jeff quadratic root 1

Average Error: 20.4 → 7.0

Time: 15.1s

Precision: binary64

Cost: 51364

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -4\right)\\ t_1 := \sqrt{b \cdot b + t_0}\\ t_2 := \frac{c \cdot 2}{t_1 - b}\\ t_3 := \frac{\left(-b\right) - t_1}{a \cdot 2}\\ t_4 := \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ t_5 := \frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{t_0}\right)}}\right)}^{3}\\ \end{array}\\ \mathbf{elif}\;t_4 \leq -1 \cdot 10^{-182}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{fma}\left(b, -2, {\left(\sqrt[3]{a \cdot \left(2 \cdot \frac{c}{b}\right)}\right)}^{3}\right)}\\ \end{array}\\ \mathbf{elif}\;t_4 \leq 10^{+266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(4 \cdot c\right)\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(a \cdot \frac{c \cdot 2}{b} - b\right) - b}\\ \end{array} \]

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 (* c (* a -4.0)))
        (t_1 (sqrt (+ (* b b) t_0)))
        (t_2 (/ (* c 2.0) (- t_1 b)))
        (t_3 (/ (- (- b) t_1) (* a 2.0)))
        (t_4 (if (>= b 0.0) t_3 t_2))
        (t_5 (/ (- (- b) b) (* a 2.0))))
   (if (<= t_4 (- INFINITY))
     (if (>= b 0.0)
       t_5
       (pow (cbrt (/ (* c -2.0) (- b (hypot b (sqrt t_0))))) 3.0))
     (if (<= t_4 -1e-182)
       t_4
       (if (<= t_4 0.0)
         (if (>= b 0.0)
           t_3
           (/ (* c 2.0) (fma b -2.0 (pow (cbrt (* a (* 2.0 (/ c b)))) 3.0))))
         (if (<= t_4 1e+266)
           (if (>= b 0.0)
             (/
              (-
               (- b)
               (sqrt
                (+
                 (* b b)
                 (fma c (* a -4.0) (fma c (* a -4.0) (* a (* 4.0 c)))))))
              (* a 2.0))
             t_2)
           (if (>= b 0.0)
             t_5
             (/ (* c 2.0) (- (- (* a (/ (* c 2.0) b)) 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 = c * (a * -4.0);
	double t_1 = sqrt(((b * b) + t_0));
	double t_2 = (c * 2.0) / (t_1 - b);
	double t_3 = (-b - t_1) / (a * 2.0);
	double tmp;
	if (b >= 0.0) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	double t_4 = tmp;
	double t_5 = (-b - b) / (a * 2.0);
	double tmp_2;
	if (t_4 <= -((double) INFINITY)) {
		double tmp_3;
		if (b >= 0.0) {
			tmp_3 = t_5;
		} else {
			tmp_3 = pow(cbrt(((c * -2.0) / (b - hypot(b, sqrt(t_0))))), 3.0);
		}
		tmp_2 = tmp_3;
	} else if (t_4 <= -1e-182) {
		tmp_2 = t_4;
	} else if (t_4 <= 0.0) {
		double tmp_4;
		if (b >= 0.0) {
			tmp_4 = t_3;
		} else {
			tmp_4 = (c * 2.0) / fma(b, -2.0, pow(cbrt((a * (2.0 * (c / b)))), 3.0));
		}
		tmp_2 = tmp_4;
	} else if (t_4 <= 1e+266) {
		double tmp_5;
		if (b >= 0.0) {
			tmp_5 = (-b - sqrt(((b * b) + fma(c, (a * -4.0), fma(c, (a * -4.0), (a * (4.0 * c))))))) / (a * 2.0);
		} else {
			tmp_5 = t_2;
		}
		tmp_2 = tmp_5;
	} else if (b >= 0.0) {
		tmp_2 = t_5;
	} else {
		tmp_2 = (c * 2.0) / (((a * ((c * 2.0) / b)) - 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 = Float64(c * Float64(a * -4.0))
	t_1 = sqrt(Float64(Float64(b * b) + t_0))
	t_2 = Float64(Float64(c * 2.0) / Float64(t_1 - b))
	t_3 = Float64(Float64(Float64(-b) - t_1) / Float64(a * 2.0))
	tmp = 0.0
	if (b >= 0.0)
		tmp = t_3;
	else
		tmp = t_2;
	end
	t_4 = tmp
	t_5 = Float64(Float64(Float64(-b) - b) / Float64(a * 2.0))
	tmp_2 = 0.0
	if (t_4 <= Float64(-Inf))
		tmp_3 = 0.0
		if (b >= 0.0)
			tmp_3 = t_5;
		else
			tmp_3 = cbrt(Float64(Float64(c * -2.0) / Float64(b - hypot(b, sqrt(t_0))))) ^ 3.0;
		end
		tmp_2 = tmp_3;
	elseif (t_4 <= -1e-182)
		tmp_2 = t_4;
	elseif (t_4 <= 0.0)
		tmp_4 = 0.0
		if (b >= 0.0)
			tmp_4 = t_3;
		else
			tmp_4 = Float64(Float64(c * 2.0) / fma(b, -2.0, (cbrt(Float64(a * Float64(2.0 * Float64(c / b)))) ^ 3.0)));
		end
		tmp_2 = tmp_4;
	elseif (t_4 <= 1e+266)
		tmp_5 = 0.0
		if (b >= 0.0)
			tmp_5 = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) + fma(c, Float64(a * -4.0), fma(c, Float64(a * -4.0), Float64(a * Float64(4.0 * c))))))) / Float64(a * 2.0));
		else
			tmp_5 = t_2;
		end
		tmp_2 = tmp_5;
	elseif (b >= 0.0)
		tmp_2 = t_5;
	else
		tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(Float64(a * Float64(Float64(c * 2.0) / b)) - 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[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-b) - t$95$1), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = If[GreaterEqual[b, 0.0], t$95$3, t$95$2]}, Block[{t$95$5 = N[(N[((-b) - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], If[GreaterEqual[b, 0.0], t$95$5, N[Power[N[Power[N[(N[(c * -2.0), $MachinePrecision] / N[(b - N[Sqrt[b ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], If[LessEqual[t$95$4, -1e-182], t$95$4, If[LessEqual[t$95$4, 0.0], If[GreaterEqual[b, 0.0], t$95$3, N[(N[(c * 2.0), $MachinePrecision] / N[(b * -2.0 + N[Power[N[Power[N[(a * N[(2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], If[LessEqual[t$95$4, 1e+266], If[GreaterEqual[b, 0.0], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision] + N[(a * N[(4.0 * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2], If[GreaterEqual[b, 0.0], t$95$5, N[(N[(c * 2.0), $MachinePrecision] / N[(N[(N[(a * N[(N[(c * 2.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 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 17.3

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

   3. Applied egg-rr 17.3

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

   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)))))) < -1e-182

   1. Initial program 3.0

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

   if -1e-182 < (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)))))) < 0.0

   1. Initial program 33.4

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

   2. Taylor expanded in b around -inf 12.2

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

   3. Simplified 10.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(b, -2, \frac{c \cdot 2}{b} \cdot a\right)}}\\ \end{array} \]
      Proof
      (fma.f64 b -2 (*.f64 (/.f64 (*.f64 c 2) b) a)): 0 points increase in error, 0 points decrease in error
      (fma.f64 b -2 (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 c 2) (/.f64 b a)))): 10 points increase in error, 16 points decrease in error
      (fma.f64 b -2 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 c (/.f64 b a)) 2))): 0 points increase in error, 0 points decrease in error
      (fma.f64 b -2 (*.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) 2)): 23 points increase in error, 18 points decrease in error
      (fma.f64 b -2 (Rewrite<= *-commutative_binary64 (*.f64 2 (/.f64 (*.f64 c a) b)))): 0 points increase in error, 0 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

   4. Applied egg-rr 10.4

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

   if 0.0 < (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)))))) < 1e266

   1. Initial program 2.3

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

   2. Applied egg-rr 2.3

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

   if 1e266 < (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 56.9

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

   2. Taylor expanded in b around inf 19.6

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

   3. Taylor expanded in b around -inf 20.1

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

   4. Simplified 15.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \left(\frac{c \cdot 2}{b} \cdot a - b\right)}}\\ \end{array} \]
      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): 10 points increase in error, 16 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): 23 points increase in error, 18 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
3. Recombined 5 regimes into one program.

4. Final simplification 7.0

   \[\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{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{c \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\right)}^{3}\\ \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 -1 \cdot 10^{-182}:\\ \;\;\;\;\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 0:\\ \;\;\;\;\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}{\mathsf{fma}\left(b, -2, {\left(\sqrt[3]{a \cdot \left(2 \cdot \frac{c}{b}\right)}\right)}^{3}\right)}\\ \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 10^{+266}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + \mathsf{fma}\left(c, a \cdot -4, \mathsf{fma}\left(c, a \cdot -4, a \cdot \left(4 \cdot c\right)\right)\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:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(a \cdot \frac{c \cdot 2}{b} - b\right) - b}\\ \end{array} \]

Alternatives

Alternative 1
Error	45.5
Cost	644

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

Alternative 2
Error	22.7
Cost	644

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

Reproduce

herbie shell --seed 2022289 
(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)))))))