quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.5% → 85.1%

Time: 23.3s

Precision: binary64

Cost: 13704

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right)}{a}\\ \mathbf{elif}\;b_2 \leq 5.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.2e-22)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3e-191)
     (/ (- (- b_2) (hypot (sqrt (- (* c a))) b_2)) a)
     (if (<= b_2 5.8e+56)
       (- (/ (- b_2) a) (/ (pow (- (* b_2 b_2) (* c a)) 0.5) a))
       (/ (* b_2 -2.0) a)))))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-22) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3e-191) {
		tmp = (-b_2 - hypot(sqrt(-(c * a)), b_2)) / a;
	} else if (b_2 <= 5.8e+56) {
		tmp = (-b_2 / a) - (pow(((b_2 * b_2) - (c * a)), 0.5) / a);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

↓

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.2e-22) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3e-191) {
		tmp = (-b_2 - Math.hypot(Math.sqrt(-(c * a)), b_2)) / a;
	} else if (b_2 <= 5.8e+56) {
		tmp = (-b_2 / a) - (Math.pow(((b_2 * b_2) - (c * a)), 0.5) / a);
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

↓

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.2e-22:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3e-191:
		tmp = (-b_2 - math.hypot(math.sqrt(-(c * a)), b_2)) / a
	elif b_2 <= 5.8e+56:
		tmp = (-b_2 / a) - (math.pow(((b_2 * b_2) - (c * a)), 0.5) / a)
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

↓

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.2e-22)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3e-191)
		tmp = Float64(Float64(Float64(-b_2) - hypot(sqrt(Float64(-Float64(c * a))), b_2)) / a);
	elseif (b_2 <= 5.8e+56)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64((Float64(Float64(b_2 * b_2) - Float64(c * a)) ^ 0.5) / a));
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

↓

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.2e-22)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3e-191)
		tmp = (-b_2 - hypot(sqrt(-(c * a)), b_2)) / a;
	elseif (b_2 <= 5.8e+56)
		tmp = (-b_2 / a) - ((((b_2 * b_2) - (c * a)) ^ 0.5) / a);
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

↓

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.2e-22], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3e-191], N[(N[((-b$95$2) - N[Sqrt[N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision] ^ 2 + b$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.8e+56], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Power[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -4.20000000000000016e-22

   1. Initial program 18.6%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 87.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      Step-by-step derivation

      [Start] 87.2	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 87.2	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -4.20000000000000016e-22 < b_2 < 3.0000000000000001e-191

   1. Initial program 75.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 81.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-a\right)}, b_2\right)}}{a} \]
      Step-by-step derivation

      [Start] 75.0	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sub-neg [=>] 75.0	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{b_2 \cdot b_2 + \left(-a \cdot c\right)}}}{a} \]
      +-commutative [=>] 75.0	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a \cdot c\right) + b_2 \cdot b_2}}}{a} \]
      add-sqr-sqrt [=>] 75.0	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}} + b_2 \cdot b_2}}{a} \]
      hypot-def [=>] 81.6	\[ \frac{\left(-b_2\right) - \color{blue}{\mathsf{hypot}\left(\sqrt{-a \cdot c}, b_2\right)}}{a} \]
      *-commutative [=>] 81.6	\[ \frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{-\color{blue}{c \cdot a}}, b_2\right)}{a} \]
      distribute-rgt-neg-in [=>] 81.6	\[ \frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{\color{blue}{c \cdot \left(-a\right)}}, b_2\right)}{a} \]

   if 3.0000000000000001e-191 < b_2 < 5.80000000000000014e56

   1. Initial program 90.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Applied egg-rr 89.9%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
      Step-by-step derivation

      [Start] 90.3	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-sqr-sqrt [=>] 89.9	\[ \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      pow2 [=>] 89.9	\[ \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]
      pow1/2 [=>] 89.9	\[ \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]
      sqrt-pow1 [=>] 89.9	\[ \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]
      metadata-eval [=>] 89.9	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 84.6%

      \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}}^{2}}{a} \]
      Step-by-step derivation

      [Start] 89.9	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{a} \]
      add-exp-log [=>] 84.6	\[ \frac{\left(-b_2\right) - {\color{blue}{\left(e^{\log \left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}\right)}}^{2}}{a} \]
      log-pow [=>] 84.6	\[ \frac{\left(-b_2\right) - {\left(e^{\color{blue}{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}}\right)}^{2}}{a} \]

   4. Applied egg-rr 90.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.25}\right)}^{2}}{a}\right)} \]
      Step-by-step derivation

      [Start] 84.6	\[ \frac{\left(-b_2\right) - {\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      div-sub [=>] 84.5	\[ \color{blue}{\frac{-b_2}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a}} \]
      neg-mul-1 [=>] 84.5	\[ \frac{\color{blue}{-1 \cdot b_2}}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      *-un-lft-identity [=>] 84.5	\[ \frac{-1 \cdot b_2}{\color{blue}{1 \cdot a}} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      times-frac [=>] 84.5	\[ \color{blue}{\frac{-1}{1} \cdot \frac{b_2}{a}} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      metadata-eval [=>] 84.5	\[ \color{blue}{-1} \cdot \frac{b_2}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      add-sqr-sqrt [=>] 84.6	\[ -1 \cdot \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      sqrt-prod [<=] 84.5	\[ -1 \cdot \frac{\color{blue}{\sqrt{b_2 \cdot b_2}}}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      sqr-neg [<=] 84.5	\[ -1 \cdot \frac{\sqrt{\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      sqrt-unprod [<=] 0.0	\[ -1 \cdot \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      add-sqr-sqrt [<=] 54.5	\[ -1 \cdot \frac{\color{blue}{-b_2}}{a} - \frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a} \]
      fma-neg [=>] 54.5	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{-b_2}{a}, -\frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a}\right)} \]
      add-sqr-sqrt [=>] 0.0	\[ \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a}, -\frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a}\right) \]
      sqrt-unprod [=>] 84.5	\[ \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a}, -\frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a}\right) \]
      sqr-neg [=>] 84.5	\[ \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b_2 \cdot b_2}}}{a}, -\frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a}\right) \]
      sqrt-prod [=>] 84.6	\[ \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a}, -\frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a}\right) \]
      add-sqr-sqrt [<=] 84.5	\[ \mathsf{fma}\left(-1, \frac{\color{blue}{b_2}}{a}, -\frac{{\left(e^{0.25 \cdot \log \left(b_2 \cdot b_2 - a \cdot c\right)}\right)}^{2}}{a}\right) \]

   5. Simplified 90.4%

      \[\leadsto \color{blue}{\left(-\frac{b_2}{a}\right) - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5}}{a}} \]
      Step-by-step derivation

      [Start] 90.0	\[ \mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.25}\right)}^{2}}{a}\right) \]
      fma-udef [=>] 90.0	\[ \color{blue}{-1 \cdot \frac{b_2}{a} + \left(-\frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.25}\right)}^{2}}{a}\right)} \]
      unsub-neg [=>] 90.0	\[ \color{blue}{-1 \cdot \frac{b_2}{a} - \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.25}\right)}^{2}}{a}} \]
      mul-1-neg [=>] 90.0	\[ \color{blue}{\left(-\frac{b_2}{a}\right)} - \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      unpow2 [=>] 90.0	\[ \left(-\frac{b_2}{a}\right) - \frac{\color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.25} \cdot {\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{0.25}}}{a} \]
      pow-sqr [=>] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{\color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, a \cdot \left(-c\right)\right)\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]
      fma-udef [=>] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{{\color{blue}{\left(b_2 \cdot b_2 + a \cdot \left(-c\right)\right)}}^{\left(2 \cdot 0.25\right)}}{a} \]
      unpow2 [<=] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{{\left(\color{blue}{{b_2}^{2}} + a \cdot \left(-c\right)\right)}^{\left(2 \cdot 0.25\right)}}{a} \]
      distribute-rgt-neg-out [=>] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{{\left({b_2}^{2} + \color{blue}{\left(-a \cdot c\right)}\right)}^{\left(2 \cdot 0.25\right)}}{a} \]
      *-commutative [<=] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{{\left({b_2}^{2} + \left(-\color{blue}{c \cdot a}\right)\right)}^{\left(2 \cdot 0.25\right)}}{a} \]
      unsub-neg [=>] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{{\color{blue}{\left({b_2}^{2} - c \cdot a\right)}}^{\left(2 \cdot 0.25\right)}}{a} \]
      unpow2 [=>] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{{\left(\color{blue}{b_2 \cdot b_2} - c \cdot a\right)}^{\left(2 \cdot 0.25\right)}}{a} \]
      metadata-eval [=>] 90.4	\[ \left(-\frac{b_2}{a}\right) - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.5}}}{a} \]

   if 5.80000000000000014e56 < b_2

   1. Initial program 65.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 98.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]

   3. Simplified 98.4%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]
      Step-by-step derivation

      [Start] 98.4	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 98.4	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

3. Recombined 4 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3 \cdot 10^{-191}:\\ \;\;\;\;\frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right)}{a}\\ \mathbf{elif}\;b_2 \leq 5.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.9%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{-23}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.8 \cdot 10^{+56}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{0.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 2
Accuracy	84.9%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+56}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 3
Accuracy	80.3%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{-c \cdot a}}{-a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Accuracy	80.3%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2 \cdot 10^{-22}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.22 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Accuracy	80.1%
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.05 \cdot 10^{-38}:\\ \;\;\;\;\frac{b_2 - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Accuracy	67.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7
Accuracy	44.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 8
Accuracy	44.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 9
Accuracy	67.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.8 \cdot 10^{-284}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 10
Accuracy	67.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.7 \cdot 10^{-284}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 11
Accuracy	23.9%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.75 \cdot 10^{-284}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 12
Accuracy	11.1%
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2023159 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))