jeff quadratic root 1

Average Error: 19.9 → 8.8

Time: 20.1s

Precision: binary64

Cost: 50596

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+282}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(b + \mathsf{hypot}\left(\sqrt{a \cdot -4} \cdot \sqrt{c}, b\right)\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \end{array} \]

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 (+ (* b b) (* c (* a -4.0)))))
        (t_1
         (if (>= b 0.0) (/ (- (- b) t_0) (* a 2.0)) (/ (* c 2.0) (- t_0 b)))))
   (if (<= t_1 (- INFINITY))
     (if (>= b 0.0) (/ (- b) a) (/ (- c) b))
     (if (<= t_1 -2e-254)
       t_1
       (if (<= t_1 2e-168)
         (if (>= b 0.0)
           (/ (- (- b) b) (* a 2.0))
           (/ (* c 2.0) (+ (* 2.0 (* c (/ a b))) (* b -2.0))))
         (if (<= t_1 5e+282)
           t_1
           (if (>= b 0.0)
             (* (+ b (hypot (* (sqrt (* a -4.0)) (sqrt c)) b)) (/ -0.5 a))
             (* -2.0 (/ c (- b (sqrt (fma c (* a -4.0) (* b b)))))))))))))

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(((b * b) + (c * (a * -4.0))));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (a * 2.0);
	} else {
		tmp = (c * 2.0) / (t_0 - b);
	}
	double t_1 = tmp;
	double tmp_2;
	if (t_1 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = -b / a;
		} else {
			tmp_3 = -c / b;
		}
		tmp_2 = tmp_3;
	} else if (t_1 <= -2e-254) {
		tmp_2 = t_1;
	} else if (t_1 <= 2e-168) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = (-b - b) / (a * 2.0);
		} else {
			tmp_4 = (c * 2.0) / ((2.0 * (c * (a / b))) + (b * -2.0));
		}
		tmp_2 = tmp_4;
	} else if (t_1 <= 5e+282) {
		tmp_2 = t_1;
	} else if (b >= 0.0) {
		tmp_2 = (b + hypot((sqrt((a * -4.0)) * sqrt(c)), b)) * (-0.5 / a);
	} else {
		tmp_2 = -2.0 * (c / (b - sqrt(fma(c, (a * -4.0), (b * b)))));
	}
	return tmp_2;
}

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(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0))))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(t_0 - b));
	end
	t_1 = tmp
	tmp_2 = 0.0
	if (t_1 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = Float64(Float64(-b) / a);
		else
			tmp_3 = Float64(Float64(-c) / b);
		end
		tmp_2 = tmp_3;
	elseif (t_1 <= -2e-254)
		tmp_2 = t_1;
	elseif (t_1 <= 2e-168)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0));
		else
			tmp_4 = Float64(Float64(c * 2.0) / Float64(Float64(2.0 * Float64(c * Float64(a / b))) + Float64(b * -2.0)));
		end
		tmp_2 = tmp_4;
	elseif (t_1 <= 5e+282)
		tmp_2 = t_1;
	elseif (b >= 0.0)
		tmp_2 = Float64(Float64(b + hypot(Float64(sqrt(Float64(a * -4.0)) * sqrt(c)), b)) * Float64(-0.5 / a));
	else
		tmp_2 = Float64(-2.0 * Float64(c / Float64(b - sqrt(fma(c, Float64(a * -4.0), Float64(b * b))))));
	end
	return tmp_2
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[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]]}, If[LessEqual[t$95$1, (-Infinity)], If[GreaterEqual[b, 0.0], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]], If[LessEqual[t$95$1, -2e-254], t$95$1, If[LessEqual[t$95$1, 2e-168], If[GreaterEqual[b, 0.0], N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[(2.0 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$1, 5e+282], t$95$1, If[GreaterEqual[b, 0.0], N[(N[(b + N[Sqrt[N[(N[Sqrt[N[(a * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(c / N[(b - N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (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 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in b around -inf 64.0

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

   3. Simplified 64.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{c}{\frac{b}{a}}, b \cdot -2\right)}}\\ \end{array} \]
      Proof
      (fma.f64 2 (/.f64 c (/.f64 b a)) (*.f64 b -2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 b -2)): 16 points increase in error, 14 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 c a) b) (Rewrite<= *-commutative_binary64 (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 c a) b)) (*.f64 -2 b))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 64.0

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

   5. Taylor expanded in c around 0 63.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \]

   6. Simplified 63.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
      Proof
      (/.f64 (neg.f64 c) b): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 c b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 c b))): 0 points increase in error, 0 points decrease in error

   7. Taylor expanded in b around inf 17.6

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

   8. Simplified 17.6

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

   if -inf.0 < (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 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < -1.9999999999999998e-254 or 2.0000000000000001e-168 < (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 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < 4.99999999999999978e282

   1. Initial program 2.6

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

   if -1.9999999999999998e-254 < (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 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))))) < 2.0000000000000001e-168

   1. Initial program 29.9

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

   2. Taylor expanded in b around inf 32.2

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

   3. Taylor expanded in b around -inf 14.8

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

   4. Applied egg-rr 12.8

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

   5. Simplified 12.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \color{blue}{c}}{2 \cdot \left(c \cdot \frac{a}{b}\right) + -2 \cdot b}\\ \end{array} \]
      Proof
      (*.f64 c (/.f64 a b)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 c a) b)): 46 points increase in error, 37 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 c (/.f64 b a))): 38 points increase in error, 47 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (/.f64 c (/.f64 b a)))): 0 points increase in error, 0 points decrease in error

   if 4.99999999999999978e282 < (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 c) (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))

   1. Initial program 60.8

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

   2. Simplified 60.7

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ } \end{array}} \]
      Proof
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b)))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (fma.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b)))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (fma.f64 c (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))) (*.f64 b b)))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (fma.f64 c (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a))) (*.f64 b b)))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c (neg.f64 (*.f64 4 a))) (*.f64 b b))))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 a)) c)) (*.f64 b b)))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.f64 (*.f64 4 a)) c))))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))) (/.f64 -1/2 a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (/.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) a)) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (*.f64 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 2 a)))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) -1) (*.f64 2 a))) (*.f64 -2 (/.f64 c (-.f64 b (sqrt.f64 (fma.f64 c (*.f64 a -4) (*.f64 b b))))))): 4 points increase in error, 18 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (+.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 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (+.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 c (*.f64 a -4) (*.f64 b b))))))): 0 points increase in error, 0 points decrease in error
      (if (>=.f64 b 0) (/.f64 (Rewrite<= distribute-neg-out_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 c (*.f64 a -4) (*.f64 b b))))))): 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 c (*.f64 a -4) (*.f64 b b))))))): 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 c (*.f64 a -4) (*.f64 b b))))))): 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 c (*.f64 a -4) (*.f64 b b))))))): 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 c (*.f64 a -4) (*.f64 b b))))))): 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 c (*.f64 a -4) (*.f64 b b))))))): 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 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 b b))))))): 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 c (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 4))) (*.f64 b b))))))): 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 c (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 a))) (*.f64 b b))))))): 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 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 c (neg.f64 (*.f64 4 a))) (*.f64 b b)))))))): 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 2 -1) (/.f64 (/.f64 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 a)) c)) (*.f64 b b))))))): 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 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 b b) (*.f64 (neg.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 (/.f64 c -1) -1) (-.f64 b (sqrt.f64 (Rewrite<= cancel-sign-sub-inv_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, 3 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))))))): 8 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-/r/_binary64 (*.f64 (/.f64 2 -1) c)) (*.f64 -1 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c))))))): 0 points increase in error, 8 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)))))))): 3 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, 3 points decrease in error

   3. Applied egg-rr 27.9

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

   4. Simplified 27.9

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

4. Final simplification 8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq -2 \cdot 10^{-254}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 2 \cdot 10^{-168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array} \leq 5 \cdot 10^{+282}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\left(b + \mathsf{hypot}\left(\sqrt{a \cdot -4} \cdot \sqrt{c}, b\right)\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{b - \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	38052

\[\begin{array}{l} t_0 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\\ t_1 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_2 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_1 - b}\\ \end{array}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-168}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;t_2 \leq 10^{+271}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	6.9
Cost	7952

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+150}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	6.9
Cost	7952

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	10.3
Cost	7888

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{t_0}{a \cdot 2}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{-97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 5
Error	7.0
Cost	7888

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{-c}{b}\\ \mathbf{if}\;b \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t_0}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+61}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	13.4
Cost	7632

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \left(-b\right) - b\\ t_2 := \frac{t_1}{a \cdot 2}\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{-97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-2}{b - t_0}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-109}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{t_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7
Error	13.4
Cost	7632

\[\begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ t_1 := \left(-b\right) - b\\ t_2 := \frac{t_1}{a \cdot 2}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{b - t_0}{c}}\\ \end{array}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{t_1}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 8
Error	13.4
Cost	7632

\[\begin{array}{l} t_0 := \left(-b\right) - b\\ t_1 := \frac{t_0}{a \cdot 2}\\ t_2 := \sqrt{c \cdot \left(a \cdot -4\right)}\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{-97}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{2 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}\\ \end{array}\\ \mathbf{elif}\;b \leq -7 \cdot 10^{-300}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t_2 - b}\\ \end{array}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-109}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{t_0}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 9
Error	18.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-109}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b + \sqrt{c \cdot \left(a \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 10
Error	22.3
Cost	1092

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

Alternative 11
Error	22.6
Cost	580

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

Alternative 12
Error	22.5
Cost	388

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

Reproduce

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