quad2p (problem 3.2.1, positive)

Average Error: 34.6 → 9.3

Time: 18.6s

Precision: binary64

Cost: 7880

Math

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

↓

\[\begin{array}{l} t_0 := b_2 \cdot b_2 - a \cdot c\\ \mathbf{if}\;b_2 \leq -8 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{a \cdot c + \left(t_0 - a \cdot c\right)} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + \sqrt{t_0}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

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 (- (* b_2 b_2) (* a c))))
   (if (<= b_2 -8e+62)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
     (if (<= b_2 1.5e-157)
       (/ (- (sqrt (+ (* a c) (- t_0 (* a c)))) b_2) a)
       (if (<= b_2 1.55e+22)
         (/ (/ (* a (- c)) (+ b_2 (sqrt t_0))) a)
         (/ (* c -0.5) 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 = (b_2 * b_2) - (a * c);
	double tmp;
	if (b_2 <= -8e+62) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.5e-157) {
		tmp = (sqrt(((a * c) + (t_0 - (a * c)))) - b_2) / a;
	} else if (b_2 <= 1.55e+22) {
		tmp = ((a * -c) / (b_2 + sqrt(t_0))) / a;
	} else {
		tmp = (c * -0.5) / 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 = (b_2 * b_2) - (a * c)
    if (b_2 <= (-8d+62)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 1.5d-157) then
        tmp = (sqrt(((a * c) + (t_0 - (a * c)))) - b_2) / a
    else if (b_2 <= 1.55d+22) then
        tmp = ((a * -c) / (b_2 + sqrt(t_0))) / a
    else
        tmp = (c * (-0.5d0)) / 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 = (b_2 * b_2) - (a * c);
	double tmp;
	if (b_2 <= -8e+62) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 1.5e-157) {
		tmp = (Math.sqrt(((a * c) + (t_0 - (a * c)))) - b_2) / a;
	} else if (b_2 <= 1.55e+22) {
		tmp = ((a * -c) / (b_2 + Math.sqrt(t_0))) / a;
	} else {
		tmp = (c * -0.5) / 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 = (b_2 * b_2) - (a * c)
	tmp = 0
	if b_2 <= -8e+62:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 1.5e-157:
		tmp = (math.sqrt(((a * c) + (t_0 - (a * c)))) - b_2) / a
	elif b_2 <= 1.55e+22:
		tmp = ((a * -c) / (b_2 + math.sqrt(t_0))) / a
	else:
		tmp = (c * -0.5) / 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 = Float64(Float64(b_2 * b_2) - Float64(a * c))
	tmp = 0.0
	if (b_2 <= -8e+62)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 1.5e-157)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * c) + Float64(t_0 - Float64(a * c)))) - b_2) / a);
	elseif (b_2 <= 1.55e+22)
		tmp = Float64(Float64(Float64(a * Float64(-c)) / Float64(b_2 + sqrt(t_0))) / a);
	else
		tmp = Float64(Float64(c * -0.5) / 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 = (b_2 * b_2) - (a * c);
	tmp = 0.0;
	if (b_2 <= -8e+62)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 1.5e-157)
		tmp = (sqrt(((a * c) + (t_0 - (a * c)))) - b_2) / a;
	elseif (b_2 <= 1.55e+22)
		tmp = ((a * -c) / (b_2 + sqrt(t_0))) / a;
	else
		tmp = (c * -0.5) / 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[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -8e+62], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-157], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] + N[(t$95$0 - N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.55e+22], N[(N[(N[(a * (-c)), $MachinePrecision] / N[(b$95$2 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -8.00000000000000028e62

   1. Initial program 40.2

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

   2. Simplified 40.2

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

      [Start] 40.2	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 40.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 40.2	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around -inf 5.5

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -8.00000000000000028e62 < b_2 < 1.5e-157

   1. Initial program 12.5

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

   2. Simplified 12.5

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

      [Start] 12.5	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 12.5	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 12.5	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 12.5

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

   if 1.5e-157 < b_2 < 1.5500000000000001e22

   1. Initial program 33.8

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

   2. Simplified 33.8

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

      [Start] 33.8	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 33.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 33.8	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Applied egg-rr 33.8

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

   4. Simplified 33.8

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

      [Start] 33.8	\[ \frac{\frac{\left(b_2 \cdot b_2 - a \cdot c\right) - \left(-b_2\right) \cdot \left(-b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      *-commutative [=>] 33.8	\[ \frac{\frac{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) - \left(-b_2\right) \cdot \left(-b_2\right)}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      sqr-neg [=>] 33.8	\[ \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - \color{blue}{b_2 \cdot b_2}}{\sqrt{b_2 \cdot b_2 - a \cdot c} - \left(-b_2\right)}}{a} \]
      *-commutative [=>] 33.8	\[ \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}} - \left(-b_2\right)}}{a} \]

   5. Applied egg-rr 34.0

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

   6. Simplified 16.5

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

      [Start] 34.0	\[ \frac{\frac{1}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}} \cdot \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      associate-*l/ [=>] 33.9	\[ \frac{\color{blue}{\frac{1 \cdot \frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}}{a} \]
      associate-*r/ [=>] 33.9	\[ \frac{\frac{\color{blue}{\frac{1 \cdot \left(b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)\right)}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      *-lft-identity [=>] 33.9	\[ \frac{\frac{\frac{\color{blue}{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      associate-/l/ [=>] 34.0	\[ \frac{\color{blue}{\frac{b_2 \cdot b_2 - \mathsf{fma}\left(b_2, b_2, c \cdot a\right)}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}}{a} \]
      fma-udef [=>] 33.9	\[ \frac{\frac{b_2 \cdot b_2 - \color{blue}{\left(b_2 \cdot b_2 + c \cdot a\right)}}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      associate--r+ [=>] 16.6	\[ \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) - c \cdot a}}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      +-inverses [=>] 16.6	\[ \frac{\frac{\color{blue}{0} - c \cdot a}{\sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}} \cdot \sqrt{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]
      rem-square-sqrt [=>] 16.5	\[ \frac{\frac{0 - c \cdot a}{\color{blue}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}}}{a} \]

   if 1.5500000000000001e22 < b_2

   1. Initial program 56.0

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

   2. Simplified 56.0

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

      [Start] 56.0	\[ \frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      +-commutative [=>] 56.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]
      unsub-neg [=>] 56.0	\[ \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Taylor expanded in b_2 around inf 5.0

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

   4. Simplified 5.0

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

      [Start] 5.0	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 5.0	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      *-commutative [=>] 5.0	\[ \frac{\color{blue}{c \cdot -0.5}}{b_2} \]

3. Recombined 4 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\sqrt{a \cdot c + \left(\left(b_2 \cdot b_2 - a \cdot c\right) - a \cdot c\right)} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.55 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.3
Cost	7820

\[\begin{array}{l} t_0 := \sqrt{b_2 \cdot b_2 - a \cdot c}\\ \mathbf{if}\;b_2 \leq -8 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 6.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{t_0 - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 7.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{a \cdot \left(-c\right)}{b_2 + t_0}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 2
Error	14.2
Cost	7440

\[\begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{if}\;b_2 \leq -2 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.18 \cdot 10^{-101}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 6.2 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 3
Error	10.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.7 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 4
Error	14.8
Cost	7312

\[\begin{array}{l} t_0 := \frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ t_1 := -2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -4 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -1.1 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 3.25 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 5
Error	53.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Error	40.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 1.95 \cdot 10^{-49}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7
Error	40.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 8
Error	39.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 2.05 \cdot 10^{-48}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 9
Error	22.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq 8.5 \cdot 10^{-216}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Alternative 10
Error	59.3
Cost	256

\[\frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2023083 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))