quad2m (problem 3.2.1, negative)

Average Accuracy: 46.1% → 85.7%

Time: 16.9s

Precision: binary64

Cost: 7824

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - c \cdot a}\\ t_1 := -0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -2.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - t_0}}{a}\\ \mathbf{elif}\;b_2 \leq -6.6 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 1.56 \cdot 10^{+115}:\\ \;\;\;\;\frac{-t_0}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

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
 (let* ((t_0 (sqrt (- (* b_2 b_2) (* c a)))) (t_1 (* -0.5 (/ c b_2))))
   (if (<= b_2 -1.9e-26)
     t_1
     (if (<= b_2 -2.8e-98)
       (/ (/ (* c (- a)) (- b_2 t_0)) a)
       (if (<= b_2 -6.6e-112)
         t_1
         (if (<= b_2 1.56e+115)
           (- (/ (- t_0) a) (/ b_2 a))
           (+ (* (/ b_2 a) -2.0) (* (/ c 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 t_0 = sqrt(((b_2 * b_2) - (c * a)));
	double t_1 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -1.9e-26) {
		tmp = t_1;
	} else if (b_2 <= -2.8e-98) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -6.6e-112) {
		tmp = t_1;
	} else if (b_2 <= 1.56e+115) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

↓

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (c * a)))
    t_1 = (-0.5d0) * (c / b_2)
    if (b_2 <= (-1.9d-26)) then
        tmp = t_1
    else if (b_2 <= (-2.8d-98)) then
        tmp = ((c * -a) / (b_2 - t_0)) / a
    else if (b_2 <= (-6.6d-112)) then
        tmp = t_1
    else if (b_2 <= 1.56d+115) then
        tmp = (-t_0 / a) - (b_2 / a)
    else
        tmp = ((b_2 / a) * (-2.0d0)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

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 t_0 = Math.sqrt(((b_2 * b_2) - (c * a)));
	double t_1 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -1.9e-26) {
		tmp = t_1;
	} else if (b_2 <= -2.8e-98) {
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	} else if (b_2 <= -6.6e-112) {
		tmp = t_1;
	} else if (b_2 <= 1.56e+115) {
		tmp = (-t_0 / a) - (b_2 / a);
	} else {
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	}
	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):
	t_0 = math.sqrt(((b_2 * b_2) - (c * a)))
	t_1 = -0.5 * (c / b_2)
	tmp = 0
	if b_2 <= -1.9e-26:
		tmp = t_1
	elif b_2 <= -2.8e-98:
		tmp = ((c * -a) / (b_2 - t_0)) / a
	elif b_2 <= -6.6e-112:
		tmp = t_1
	elif b_2 <= 1.56e+115:
		tmp = (-t_0 / a) - (b_2 / a)
	else:
		tmp = ((b_2 / a) * -2.0) + ((c / 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)
	t_0 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	t_1 = Float64(-0.5 * Float64(c / b_2))
	tmp = 0.0
	if (b_2 <= -1.9e-26)
		tmp = t_1;
	elseif (b_2 <= -2.8e-98)
		tmp = Float64(Float64(Float64(c * Float64(-a)) / Float64(b_2 - t_0)) / a);
	elseif (b_2 <= -6.6e-112)
		tmp = t_1;
	elseif (b_2 <= 1.56e+115)
		tmp = Float64(Float64(Float64(-t_0) / a) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + Float64(Float64(c / b_2) * 0.5));
	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)
	t_0 = sqrt(((b_2 * b_2) - (c * a)));
	t_1 = -0.5 * (c / b_2);
	tmp = 0.0;
	if (b_2 <= -1.9e-26)
		tmp = t_1;
	elseif (b_2 <= -2.8e-98)
		tmp = ((c * -a) / (b_2 - t_0)) / a;
	elseif (b_2 <= -6.6e-112)
		tmp = t_1;
	elseif (b_2 <= 1.56e+115)
		tmp = (-t_0 / a) - (b_2 / a);
	else
		tmp = ((b_2 / a) * -2.0) + ((c / b_2) * 0.5);
	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_] := Block[{t$95$0 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -1.9e-26], t$95$1, If[LessEqual[b$95$2, -2.8e-98], N[(N[(N[(c * (-a)), $MachinePrecision] / N[(b$95$2 - t$95$0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -6.6e-112], t$95$1, If[LessEqual[b$95$2, 1.56e+115], N[(N[((-t$95$0) / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.90000000000000007e-26 or -2.7999999999999999e-98 < b_2 < -6.6000000000000002e-112

   1. Initial program 13.5%

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

   2. Taylor expanded in b_2 around -inf 89.1%

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

   if -1.90000000000000007e-26 < b_2 < -2.7999999999999999e-98

   1. Initial program 39.5%

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

   2. Applied egg-rr 70.9%

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

      [Start] 39.5	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      flip-- [=>] 39.5	\[ \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      frac-2neg [=>] 39.5	\[ \frac{\color{blue}{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{a} \]
      add-sqr-sqrt [<=] 39.5	\[ \frac{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - \color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}\right)}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      associate--r- [=>] 70.9	\[ \frac{\frac{-\color{blue}{\left(\left(\left(-b_2\right) \cdot \left(-b_2\right) - b_2 \cdot b_2\right) + a \cdot c\right)}}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      +-commutative [=>] 70.9	\[ \frac{\frac{-\color{blue}{\left(a \cdot c + \left(\left(-b_2\right) \cdot \left(-b_2\right) - b_2 \cdot b_2\right)\right)}}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqr-neg [=>] 70.9	\[ \frac{\frac{-\left(a \cdot c + \left(\color{blue}{b_2 \cdot b_2} - b_2 \cdot b_2\right)\right)}{-\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      add-sqr-sqrt [=>] 70.9	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqrt-unprod [=>] 70.9	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqr-neg [=>] 70.9	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\sqrt{\color{blue}{b_2 \cdot b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      sqrt-prod [=>] 0.0	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]
      add-sqr-sqrt [<=] 34.3	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{-\left(\color{blue}{b_2} + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a} \]

   3. Simplified 70.9%

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

      [Start] 70.9	\[ \frac{\frac{-\left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [=>] 70.9	\[ \frac{\frac{\color{blue}{0 - \left(a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-commutative [=>] 70.9	\[ \frac{\frac{0 - \color{blue}{\left(\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      +-inverses [=>] 70.9	\[ \frac{\frac{0 - \left(\color{blue}{0} + a \cdot c\right)}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      associate--r+ [=>] 70.9	\[ \frac{\frac{\color{blue}{\left(0 - 0\right) - a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      metadata-eval [=>] 70.9	\[ \frac{\frac{\color{blue}{0} - a \cdot c}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      neg-sub0 [<=] 70.9	\[ \frac{\frac{\color{blue}{-a \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      distribute-lft-neg-in [=>] 70.9	\[ \frac{\frac{\color{blue}{\left(-a\right) \cdot c}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 70.9	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]
      *-commutative [=>] 70.9	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}}{a} \]

   if -6.6000000000000002e-112 < b_2 < 1.56e115

   1. Initial program 81.8%

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

   2. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]
      Proof

      [Start] 81.8	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      div-sub [=>] 81.8	\[ \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \]
      neg-sub0 [=>] 81.8	\[ \frac{\color{blue}{0 - b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      div-sub [=>] 81.8	\[ \color{blue}{\left(\frac{0}{a} - \frac{b_2}{a}\right)} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-sqr-sqrt [=>] 62.0	\[ \left(\frac{0}{a} - \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sqrt-prod [<=] 81.3	\[ \left(\frac{0}{a} - \frac{\color{blue}{\sqrt{b_2 \cdot b_2}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sqr-neg [<=] 81.3	\[ \left(\frac{0}{a} - \frac{\sqrt{\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sqrt-unprod [<=] 19.5	\[ \left(\frac{0}{a} - \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-sqr-sqrt [<=] 51.1	\[ \left(\frac{0}{a} - \frac{\color{blue}{-b_2}}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      associate--l- [=>] 51.1	\[ \color{blue}{\frac{0}{a} - \left(\frac{-b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]
      add-sqr-sqrt [=>] 19.5	\[ \frac{0}{a} - \left(\frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      sqrt-unprod [=>] 81.3	\[ \frac{0}{a} - \left(\frac{\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      sqr-neg [=>] 81.3	\[ \frac{0}{a} - \left(\frac{\sqrt{\color{blue}{b_2 \cdot b_2}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      sqrt-prod [=>] 62.0	\[ \frac{0}{a} - \left(\frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      add-sqr-sqrt [<=] 81.8	\[ \frac{0}{a} - \left(\frac{\color{blue}{b_2}}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}} \]
      Proof

      [Start] 81.8	\[ \frac{0}{a} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      div0 [=>] 81.8	\[ \color{blue}{0} - \left(\frac{b_2}{a} + \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]
      +-commutative [=>] 81.8	\[ 0 - \color{blue}{\left(\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} + \frac{b_2}{a}\right)} \]
      associate--r+ [=>] 81.8	\[ \color{blue}{\left(0 - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) - \frac{b_2}{a}} \]
      neg-sub0 [<=] 81.8	\[ \color{blue}{\left(-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} - \frac{b_2}{a} \]
      distribute-neg-frac [=>] 81.8	\[ \color{blue}{\frac{-\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} - \frac{b_2}{a} \]
      *-commutative [=>] 81.8	\[ \frac{-\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} - \frac{b_2}{a} \]

   if 1.56e115 < b_2

   1. Initial program 21.5%

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

   2. Taylor expanded in b_2 around inf 94.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-26}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{a}\\ \mathbf{elif}\;b_2 \leq -6.6 \cdot 10^{-112}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.56 \cdot 10^{+115}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.0%
Cost	7560

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.23 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.56 \cdot 10^{+115}:\\ \;\;\;\;\frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2
Accuracy	85.0%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.23 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.56 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(-\sqrt{b_2 \cdot b_2 - c \cdot a}\right) - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3
Accuracy	79.4%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.3 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4
Accuracy	65.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.6 \cdot 10^{-306}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2 + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 5
Accuracy	65.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.7 \cdot 10^{-265}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 6
Accuracy	37.4%
Cost	320

\[-0.5 \cdot \frac{c}{b_2} \]

Alternative 7
Accuracy	3.8%
Cost	64

\[-2 \]

Alternative 8
Accuracy	11.9%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))