quadm (p42, negative)

Average Error: 34.7 → 10.8

Time: 10.2s

Precision: binary64

Cost: 14028

Math

\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

↓

\[\begin{array}{l} t_0 := -\frac{c}{b}\\ t_1 := \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{if}\;b \leq -0.0125:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-101}:\\ \;\;\;\;t_0 + \left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (/ c b)))
        (t_1 (/ (+ b (sqrt (+ (* b b) (* (* a c) -4.0)))) (* a -2.0))))
   (if (<= b -0.0125)
     t_0
     (if (<= b -6.4e-37)
       t_1
       (if (<= b -4e-101)
         (+ t_0 (- (/ (* a (pow c 2.0)) (pow b 3.0))))
         (if (<= b 4.9e+82) t_1 (- (/ c b) (/ b a))))))))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double t_0 = -(c / b);
	double t_1 = (b + sqrt(((b * b) + ((a * c) * -4.0)))) / (a * -2.0);
	double tmp;
	if (b <= -0.0125) {
		tmp = t_0;
	} else if (b <= -6.4e-37) {
		tmp = t_1;
	} else if (b <= -4e-101) {
		tmp = t_0 + -((a * pow(c, 2.0)) / pow(b, 3.0));
	} else if (b <= 4.9e+82) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -(c / b)
    t_1 = (b + sqrt(((b * b) + ((a * c) * (-4.0d0))))) / (a * (-2.0d0))
    if (b <= (-0.0125d0)) then
        tmp = t_0
    else if (b <= (-6.4d-37)) then
        tmp = t_1
    else if (b <= (-4d-101)) then
        tmp = t_0 + -((a * (c ** 2.0d0)) / (b ** 3.0d0))
    else if (b <= 4.9d+82) then
        tmp = t_1
    else
        tmp = (c / b) - (b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double t_0 = -(c / b);
	double t_1 = (b + Math.sqrt(((b * b) + ((a * c) * -4.0)))) / (a * -2.0);
	double tmp;
	if (b <= -0.0125) {
		tmp = t_0;
	} else if (b <= -6.4e-37) {
		tmp = t_1;
	} else if (b <= -4e-101) {
		tmp = t_0 + -((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0));
	} else if (b <= 4.9e+82) {
		tmp = t_1;
	} else {
		tmp = (c / b) - (b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

↓

def code(a, b, c):
	t_0 = -(c / b)
	t_1 = (b + math.sqrt(((b * b) + ((a * c) * -4.0)))) / (a * -2.0)
	tmp = 0
	if b <= -0.0125:
		tmp = t_0
	elif b <= -6.4e-37:
		tmp = t_1
	elif b <= -4e-101:
		tmp = t_0 + -((a * math.pow(c, 2.0)) / math.pow(b, 3.0))
	elif b <= 4.9e+82:
		tmp = t_1
	else:
		tmp = (c / b) - (b / a)
	return tmp

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	t_0 = Float64(-Float64(c / b))
	t_1 = Float64(Float64(b + sqrt(Float64(Float64(b * b) + Float64(Float64(a * c) * -4.0)))) / Float64(a * -2.0))
	tmp = 0.0
	if (b <= -0.0125)
		tmp = t_0;
	elseif (b <= -6.4e-37)
		tmp = t_1;
	elseif (b <= -4e-101)
		tmp = Float64(t_0 + Float64(-Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	elseif (b <= 4.9e+82)
		tmp = t_1;
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	t_0 = -(c / b);
	t_1 = (b + sqrt(((b * b) + ((a * c) * -4.0)))) / (a * -2.0);
	tmp = 0.0;
	if (b <= -0.0125)
		tmp = t_0;
	elseif (b <= -6.4e-37)
		tmp = t_1;
	elseif (b <= -4e-101)
		tmp = t_0 + -((a * (c ^ 2.0)) / (b ^ 3.0));
	elseif (b <= 4.9e+82)
		tmp = t_1;
	else
		tmp = (c / b) - (b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := Block[{t$95$0 = (-N[(c / b), $MachinePrecision])}, Block[{t$95$1 = N[(N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * -2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0125], t$95$0, If[LessEqual[b, -6.4e-37], t$95$1, If[LessEqual[b, -4e-101], N[(t$95$0 + (-N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[b, 4.9e+82], t$95$1, N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	34.7
Target	21.3
Herbie	10.8

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if b < -0.012500000000000001

   1. Initial program 56.3

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Simplified 56.3

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}} \]
      Proof

      [Start] 56.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best-simplify-65 [=>] 56.3	\[ \frac{\color{blue}{-\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)\right)}}{2 \cdot a} \]
      rational_best-simplify-54 [=>] 56.3	\[ \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)}{-2 \cdot a}} \]
      rational_best-simplify-63 [=>] 56.3	\[ \frac{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]
      rational_best-simplify-61 [=>] 56.3	\[ \frac{b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 56.3	\[ \frac{b + \sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}}}{-2 \cdot a} \]
      metadata-eval [=>] 56.3	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot \color{blue}{-4}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 56.3	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{\color{blue}{a \cdot \left(-2\right)}} \]
      metadata-eval [=>] 56.3	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot \color{blue}{-2}} \]

   3. Taylor expanded in b around -inf 5.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

   4. Simplified 5.7

      \[\leadsto \color{blue}{-\frac{c}{b}} \]
      Proof

      [Start] 5.7	\[ -1 \cdot \frac{c}{b} \]
      rational_best-simplify-3 [=>] 5.7	\[ \color{blue}{\frac{c}{b} \cdot -1} \]
      rational_best-simplify-17 [=>] 5.7	\[ \color{blue}{-\frac{c}{b}} \]

   if -0.012500000000000001 < b < -6.3999999999999998e-37 or -4.00000000000000021e-101 < b < 4.9000000000000001e82

   1. Initial program 14.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Simplified 14.4

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}} \]
      Proof

      [Start] 14.4	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best-simplify-65 [=>] 14.4	\[ \frac{\color{blue}{-\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)\right)}}{2 \cdot a} \]
      rational_best-simplify-54 [=>] 14.4	\[ \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)}{-2 \cdot a}} \]
      rational_best-simplify-63 [=>] 14.4	\[ \frac{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]
      rational_best-simplify-61 [=>] 14.4	\[ \frac{b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 14.4	\[ \frac{b + \sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}}}{-2 \cdot a} \]
      metadata-eval [=>] 14.4	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot \color{blue}{-4}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 14.4	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{\color{blue}{a \cdot \left(-2\right)}} \]
      metadata-eval [=>] 14.4	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot \color{blue}{-2}} \]

   if -6.3999999999999998e-37 < b < -4.00000000000000021e-101

   1. Initial program 35.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Simplified 35.8

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}} \]
      Proof

      [Start] 35.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best-simplify-65 [=>] 35.8	\[ \frac{\color{blue}{-\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)\right)}}{2 \cdot a} \]
      rational_best-simplify-54 [=>] 35.8	\[ \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)}{-2 \cdot a}} \]
      rational_best-simplify-63 [=>] 35.8	\[ \frac{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]
      rational_best-simplify-61 [=>] 35.8	\[ \frac{b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 35.8	\[ \frac{b + \sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}}}{-2 \cdot a} \]
      metadata-eval [=>] 35.8	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot \color{blue}{-4}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 35.8	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{\color{blue}{a \cdot \left(-2\right)}} \]
      metadata-eval [=>] 35.8	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot \color{blue}{-2}} \]

   3. Taylor expanded in b around -inf 32.8

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}} \]

   4. Simplified 32.8

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

      [Start] 32.8	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b} \]
      rational_best-simplify-1 [=>] 32.8	\[ \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]
      rational_best-simplify-3 [=>] 32.8	\[ \color{blue}{\frac{c}{b} \cdot -1} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      rational_best-simplify-17 [=>] 32.8	\[ \color{blue}{\left(-\frac{c}{b}\right)} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      rational_best-simplify-3 [=>] 32.8	\[ \left(-\frac{c}{b}\right) + \color{blue}{\frac{{c}^{2} \cdot a}{{b}^{3}} \cdot -1} \]
      rational_best-simplify-17 [=>] 32.8	\[ \left(-\frac{c}{b}\right) + \color{blue}{\left(-\frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      rational_best-simplify-3 [=>] 32.8	\[ \left(-\frac{c}{b}\right) + \left(-\frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}\right) \]

   if 4.9000000000000001e82 < b

   1. Initial program 44.3

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Simplified 44.3

      \[\leadsto \color{blue}{\frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}} \]
      Proof

      [Start] 44.3	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      rational_best-simplify-65 [=>] 44.3	\[ \frac{\color{blue}{-\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)\right)}}{2 \cdot a} \]
      rational_best-simplify-54 [=>] 44.3	\[ \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(-b\right)}{-2 \cdot a}} \]
      rational_best-simplify-63 [=>] 44.3	\[ \frac{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]
      rational_best-simplify-61 [=>] 44.3	\[ \frac{b + \sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 44.3	\[ \frac{b + \sqrt{b \cdot b + \color{blue}{\left(a \cdot c\right) \cdot \left(-4\right)}}}{-2 \cdot a} \]
      metadata-eval [=>] 44.3	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot \color{blue}{-4}}}{-2 \cdot a} \]
      rational_best-simplify-52 [=>] 44.3	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{\color{blue}{a \cdot \left(-2\right)}} \]
      metadata-eval [=>] 44.3	\[ \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot \color{blue}{-2}} \]

   3. Taylor expanded in b around inf 5.0

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

   4. Simplified 5.0

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

      [Start] 5.0	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational_best-simplify-1 [=>] 5.0	\[ \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
      rational_best-simplify-3 [=>] 5.0	\[ \color{blue}{\frac{b}{a} \cdot -1} + \frac{c}{b} \]
      rational_best-simplify-17 [=>] 5.0	\[ \color{blue}{\left(-\frac{b}{a}\right)} + \frac{c}{b} \]

   5. Taylor expanded in b around 0 5.0

      \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

   6. Simplified 5.0

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Proof

      [Start] 5.0	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      rational_best-simplify-3 [=>] 5.0	\[ \frac{c}{b} + \color{blue}{\frac{b}{a} \cdot -1} \]
      rational_best-simplify-18 [<=] 5.0	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      rational_best-simplify-61 [<=] 5.0	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0125:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -6.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-101}:\\ \;\;\;\;\left(-\frac{c}{b}\right) + \left(-\frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{+82}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.0
Cost	7888

\[\begin{array}{l} t_0 := -\frac{c}{b}\\ t_1 := \frac{b + \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a \cdot -2}\\ \mathbf{if}\;b \leq -0.011:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -3.5 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -2.4 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Error	13.7
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-101}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.72 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right)} + b}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Error	13.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-101}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.72 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(a \cdot c\right)}}{a \cdot -2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Error	20.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-156}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-126}:\\ \;\;\;\;-0.5 \cdot \sqrt{\frac{c}{a} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Error	40.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 6
Error	23.0
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-243}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array} \]

Alternative 7
Error	56.5
Cost	192

\[\frac{c}{b} \]

Reproduce

herbie shell --seed 2023104 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))