The quadratic formula (r2)

Average Accuracy: 47.8% → 88.1%

Time: 17.4s

Precision: binary64

Cost: 13832

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-7)
   (/ (- c) b)
   (if (<= b 4.4e-274)
     (/ (* c 2.0) (- (hypot (sqrt (* c (* a -4.0))) b) b))
     (if (<= b 7.2e+109)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 2.0 a))
       (- (/ b a))))))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-7) {
		tmp = -c / b;
	} else if (b <= 4.4e-274) {
		tmp = (c * 2.0) / (hypot(sqrt((c * (a * -4.0))), b) - b);
	} else if (b <= 7.2e+109) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-7) {
		tmp = -c / b;
	} else if (b <= 4.4e-274) {
		tmp = (c * 2.0) / (Math.hypot(Math.sqrt((c * (a * -4.0))), b) - b);
	} else if (b <= 7.2e+109) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -(b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-7:
		tmp = -c / b
	elif b <= 4.4e-274:
		tmp = (c * 2.0) / (math.hypot(math.sqrt((c * (a * -4.0))), b) - b)
	elif b <= 7.2e+109:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a)
	else:
		tmp = -(b / a)
	return tmp

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-7)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 4.4e-274)
		tmp = Float64(Float64(c * 2.0) / Float64(hypot(sqrt(Float64(c * Float64(a * -4.0))), b) - b));
	elseif (b <= 7.2e+109)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(-Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-7)
		tmp = -c / b;
	elseif (b <= 4.4e-274)
		tmp = (c * 2.0) / (hypot(sqrt((c * (a * -4.0))), b) - b);
	elseif (b <= 7.2e+109)
		tmp = (-b - sqrt(((b * b) - (4.0 * (c * a))))) / (2.0 * a);
	else
		tmp = -(b / a);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_, c_] := If[LessEqual[b, -8.5e-7], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 4.4e-274], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e+109], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], (-N[(b / a), $MachinePrecision])]]]

Target

Original	47.8%
Target	68.0%
Herbie	88.1%

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

Derivation

1. Split input into 4 regimes

2. if b < -8.50000000000000014e-7

   1. Initial program 13.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf 91.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   3. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
      Proof

      [Start] 91.1	\[ -1 \cdot \frac{c}{b} \]
      associate-*r/ [=>] 91.1	\[ \color{blue}{\frac{-1 \cdot c}{b}} \]
      neg-mul-1 [<=] 91.1	\[ \frac{\color{blue}{-c}}{b} \]

   if -8.50000000000000014e-7 < b < 4.3999999999999999e-274

   1. Initial program 63.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\left(b + \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right) \cdot \frac{1}{a \cdot -2}} \]
      Proof

      [Start] 63.4	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      frac-2neg [=>] 63.4	\[ \color{blue}{\frac{-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{-2 \cdot a}} \]
      div-inv [=>] 63.4	\[ \color{blue}{\left(-\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\right) \cdot \frac{1}{-2 \cdot a}} \]

   3. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(\sqrt{-4 \cdot \left(c \cdot a\right)}, b\right)\right)}} \]
      Proof

      [Start] 62.8	\[ \left(b + \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right) \cdot \frac{1}{a \cdot -2} \]
      un-div-inv [=>] 62.8	\[ \color{blue}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}{a \cdot -2}} \]
      flip-+ [=>] 62.8	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right) \cdot \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}}}{a \cdot -2} \]
      associate-/l/ [=>] 53.2	\[ \color{blue}{\frac{b \cdot b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right) \cdot \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)}} \]
      hypot-udef [=>] 53.2	\[ \frac{b \cdot b - \color{blue}{\sqrt{b \cdot b + \sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}}} \cdot \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)} \]
      hypot-udef [=>] 53.2	\[ \frac{b \cdot b - \sqrt{b \cdot b + \sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}} \cdot \color{blue}{\sqrt{b \cdot b + \sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}}}}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)} \]
      add-sqr-sqrt [<=] 53.2	\[ \frac{b \cdot b - \color{blue}{\left(b \cdot b + \sqrt{\left(-4 \cdot c\right) \cdot a} \cdot \sqrt{\left(-4 \cdot c\right) \cdot a}\right)}}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)} \]
      add-sqr-sqrt [<=] 53.3	\[ \frac{b \cdot b - \left(b \cdot b + \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)} \]
      +-commutative [=>] 53.3	\[ \frac{b \cdot b - \color{blue}{\left(\left(-4 \cdot c\right) \cdot a + b \cdot b\right)}}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)} \]
      associate-*l* [=>] 53.2	\[ \frac{b \cdot b - \left(\color{blue}{-4 \cdot \left(c \cdot a\right)} + b \cdot b\right)}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)} \]
      fma-def [=>] 53.2	\[ \frac{b \cdot b - \color{blue}{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(-4 \cdot c\right) \cdot a}\right)\right)} \]

   4. Simplified 68.4%

      \[\leadsto \color{blue}{\frac{\frac{c \cdot \left(a \cdot -4\right) + 0}{a \cdot 2}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}} \]
      Proof

      [Start] 53.2	\[ \frac{b \cdot b - \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}{\left(a \cdot -2\right) \cdot \left(b - \mathsf{hypot}\left(\sqrt{-4 \cdot \left(c \cdot a\right)}, b\right)\right)} \]
      associate-/r* [=>] 62.7	\[ \color{blue}{\frac{\frac{b \cdot b - \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}{a \cdot -2}}{b - \mathsf{hypot}\left(\sqrt{-4 \cdot \left(c \cdot a\right)}, b\right)}} \]

   5. Applied egg-rr 18.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c \cdot a}{a} \cdot -2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)} - 1} \]
      Proof

      [Start] 68.4	\[ \frac{\frac{c \cdot \left(a \cdot -4\right) + 0}{a \cdot 2}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)} \]
      expm1-log1p-u [=>] 51.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{c \cdot \left(a \cdot -4\right) + 0}{a \cdot 2}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)\right)} \]
      expm1-udef [=>] 18.5	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{c \cdot \left(a \cdot -4\right) + 0}{a \cdot 2}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)} - 1} \]
      +-rgt-identity [=>] 18.5	\[ e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{c \cdot \left(a \cdot -4\right)}}{a \cdot 2}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)} - 1 \]
      associate-*r* [=>] 18.5	\[ e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -4}}{a \cdot 2}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)} - 1 \]
      times-frac [=>] 18.5	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{c \cdot a}{a} \cdot \frac{-4}{2}}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)} - 1 \]
      metadata-eval [=>] 18.5	\[ e^{\mathsf{log1p}\left(\frac{\frac{c \cdot a}{a} \cdot \color{blue}{-2}}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)} - 1 \]

   6. Simplified 80.4%

      \[\leadsto \color{blue}{\frac{c \cdot 2}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}} \]
      Proof

      [Start] 18.5	\[ e^{\mathsf{log1p}\left(\frac{\frac{c \cdot a}{a} \cdot -2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)} - 1 \]
      expm1-def [=>] 51.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{c \cdot a}{a} \cdot -2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}\right)\right)} \]
      expm1-log1p [=>] 68.3	\[ \color{blue}{\frac{\frac{c \cdot a}{a} \cdot -2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}} \]
      *-lft-identity [<=] 68.3	\[ \color{blue}{1 \cdot \frac{\frac{c \cdot a}{a} \cdot -2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)}} \]
      metadata-eval [<=] 68.3	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\frac{c \cdot a}{a} \cdot -2}{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)} \]
      times-frac [<=] 68.3	\[ \color{blue}{\frac{-1 \cdot \left(\frac{c \cdot a}{a} \cdot -2\right)}{-1 \cdot \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)}} \]
      neg-mul-1 [<=] 68.3	\[ \frac{-1 \cdot \left(\frac{c \cdot a}{a} \cdot -2\right)}{\color{blue}{-\left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)}} \]
      neg-mul-1 [<=] 68.3	\[ \frac{\color{blue}{-\frac{c \cdot a}{a} \cdot -2}}{-\left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      distribute-rgt-neg-in [=>] 68.3	\[ \frac{\color{blue}{\frac{c \cdot a}{a} \cdot \left(--2\right)}}{-\left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      associate-/l* [=>] 80.4	\[ \frac{\color{blue}{\frac{c}{\frac{a}{a}}} \cdot \left(--2\right)}{-\left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      *-inverses [=>] 80.4	\[ \frac{\frac{c}{\color{blue}{1}} \cdot \left(--2\right)}{-\left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      metadata-eval [=>] 80.4	\[ \frac{\frac{c}{1} \cdot \color{blue}{2}}{-\left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      /-rgt-identity [=>] 80.4	\[ \frac{\color{blue}{c} \cdot 2}{-\left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)} \]
      neg-sub0 [=>] 80.4	\[ \frac{c \cdot 2}{\color{blue}{0 - \left(b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)}} \]
      sub-neg [=>] 80.4	\[ \frac{c \cdot 2}{0 - \color{blue}{\left(b + \left(-\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)\right)}} \]
      +-commutative [=>] 80.4	\[ \frac{c \cdot 2}{0 - \color{blue}{\left(\left(-\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right) + b\right)}} \]
      associate--r+ [=>] 80.4	\[ \frac{c \cdot 2}{\color{blue}{\left(0 - \left(-\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)\right)\right) - b}} \]

   if 4.3999999999999999e-274 < b < 7.2e109

   1. Initial program 87.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   if 7.2e109 < b

   1. Initial program 22.8%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around inf 94.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]

   3. Simplified 94.9%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Proof

      [Start] 94.9	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 94.9	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 94.9	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{c \cdot 2}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.2%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	79.4%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-116}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-58}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Accuracy	39.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 4
Accuracy	65.1%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-212}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 5
Accuracy	2.6%
Cost	192

\[\frac{b}{a} \]

Alternative 6
Accuracy	11.8%
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023146 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))