quad2m (problem 3.2.1, negative)

Average Accuracy: 42.6% → 75.8%

Time: 15.2s

Precision: binary64

Cost: 13768

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \left(\frac{c}{b_2} \cdot \left(0.125 \cdot \frac{a}{b_2}\right) + 0.5\right)\right)\\ \end{array} \]

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 -9e-122)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6.8e+30)
     (/ (- (- b_2) (sqrt (fma b_2 b_2 (* c (- a))))) a)
     (fma
      -2.0
      (/ b_2 a)
      (* (/ c b_2) (+ (* (/ c b_2) (* 0.125 (/ a b_2))) 0.5))))))

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 <= -9e-122) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.8e+30) {
		tmp = (-b_2 - sqrt(fma(b_2, b_2, (c * -a)))) / a;
	} else {
		tmp = fma(-2.0, (b_2 / a), ((c / b_2) * (((c / b_2) * (0.125 * (a / b_2))) + 0.5)));
	}
	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 <= -9e-122)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6.8e+30)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(c * Float64(-a))))) / a);
	else
		tmp = fma(-2.0, Float64(b_2 / a), Float64(Float64(c / b_2) * Float64(Float64(Float64(c / b_2) * Float64(0.125 * Float64(a / b_2))) + 0.5)));
	end
	return 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, -9e-122], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.8e+30], N[(N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * N[(N[(N[(c / b$95$2), $MachinePrecision] * N[(0.125 * N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.99999999999999959e-122

   1. Initial program 19.9%

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

   2. Taylor expanded in b_2 around -inf 82.5%

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

   if -8.99999999999999959e-122 < b_2 < 6.8000000000000005e30

   1. Initial program 79.4%

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

   2. Applied egg-rr 79.4%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}}{a} \]
      Proof

      [Start] 79.4	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      fma-neg [=>] 79.4	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}}{a} \]
      *-commutative [=>] 79.4	\[ \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)}}{a} \]
      distribute-rgt-neg-in [=>] 79.4	\[ \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)}}{a} \]

   if 6.8000000000000005e30 < b_2

   1. Initial program 31.5%

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

   2. Taylor expanded in b_2 around inf 49.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + 0.5 \cdot \frac{c}{b_2}\right)} \]

   3. Simplified 50.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{0.125}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}\right)\right)} \]
      Proof

      [Start] 49.1	\[ -2 \cdot \frac{b_2}{a} + \left(0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + 0.5 \cdot \frac{c}{b_2}\right) \]
      fma-def [=>] 49.1	\[ \color{blue}{\mathsf{fma}\left(-2, \frac{b_2}{a}, 0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + 0.5 \cdot \frac{c}{b_2}\right)} \]
      +-commutative [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{0.5 \cdot \frac{c}{b_2} + 0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}}\right) \]
      fma-def [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b_2}, 0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}}\right)}\right) \]
      associate-*r/ [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(0.5, \frac{c}{b_2}, \color{blue}{\frac{0.125 \cdot \left({c}^{2} \cdot a\right)}{{b_2}^{3}}}\right)\right) \]
      associate-/l* [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(0.5, \frac{c}{b_2}, \color{blue}{\frac{0.125}{\frac{{b_2}^{3}}{{c}^{2} \cdot a}}}\right)\right) \]
      unpow2 [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{0.125}{\frac{{b_2}^{3}}{\color{blue}{\left(c \cdot c\right)} \cdot a}}\right)\right) \]
      associate-*l* [=>] 50.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{0.125}{\frac{{b_2}^{3}}{\color{blue}{c \cdot \left(c \cdot a\right)}}}\right)\right) \]

   4. Applied egg-rr 54.5%

      \[\leadsto \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\log \left({\left(e^{0.125}\right)}^{\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}\right)} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)}\right) \]
      Proof

      [Start] 50.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{0.125}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}\right)\right) \]
      fma-udef [=>] 50.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{0.5 \cdot \frac{c}{b_2} + \frac{0.125}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}}\right) \]
      +-commutative [=>] 50.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\frac{0.125}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}} + 0.5 \cdot \frac{c}{b_2}}\right) \]
      add-log-exp [=>] 49.8	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\log \left(e^{\frac{0.125}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}}\right)} + 0.5 \cdot \frac{c}{b_2}\right) \]
      add-log-exp [=>] 46.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left(e^{\frac{0.125}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}}\right) + \color{blue}{\log \left(e^{0.5 \cdot \frac{c}{b_2}}\right)}\right) \]
      sum-log [=>] 46.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\log \left(e^{\frac{0.125}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)}\right) \]
      div-inv [=>] 46.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left(e^{\color{blue}{0.125 \cdot \frac{1}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}}} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      exp-prod [=>] 46.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left(\color{blue}{{\left(e^{0.125}\right)}^{\left(\frac{1}{\frac{{b_2}^{3}}{c \cdot \left(c \cdot a\right)}}\right)}} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      clear-num [<=] 46.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left({\left(e^{0.125}\right)}^{\color{blue}{\left(\frac{c \cdot \left(c \cdot a\right)}{{b_2}^{3}}\right)}} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      associate-*r* [=>] 46.8	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left({\left(e^{0.125}\right)}^{\left(\frac{\color{blue}{\left(c \cdot c\right) \cdot a}}{{b_2}^{3}}\right)} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      unpow3 [=>] 46.8	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left({\left(e^{0.125}\right)}^{\left(\frac{\left(c \cdot c\right) \cdot a}{\color{blue}{\left(b_2 \cdot b_2\right) \cdot b_2}}\right)} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      times-frac [=>] 50.4	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left({\left(e^{0.125}\right)}^{\color{blue}{\left(\frac{c \cdot c}{b_2 \cdot b_2} \cdot \frac{a}{b_2}\right)}} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      frac-times [<=] 54.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left({\left(e^{0.125}\right)}^{\left(\color{blue}{\left(\frac{c}{b_2} \cdot \frac{c}{b_2}\right)} \cdot \frac{a}{b_2}\right)} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      pow2 [=>] 54.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left({\left(e^{0.125}\right)}^{\left(\color{blue}{{\left(\frac{c}{b_2}\right)}^{2}} \cdot \frac{a}{b_2}\right)} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]

   5. Simplified 61.9%

      \[\leadsto \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\mathsf{fma}\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}, 0.125, 0.5 \cdot \frac{c}{b_2}\right)}\right) \]
      Proof

      [Start] 54.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \log \left({\left(e^{0.125}\right)}^{\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}\right)} \cdot e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      log-prod [=>] 54.5	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\log \left({\left(e^{0.125}\right)}^{\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}\right)}\right) + \log \left(e^{0.5 \cdot \frac{c}{b_2}}\right)}\right) \]
      log-pow [=>] 54.4	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}\right) \cdot \log \left(e^{0.125}\right)} + \log \left(e^{0.5 \cdot \frac{c}{b_2}}\right)\right) \]
      fma-def [=>] 54.4	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\mathsf{fma}\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}, \log \left(e^{0.125}\right), \log \left(e^{0.5 \cdot \frac{c}{b_2}}\right)\right)}\right) \]
      rem-log-exp [=>] 54.4	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}, \color{blue}{0.125}, \log \left(e^{0.5 \cdot \frac{c}{b_2}}\right)\right)\right) \]
      rem-log-exp [=>] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}, 0.125, \color{blue}{0.5 \cdot \frac{c}{b_2}}\right)\right) \]

   6. Taylor expanded in c around 0 49.1%

      \[\leadsto \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + 0.5 \cdot \frac{c}{b_2}}\right) \]

   7. Simplified 62.3%

      \[\leadsto \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\frac{c}{b_2} \cdot \left(\frac{c}{b_2} \cdot \left(0.125 \cdot \frac{a}{b_2}\right) + 0.5\right)}\right) \]
      Proof

      [Start] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, 0.125 \cdot \frac{{c}^{2} \cdot a}{{b_2}^{3}} + 0.5 \cdot \frac{c}{b_2}\right) \]
      *-commutative [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\frac{{c}^{2} \cdot a}{{b_2}^{3}} \cdot 0.125} + 0.5 \cdot \frac{c}{b_2}\right) \]
      *-commutative [<=] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{{c}^{2} \cdot a}{{b_2}^{3}} \cdot 0.125 + \color{blue}{\frac{c}{b_2} \cdot 0.5}\right) \]
      fma-def [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b_2}^{3}}, 0.125, \frac{c}{b_2} \cdot 0.5\right)}\right) \]
      unpow3 [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\frac{{c}^{2} \cdot a}{\color{blue}{\left(b_2 \cdot b_2\right) \cdot b_2}}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      unpow2 [<=] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\frac{{c}^{2} \cdot a}{\color{blue}{{b_2}^{2}} \cdot b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      unpow2 [=>] 49.1	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\frac{\color{blue}{\left(c \cdot c\right)} \cdot a}{{b_2}^{2} \cdot b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      times-frac [=>] 52.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\color{blue}{\frac{c \cdot c}{{b_2}^{2}} \cdot \frac{a}{b_2}}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      associate-/l* [=>] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\color{blue}{\frac{c}{\frac{{b_2}^{2}}{c}}} \cdot \frac{a}{b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      unpow2 [=>] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\frac{c}{\frac{\color{blue}{b_2 \cdot b_2}}{c}} \cdot \frac{a}{b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      associate-*l/ [<=] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\frac{c}{\color{blue}{\frac{b_2}{c} \cdot b_2}} \cdot \frac{a}{b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      associate-/l/ [<=] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b_2}}{\frac{b_2}{c}}} \cdot \frac{a}{b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      associate-/l* [<=] 59.7	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\color{blue}{\frac{\frac{c}{b_2} \cdot c}{b_2}} \cdot \frac{a}{b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      associate-*r/ [<=] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\color{blue}{\left(\frac{c}{b_2} \cdot \frac{c}{b_2}\right)} \cdot \frac{a}{b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      unpow2 [<=] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \mathsf{fma}\left(\color{blue}{{\left(\frac{c}{b_2}\right)}^{2}} \cdot \frac{a}{b_2}, 0.125, \frac{c}{b_2} \cdot 0.5\right)\right) \]
      fma-def [<=] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{\left({\left(\frac{c}{b_2}\right)}^{2} \cdot \frac{a}{b_2}\right) \cdot 0.125 + \frac{c}{b_2} \cdot 0.5}\right) \]
      associate-*r* [<=] 61.9	\[ \mathsf{fma}\left(-2, \frac{b_2}{a}, \color{blue}{{\left(\frac{c}{b_2}\right)}^{2} \cdot \left(\frac{a}{b_2} \cdot 0.125\right)} + \frac{c}{b_2} \cdot 0.5\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \left(\frac{c}{b_2} \cdot \left(0.125 \cdot \frac{a}{b_2}\right) + 0.5\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	75.8%
Cost	7880

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{c}{b_2} \cdot \left(\frac{c}{b_2} \cdot \left(0.125 \cdot \frac{a}{b_2}\right) + 0.5\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	75.9%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.8 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Accuracy	72.1%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.9 \cdot 10^{-73}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4
Accuracy	59.2%
Cost	836

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

Alternative 5
Accuracy	35.2%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 6
Accuracy	59.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.2 \cdot 10^{-203}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 7
Accuracy	15.3%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.6 \cdot 10^{-300}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 8
Accuracy	11.1%
Cost	192

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

Reproduce

herbie shell --seed 2023153 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))