.0 * Float64(c / Float64(b - (Float64(Float64(cbrt(c) * cbrt(a)) * cbrt(-4.0)) ^ 1.5))));
		end
		tmp_1 = tmp_4;
	elseif (b >= 0.0)
		tmp_1 = Float64(-0.5 * Float64(Float64(b * 2.0) / a));
	else
		tmp_1 = Float64(b / a);
	end
	return tmp_1
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(b * b + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -7.804026923526817e+154], If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[(N[(N[(a * N[(N[(2.0 * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, -2.1e-188], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + N[Power[N[Power[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(c / N[(b - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[b, 2.7e+39], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b + t$95$0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(c / N[(b - N[Power[N[(N[(N[Power[c, 1/3], $MachinePrecision] * N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[-4.0, 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(-0.5 * N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(b / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -7.804026923526817e154

   1. Initial program 38.3

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Taylor expanded in b around inf 38.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   3. Taylor expanded in b around -inf 6.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(2 \cdot \frac{c \cdot a}{b} + -1 \cdot b\right)}}\\ \end{array} \]

   4. Simplified 1.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\frac{c \cdot 2}{b} \cdot a - b\right)}}\\ \end{array} \]
      Proof
      (-.f64 (*.f64 (/.f64 (*.f64 c 2) b) a) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 c 2) (/.f64 b a))) b): 15 points increase in error, 17 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c (/.f64 b a)) 2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) 2) b): 19 points increase in error, 15 points decrease in error
      (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 c a) b))) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (neg.f64 b))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b))): 0 points increase in error, 0 points decrease in error

   if -7.804026923526817e154 < b < -2.0999999999999999e-188

   1. Initial program 7.0

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Simplified 6.9

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ } \end{array}} \]
      Proof
      (if (>=.f64 b 0) (*.f64 -1/2 (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4)))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4)))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c))))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 b a) (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (-.f64 (/.f64 b a) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 b a) (neg.f64 (neg.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 b a) (Rewrite=> remove-double-neg_binary64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 b a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 b))) a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 (neg.f64 b) a))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 (neg.f64 b) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (*.f64 (/.f64 -1 2) (/.f64 (neg.f64 b) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (neg.f64 b)) (*.f64 2 a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (neg.f64 b))) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (/.f64 (Rewrite=> remove-double-neg_binary64 b) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) (/.f64 b (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 2 a)) (/.f64 b (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (neg.f64 b))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (Rewrite<= metadata-eval (/.f64 2 -1)) (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 c 1)) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c (Rewrite<= metadata-eval (*.f64 -1 -1))) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 c -1) -1)) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c)))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))))))): 1 points increase in error, 1 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (Rewrite=> associate-/l/_binary64 (/.f64 (/.f64 c -1) (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) -1))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 (/.f64 c -1)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 2 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 c) -1)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 2 (/.f64 -1 c))) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 12 points increase in error, 2 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 2 -1) c)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 2 points increase in error, 12 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (/.f64 2 -1) -1) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 2 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 (Rewrite=> metadata-eval -2) -1) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (Rewrite=> metadata-eval 2) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 2 points decrease in error

   3. Applied egg-rr 6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{{\left({\left(\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)\right)}^{0.25}\right)}^{2}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ \end{array} \]

   4. Taylor expanded in b around 0 6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + {\color{blue}{\left({\left(-4 \cdot \left(c \cdot a\right)\right)}^{0.25}\right)}}^{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ \end{array} \]

   5. Simplified 6.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + {\color{blue}{\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}}^{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ \end{array} \]
      Proof
      (pow.f64 (*.f64 a (*.f64 c -4)) 1/4): 0 points increase in error, 0 points decrease in error
      (pow.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) -4)) 1/4): 2 points increase in error, 1 points decrease in error
      (pow.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 c a)) -4) 1/4): 0 points increase in error, 0 points decrease in error
      (pow.f64 (Rewrite<= *-commutative_binary64 (*.f64 -4 (*.f64 c a))) 1/4): 0 points increase in error, 0 points decrease in error

   if -2.0999999999999999e-188 < b < 2.70000000000000003e39

   1. Initial program 11.3

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Simplified 11.2

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ } \end{array}} \]
      Proof
      (if (>=.f64 b 0) (*.f64 -1/2 (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4)))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4)))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c))))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 b a) (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (-.f64 (/.f64 b a) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 b a) (neg.f64 (neg.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 b a) (Rewrite=> remove-double-neg_binary64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 b a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 b))) a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 (neg.f64 b) a))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 (neg.f64 b) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (*.f64 (/.f64 -1 2) (/.f64 (neg.f64 b) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (neg.f64 b)) (*.f64 2 a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (neg.f64 b))) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (/.f64 (Rewrite=> remove-double-neg_binary64 b) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) (/.f64 b (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 2 a)) (/.f64 b (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (neg.f64 b))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (Rewrite<= metadata-eval (/.f64 2 -1)) (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 c 1)) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c (Rewrite<= metadata-eval (*.f64 -1 -1))) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 c -1) -1)) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c)))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))))))): 1 points increase in error, 1 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (Rewrite=> associate-/l/_binary64 (/.f64 (/.f64 c -1) (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) -1))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 (/.f64 c -1)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 2 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 c) -1)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 2 (/.f64 -1 c))) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 12 points increase in error, 2 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 2 -1) c)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 2 points increase in error, 12 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (/.f64 2 -1) -1) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 2 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 (Rewrite=> metadata-eval -2) -1) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (Rewrite=> metadata-eval 2) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 2 points decrease in error

   3. Applied egg-rr 11.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{b - {\left(\sqrt[3]{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}^{1.5}}}\\ \end{array} \]

   4. Taylor expanded in b around 0 21.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{b} - {\left({\left(c \cdot \left(1 \cdot a\right)\right)}^{0.3333333333333333} \cdot \sqrt[3]{-4}\right)}^{1.5}}\\ \end{array} \]

   5. Simplified 11.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{b} - {\left(\sqrt[3]{c \cdot a} \cdot \sqrt[3]{-4}\right)}^{1.5}}\\ \end{array} \]
      Proof
      (*.f64 (cbrt.f64 (*.f64 c a)) (cbrt.f64 -4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cbrt.f64 (*.f64 c (Rewrite<= *-lft-identity_binary64 (*.f64 1 a)))) (cbrt.f64 -4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow1/3_binary64 (pow.f64 (*.f64 c (*.f64 1 a)) 1/3)) (cbrt.f64 -4)): 163 points increase in error, 34 points decrease in error

   6. Applied egg-rr 8.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - {\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{-4}\right)}^{1.5}}\\ \end{array} \]

   if 2.70000000000000003e39 < b

   1. Initial program 35.9

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   2. Simplified 35.9

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ } \end{array}} \]
      Proof
      (if (>=.f64 b 0) (*.f64 -1/2 (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4)))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4)))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c))))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (+.f64 b (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 b (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> div-sub_binary64 (-.f64 (/.f64 b a) (/.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (-.f64 (/.f64 b a) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 b a) (neg.f64 (neg.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 b a) (Rewrite=> remove-double-neg_binary64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 b a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 b))) a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (+.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 (neg.f64 b) a))))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (/.f64 -1 2) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a) (/.f64 (neg.f64 b) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (*.f64 (/.f64 -1 2) (/.f64 (neg.f64 b) a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (neg.f64 b)) (*.f64 2 a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (neg.f64 b))) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (*.f64 (/.f64 -1 2) (/.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))) a)) (/.f64 (Rewrite=> remove-double-neg_binary64 b) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))) (/.f64 b (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 2 a)) (/.f64 b (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) b) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (neg.f64 b))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (neg.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (*.f64 2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (Rewrite<= metadata-eval (/.f64 2 -1)) (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 c 1)) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c (Rewrite<= metadata-eval (*.f64 -1 -1))) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 c -1) -1)) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) -4))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 (*.f64 a c) (Rewrite<= metadata-eval (neg.f64 4))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 4))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 a c)))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 a) c))))))))): 1 points increase in error, 1 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (Rewrite=> associate-/l/_binary64 (/.f64 (/.f64 c -1) (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) -1))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 2 -1) (/.f64 (/.f64 c -1) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 2 (/.f64 c -1)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 0 points increase in error, 2 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 c) -1)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 2 (/.f64 -1 c))) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 12 points increase in error, 2 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (/.f64 (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 2 -1) c)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 2 points increase in error, 12 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (/.f64 2 -1) -1) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))))): 2 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (/.f64 (Rewrite=> metadata-eval -2) -1) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (*.f64 (Rewrite=> metadata-eval 2) (/.f64 c (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (-.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a)) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 2 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 2 points decrease in error

   3. Taylor expanded in b around inf 11.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\left(b + -2 \cdot \frac{c \cdot a}{b}\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ \end{array} \]

   4. Simplified 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}}\\ \end{array} \]
      Proof
      (fma.f64 -2 (/.f64 c (/.f64 b a)) b): 0 points increase in error, 0 points decrease in error
      (fma.f64 -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) b): 18 points increase in error, 16 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -2 (/.f64 (*.f64 c a) b)) b)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 b (*.f64 -2 (/.f64 (*.f64 c a) b)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in b around -inf 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \end{array} \]

   6. Simplified 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \color{blue}{\frac{c}{\mathsf{fma}\left(b, 2, -2 \cdot \frac{c}{\frac{b}{a}}\right)}}\\ \end{array} \]
      Proof
      (fma.f64 b 2 (*.f64 -2 (/.f64 c (/.f64 b a)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b 2 (*.f64 -2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)))): 18 points increase in error, 16 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 b 2) (*.f64 -2 (/.f64 (*.f64 c a) b)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 b)) (*.f64 -2 (/.f64 (*.f64 c a) b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -2 (/.f64 (*.f64 c a) b)) (*.f64 2 b))): 0 points increase in error, 0 points decrease in error

   7. Taylor expanded in c around inf 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

   8. Taylor expanded in b around inf 6.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{2 \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

   9. Simplified 6.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{\color{blue}{b \cdot 2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]
      Proof
      (*.f64 b 2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 2 b)): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.804026923526817 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-188}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + {\left({\left(a \cdot \left(c \cdot -4\right)\right)}^{0.25}\right)}^{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - {\left(\left(\sqrt[3]{c} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{-4}\right)}^{1.5}}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.3
Cost	38052

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_1 - b}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.3
Cost	14028

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)}\\ \mathbf{if}\;b \leq -7.804026923526817 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b + t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - t_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

Alternative 3
Error	7.3
Cost	7952

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \left(-b\right) - b\\ \mathbf{if}\;b \leq -7.804026923526817 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

Alternative 4
Error	8.3
Cost	7952

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -7.804026923526817 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-71}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

Alternative 5
Error	15.1
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -7.804026923526817 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 6
Error	18.1
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \mathsf{fma}\left(-2, \frac{a}{\frac{b}{c}}, b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(b, -2, a \cdot \frac{2 \cdot c}{b}\right)}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

Alternative 7
Error	18.1
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{-61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \]

Alternative 8
Error	18.1
Cost	7368

\[\begin{array}{l} t_0 := \frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{if}\;b \leq -2.3 \cdot 10^{-61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

Alternative 9
Error	23.7
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} + b \cdot -2}\\ \end{array} \]

Alternative 10
Error	22.9
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(a \cdot \frac{2 \cdot c}{b} - b\right) - b}\\ \end{array} \]

Alternative 11
Error	23.1
Cost	644

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{b \cdot -2}\\ \end{array} \]

Alternative 12
Error	45.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-0.5 \cdot \frac{b \cdot 2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022316 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))