The quadratic formula (r2)

Percentage Accurate: 52.4% → 80.5%

Time: 12.1s

Precision: binary64

Cost: 20100

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -150000000:\\ \;\;\;\;\frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot a\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

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 -150000000.0)
   (- (/ (- c) b) (* (pow (/ (pow (cbrt c) 2.0) b) 3.0) a))
   (if (<= b 2e-90)
     (* -0.5 (/ 1.0 (* a (/ 1.0 (+ b (hypot b (sqrt (* c (* a -4.0)))))))))
     (/ (- 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 <= -150000000.0) {
		tmp = (-c / b) - (pow((pow(cbrt(c), 2.0) / b), 3.0) * a);
	} else if (b <= 2e-90) {
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + hypot(b, sqrt((c * (a * -4.0))))))));
	} 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 <= -150000000.0) {
		tmp = (-c / b) - (Math.pow((Math.pow(Math.cbrt(c), 2.0) / b), 3.0) * a);
	} else if (b <= 2e-90) {
		tmp = -0.5 * (1.0 / (a * (1.0 / (b + Math.hypot(b, Math.sqrt((c * (a * -4.0))))))));
	} 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 <= -150000000.0)
		tmp = Float64(Float64(Float64(-c) / b) - Float64((Float64((cbrt(c) ^ 2.0) / b) ^ 3.0) * a));
	elseif (b <= 2e-90)
		tmp = Float64(-0.5 * Float64(1.0 / Float64(a * Float64(1.0 / Float64(b + hypot(b, sqrt(Float64(c * Float64(a * -4.0)))))))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return 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, -150000000.0], N[(N[((-c) / b), $MachinePrecision] - N[(N[Power[N[(N[Power[N[Power[c, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / b), $MachinePrecision], 3.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-90], N[(-0.5 * N[(1.0 / N[(a * N[(1.0 / N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Target

Original	52.4%
Target	70.2%
Herbie	80.5%

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

Derivation

1. Split input into 3 regimes

2. if b < -1.5e8

   1. Initial program 13.9%

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

   2. Taylor expanded in b around -inf 79.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
      Step-by-step derivation

      [Start] 79.3%	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b} \]
      +-commutative [=>] 79.3%	\[ \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      mul-1-neg [=>] 79.3%	\[ -1 \cdot \frac{c}{b} + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      unsub-neg [=>] 79.3%	\[ \color{blue}{-1 \cdot \frac{c}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      associate-*r/ [=>] 79.3%	\[ \color{blue}{\frac{-1 \cdot c}{b}} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      neg-mul-1 [<=] 79.3%	\[ \frac{\color{blue}{-c}}{b} - \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      associate-/l* [=>] 81.8%	\[ \frac{-c}{b} - \color{blue}{\frac{{c}^{2}}{\frac{{b}^{3}}{a}}} \]
      unpow2 [=>] 81.8%	\[ \frac{-c}{b} - \frac{\color{blue}{c \cdot c}}{\frac{{b}^{3}}{a}} \]

   4. Applied egg-rr 94.0%

      \[\leadsto \frac{-c}{b} - \color{blue}{{\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{2} \cdot \frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}} \]
      Step-by-step derivation

      [Start] 81.8%	\[ \frac{-c}{b} - \frac{c \cdot c}{\frac{{b}^{3}}{a}} \]
      add-cube-cbrt [=>] 81.8%	\[ \frac{-c}{b} - \color{blue}{\left(\sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}} \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}}\right) \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}}} \]
      pow2 [=>] 81.8%	\[ \frac{-c}{b} - \color{blue}{{\left(\sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}}\right)}^{2}} \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
      cbrt-div [=>] 81.8%	\[ \frac{-c}{b} - {\color{blue}{\left(\frac{\sqrt[3]{c \cdot c}}{\sqrt[3]{\frac{{b}^{3}}{a}}}\right)}}^{2} \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
      cbrt-prod [=>] 81.8%	\[ \frac{-c}{b} - {\left(\frac{\color{blue}{\sqrt[3]{c} \cdot \sqrt[3]{c}}}{\sqrt[3]{\frac{{b}^{3}}{a}}}\right)}^{2} \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
      pow2 [=>] 81.8%	\[ \frac{-c}{b} - {\left(\frac{\color{blue}{{\left(\sqrt[3]{c}\right)}^{2}}}{\sqrt[3]{\frac{{b}^{3}}{a}}}\right)}^{2} \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
      cbrt-div [=>] 81.8%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\color{blue}{\frac{\sqrt[3]{{b}^{3}}}{\sqrt[3]{a}}}}\right)}^{2} \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
      rem-cbrt-cube [=>] 81.8%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{\color{blue}{b}}{\sqrt[3]{a}}}\right)}^{2} \cdot \sqrt[3]{\frac{c \cdot c}{\frac{{b}^{3}}{a}}} \]
      cbrt-div [=>] 81.8%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{2} \cdot \color{blue}{\frac{\sqrt[3]{c \cdot c}}{\sqrt[3]{\frac{{b}^{3}}{a}}}} \]
      cbrt-prod [=>] 94.0%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{2} \cdot \frac{\color{blue}{\sqrt[3]{c} \cdot \sqrt[3]{c}}}{\sqrt[3]{\frac{{b}^{3}}{a}}} \]
      pow2 [=>] 94.0%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{2} \cdot \frac{\color{blue}{{\left(\sqrt[3]{c}\right)}^{2}}}{\sqrt[3]{\frac{{b}^{3}}{a}}} \]
      cbrt-div [=>] 94.0%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{2} \cdot \frac{{\left(\sqrt[3]{c}\right)}^{2}}{\color{blue}{\frac{\sqrt[3]{{b}^{3}}}{\sqrt[3]{a}}}} \]
      rem-cbrt-cube [=>] 94.0%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{2} \cdot \frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{\color{blue}{b}}{\sqrt[3]{a}}} \]

   5. Simplified 94.1%

      \[\leadsto \frac{-c}{b} - \color{blue}{{\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot a} \]
      Step-by-step derivation

      [Start] 94.0%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{2} \cdot \frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}} \]
      pow-plus [=>] 94.0%	\[ \frac{-c}{b} - \color{blue}{{\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{\left(2 + 1\right)}} \]
      metadata-eval [=>] 94.0%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{\frac{b}{\sqrt[3]{a}}}\right)}^{\color{blue}{3}} \]
      associate-/r/ [=>] 94.0%	\[ \frac{-c}{b} - {\color{blue}{\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b} \cdot \sqrt[3]{a}\right)}}^{3} \]
      cube-prod [=>] 94.1%	\[ \frac{-c}{b} - \color{blue}{{\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot {\left(\sqrt[3]{a}\right)}^{3}} \]
      rem-cube-cbrt [=>] 94.1%	\[ \frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot \color{blue}{a} \]

   if -1.5e8 < b < 1.99999999999999999e-90

   1. Initial program 67.7%

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

   2. Simplified 67.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]
      Step-by-step derivation

      [Start] 67.7%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 67.7%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 67.7%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 67.6%	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 67.6%	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 67.6%	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 67.6%	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 67.6%	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 67.6%	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 67.6%	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 67.7%	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 67.7%	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 67.7%	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 67.7%	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 67.7%	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 67.7%	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 67.7%	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 67.6%

      \[\leadsto -0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{-1}} \]
      Step-by-step derivation

      [Start] 67.7%	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      clear-num [=>] 67.6%	\[ -0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]
      inv-pow [=>] 67.6%	\[ -0.5 \cdot \color{blue}{{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{-1}} \]

   4. Simplified 67.6%

      \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}} \]
      Step-by-step derivation

      [Start] 67.6%	\[ -0.5 \cdot {\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}\right)}^{-1} \]
      unpow-1 [=>] 67.6%	\[ -0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]
      fma-udef [=>] 67.6%	\[ -0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}}} \]
      *-commutative [=>] 67.6%	\[ -0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{a \cdot \color{blue}{\left(-4 \cdot c\right)} + b \cdot b}}} \]
      associate-*l* [<=] 67.6%	\[ -0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{\color{blue}{\left(a \cdot -4\right) \cdot c} + b \cdot b}}} \]
      +-commutative [<=] 67.6%	\[ -0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{\color{blue}{b \cdot b + \left(a \cdot -4\right) \cdot c}}}} \]
      fma-def [=>] 67.6%	\[ -0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}}} \]
      *-commutative [=>] 67.6%	\[ -0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(a \cdot -4\right)}\right)}}} \]

   5. Applied egg-rr 69.2%

      \[\leadsto -0.5 \cdot \frac{1}{\color{blue}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}} \]
      Step-by-step derivation

      [Start] 67.6%	\[ -0.5 \cdot \frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}} \]
      div-inv [=>] 67.6%	\[ -0.5 \cdot \frac{1}{\color{blue}{a \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}}} \]
      fma-udef [=>] 67.6%	\[ -0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -4\right)}}}} \]
      add-sqr-sqrt [=>] 67.6%	\[ -0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}}}}} \]
      hypot-def [=>] 69.2%	\[ -0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}} \]

   if 1.99999999999999999e-90 < b

   1. Initial program 59.1%

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

   2. Taylor expanded in b around inf 88.9%

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

   3. Simplified 88.9%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
      Step-by-step derivation

      [Start] 88.9%	\[ -1 \cdot \frac{b}{a} \]
      associate-*r/ [=>] 88.9%	\[ \color{blue}{\frac{-1 \cdot b}{a}} \]
      mul-1-neg [=>] 88.9%	\[ \frac{\color{blue}{-b}}{a} \]

3. Recombined 3 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -150000000:\\ \;\;\;\;\frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot a\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.5%
Cost	20100

\[\begin{array}{l} \mathbf{if}\;b \leq -150000000:\\ \;\;\;\;\frac{-c}{b} - {\left(\frac{{\left(\sqrt[3]{c}\right)}^{2}}{b}\right)}^{3} \cdot a\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 2
Accuracy	80.9%
Cost	14088

\[\begin{array}{l} \mathbf{if}\;b \leq -64000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-90}:\\ \;\;\;\;-0.5 \cdot \frac{1}{a \cdot \frac{1}{b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 3
Accuracy	82.2%
Cost	7688

\[\begin{array}{l} \mathbf{if}\;b \leq -440000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Accuracy	79.2%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b \leq -85000:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-96}:\\ \;\;\;\;-\frac{b + \sqrt{-4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Accuracy	66.7%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{1}{0.5 \cdot \frac{b}{c} + -0.5 \cdot \frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 6
Accuracy	67.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 7
Accuracy	66.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 8
Accuracy	34.9%
Cost	256

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

Reproduce

herbie shell --seed 2023178 
(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)))