quad2p (problem 3.2.1, positive)

Average Error: 34.1 → 6.6

Time: 19.1s

Precision: binary64

Cost: 7692

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq -9.2 \cdot 10^{-284}:\\ \;\;\;\;\frac{t_0}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - t_0}{c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

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) (* a c)))))
   (if (<= b_2 -1.3e+154)
     (/ (* b_2 -2.0) a)
     (if (<= b_2 -9.2e-284)
       (- (/ t_0 a) (/ b_2 a))
       (if (<= b_2 1.8e+94)
         (/ 1.0 (/ (- (- b_2) t_0) c))
         (* -0.5 (/ c b_2)))))))

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) - (a * c)));
	double tmp;
	if (b_2 <= -1.3e+154) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= -9.2e-284) {
		tmp = (t_0 / a) - (b_2 / a);
	} else if (b_2 <= 1.8e+94) {
		tmp = 1.0 / ((-b_2 - t_0) / c);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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) :: tmp
    t_0 = sqrt(((b_2 * b_2) - (a * c)))
    if (b_2 <= (-1.3d+154)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= (-9.2d-284)) then
        tmp = (t_0 / a) - (b_2 / a)
    else if (b_2 <= 1.8d+94) then
        tmp = 1.0d0 / ((-b_2 - t_0) / c)
    else
        tmp = (-0.5d0) * (c / b_2)
    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) - (a * c)));
	double tmp;
	if (b_2 <= -1.3e+154) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= -9.2e-284) {
		tmp = (t_0 / a) - (b_2 / a);
	} else if (b_2 <= 1.8e+94) {
		tmp = 1.0 / ((-b_2 - t_0) / c);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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) - (a * c)))
	tmp = 0
	if b_2 <= -1.3e+154:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= -9.2e-284:
		tmp = (t_0 / a) - (b_2 / a)
	elif b_2 <= 1.8e+94:
		tmp = 1.0 / ((-b_2 - t_0) / c)
	else:
		tmp = -0.5 * (c / b_2)
	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(a * c)))
	tmp = 0.0
	if (b_2 <= -1.3e+154)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= -9.2e-284)
		tmp = Float64(Float64(t_0 / a) - Float64(b_2 / a));
	elseif (b_2 <= 1.8e+94)
		tmp = Float64(1.0 / Float64(Float64(Float64(-b_2) - t_0) / c));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	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) - (a * c)));
	tmp = 0.0;
	if (b_2 <= -1.3e+154)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= -9.2e-284)
		tmp = (t_0 / a) - (b_2 / a);
	elseif (b_2 <= 1.8e+94)
		tmp = 1.0 / ((-b_2 - t_0) / c);
	else
		tmp = -0.5 * (c / b_2);
	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[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.3e+154], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -9.2e-284], N[(N[(t$95$0 / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.8e+94], N[(1.0 / N[(N[((-b$95$2) - t$95$0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.29999999999999994e154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 64.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 64.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 2.5

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

   4. Simplified 2.5

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

      [Start] 2.5	\[ \frac{-2 \cdot b_2}{a} \]
      *-commutative [=>] 2.5	\[ \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -1.29999999999999994e154 < b_2 < -9.2e-284

   1. Initial program 8.0

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

   2. Simplified 8.0

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

      [Start] 8.0	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 8.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 8.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 8.0

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

   if -9.2e-284 < b_2 < 1.79999999999999996e94

   1. Initial program 32.1

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

   2. Simplified 32.1

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

      [Start] 32.1	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 32.1	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 32.1	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 32.2

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

   4. Simplified 32.2

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

      [Start] 32.2	\[ \frac{\frac{\left(b_2 \cdot b_2 - a \cdot c\right) - \left(-b_2\right) \cdot \left(-b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      associate--l- [=>] 32.2	\[ \frac{\frac{\color{blue}{b_2 \cdot b_2 - \left(a \cdot c + \left(-b_2\right) \cdot \left(-b_2\right)\right)}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      sqr-neg [=>] 32.2	\[ \frac{\frac{b_2 \cdot b_2 - \left(a \cdot c + \color{blue}{b_2 \cdot b_2}\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      *-commutative [=>] 32.2	\[ \frac{\frac{b_2 \cdot b_2 - \left(\color{blue}{c \cdot a} + b_2 \cdot b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      *-commutative [=>] 32.2	\[ \frac{\frac{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} - \left(-b_2\right)}}{a} \]

   5. Applied egg-rr 32.2

      \[\leadsto \frac{\color{blue}{\left(b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)\right) \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \left(-b_2\right)}}}{a} \]

   6. Simplified 16.1

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

      [Start] 32.2	\[ \frac{\left(b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)\right) \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \left(-b_2\right)}}{a} \]
      associate-*r/ [=>] 32.2	\[ \frac{\color{blue}{\frac{\left(b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)\right) \cdot 1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \left(-b_2\right)}}}{a} \]
      *-rgt-identity [=>] 32.2	\[ \frac{\frac{\color{blue}{b_2 \cdot b_2 - \left(c \cdot a + b_2 \cdot b_2\right)}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \left(-b_2\right)}}{a} \]
      +-commutative [=>] 32.2	\[ \frac{\frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \left(-b_2\right)}}{a} \]
      associate--r+ [=>] 16.1	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) - c \cdot a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \left(-b_2\right)}}{a} \]
      +-inverses [=>] 16.1	\[ \frac{\frac{\color{blue}{0} - c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \left(-b_2\right)}}{a} \]
      neg-sub0 [=>] 16.1	\[ \frac{\frac{0 - c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - \color{blue}{\left(0 - b_2\right)}}}{a} \]
      associate-+l- [<=] 16.1	\[ \frac{\frac{0 - c \cdot a}{\color{blue}{\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - 0\right) + b_2}}}{a} \]
      +-commutative [<=] 16.1	\[ \frac{\frac{0 - c \cdot a}{\color{blue}{b_2 + \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - 0\right)}}}{a} \]
      --rgt-identity [=>] 16.1	\[ \frac{\frac{0 - c \cdot a}{b_2 + \color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]

   7. Applied egg-rr 16.3

      \[\leadsto \color{blue}{{\left(a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{0 - c \cdot a}\right)}^{-1}} \]

   8. Simplified 16.3

      \[\leadsto \color{blue}{\frac{1}{a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{\left(-c\right) \cdot a}}} \]
      Proof

      [Start] 16.3	\[ {\left(a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{0 - c \cdot a}\right)}^{-1} \]
      unpow-1 [=>] 16.3	\[ \color{blue}{\frac{1}{a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{0 - c \cdot a}}} \]
      sub0-neg [=>] 16.3	\[ \frac{1}{a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{\color{blue}{-c \cdot a}}} \]
      mul-1-neg [<=] 16.3	\[ \frac{1}{a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{\color{blue}{-1 \cdot \left(c \cdot a\right)}}} \]
      associate-*r* [=>] 16.3	\[ \frac{1}{a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{\color{blue}{\left(-1 \cdot c\right) \cdot a}}} \]
      neg-mul-1 [<=] 16.3	\[ \frac{1}{a \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{\color{blue}{\left(-c\right)} \cdot a}} \]

   9. Applied egg-rr 20.6

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot \left(-\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\right)}{c \cdot a}}} \]

   10. Simplified 9.5

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{c}}} \]
       Proof

       [Start] 20.6	\[ \frac{1}{\frac{a \cdot \left(-\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\right)}{c \cdot a}} \]
       *-commutative [=>] 20.6	\[ \frac{1}{\frac{a \cdot \left(-\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)\right)}{\color{blue}{a \cdot c}}} \]
       times-frac [=>] 9.5	\[ \frac{1}{\color{blue}{\frac{a}{a} \cdot \frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}{c}}} \]
       *-inverses [=>] 9.5	\[ \frac{1}{\color{blue}{1} \cdot \frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}{c}} \]
       distribute-frac-neg [=>] 9.5	\[ \frac{1}{1 \cdot \color{blue}{\left(-\frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{c}\right)}} \]
       distribute-rgt-neg-in [<=] 9.5	\[ \frac{1}{\color{blue}{-1 \cdot \frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{c}}} \]
       *-lft-identity [=>] 9.5	\[ \frac{1}{-\color{blue}{\frac{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}{c}}} \]
       distribute-frac-neg [<=] 9.5	\[ \frac{1}{\color{blue}{\frac{-\left(b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}\right)}{c}}} \]
       distribute-neg-in [=>] 9.5	\[ \frac{1}{\frac{\color{blue}{\left(-b_2\right) + \left(-\sqrt{b_2 \cdot b_2 - c \cdot a}\right)}}{c}} \]
       unsub-neg [=>] 9.5	\[ \frac{1}{\frac{\color{blue}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}{c}} \]

   if 1.79999999999999996e94 < b_2

   1. Initial program 59.2

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

   2. Simplified 59.2

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

      [Start] 59.2	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 59.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 59.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 2.6

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq -9.2 \cdot 10^{-284}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.8
Cost	7496

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 2
Error	9.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 3
Error	13.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.2 \cdot 10^{-22}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 1.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{a \cdot \left(-c\right)} - b_2\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 4
Error	13.7
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.2 \cdot 10^{-26}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 2.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Error	14.1
Cost	7048

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.35 \cdot 10^{-26}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Error	22.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.85 \cdot 10^{-272}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7
Error	39.9
Cost	320

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

Reproduce

herbie shell --seed 2023187 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))