quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.7% → 86.5%

Time: 16.2s

Precision: binary64

Cost: 14932

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(-a\right)\\ t_1 := \sqrt{t_0}\\ t_2 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -53000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b_2 \leq -2.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{t_0}{b_2 - \mathsf{hypot}\left(b_2, t_1\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -3 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(-b_2\right) - \mathsf{hypot}\left(t_1, b_2\right)}{a}\\ \mathbf{elif}\;b_2 \leq 4.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + -0.5 \cdot \left(b_2 \cdot \left(2 \cdot \frac{c - c}{{b_2}^{2}} - \frac{c}{{b_2}^{2}}\right)\right)\\ \end{array} \]

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
 (let* ((t_0 (* c (- a))) (t_1 (sqrt t_0)) (t_2 (/ (* -0.5 c) b_2)))
   (if (<= b_2 -53000.0)
     t_2
     (if (<= b_2 -2.2e-67)
       (/ (/ t_0 (- b_2 (hypot b_2 t_1))) a)
       (if (<= b_2 -3e-81)
         t_2
         (if (<= b_2 5e-152)
           (/ (- (- b_2) (hypot t_1 b_2)) a)
           (if (<= b_2 4.5e+142)
             (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
             (+
              (* -2.0 (/ b_2 a))
              (*
               -0.5
               (*
                b_2
                (-
                 (* 2.0 (/ (- c c) (pow b_2 2.0)))
                 (/ c (pow b_2 2.0)))))))))))))

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 = c * -a;
	double t_1 = sqrt(t_0);
	double t_2 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -53000.0) {
		tmp = t_2;
	} else if (b_2 <= -2.2e-67) {
		tmp = (t_0 / (b_2 - hypot(b_2, t_1))) / a;
	} else if (b_2 <= -3e-81) {
		tmp = t_2;
	} else if (b_2 <= 5e-152) {
		tmp = (-b_2 - hypot(t_1, b_2)) / a;
	} else if (b_2 <= 4.5e+142) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (-0.5 * (b_2 * ((2.0 * ((c - c) / pow(b_2, 2.0))) - (c / pow(b_2, 2.0)))));
	}
	return tmp;
}

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 = c * -a;
	double t_1 = Math.sqrt(t_0);
	double t_2 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -53000.0) {
		tmp = t_2;
	} else if (b_2 <= -2.2e-67) {
		tmp = (t_0 / (b_2 - Math.hypot(b_2, t_1))) / a;
	} else if (b_2 <= -3e-81) {
		tmp = t_2;
	} else if (b_2 <= 5e-152) {
		tmp = (-b_2 - Math.hypot(t_1, b_2)) / a;
	} else if (b_2 <= 4.5e+142) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (-0.5 * (b_2 * ((2.0 * ((c - c) / Math.pow(b_2, 2.0))) - (c / Math.pow(b_2, 2.0)))));
	}
	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 = c * -a
	t_1 = math.sqrt(t_0)
	t_2 = (-0.5 * c) / b_2
	tmp = 0
	if b_2 <= -53000.0:
		tmp = t_2
	elif b_2 <= -2.2e-67:
		tmp = (t_0 / (b_2 - math.hypot(b_2, t_1))) / a
	elif b_2 <= -3e-81:
		tmp = t_2
	elif b_2 <= 5e-152:
		tmp = (-b_2 - math.hypot(t_1, b_2)) / a
	elif b_2 <= 4.5e+142:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (-0.5 * (b_2 * ((2.0 * ((c - c) / math.pow(b_2, 2.0))) - (c / math.pow(b_2, 2.0)))))
	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(c * Float64(-a))
	t_1 = sqrt(t_0)
	t_2 = Float64(Float64(-0.5 * c) / b_2)
	tmp = 0.0
	if (b_2 <= -53000.0)
		tmp = t_2;
	elseif (b_2 <= -2.2e-67)
		tmp = Float64(Float64(t_0 / Float64(b_2 - hypot(b_2, t_1))) / a);
	elseif (b_2 <= -3e-81)
		tmp = t_2;
	elseif (b_2 <= 5e-152)
		tmp = Float64(Float64(Float64(-b_2) - hypot(t_1, b_2)) / a);
	elseif (b_2 <= 4.5e+142)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(-0.5 * Float64(b_2 * Float64(Float64(2.0 * Float64(Float64(c - c) / (b_2 ^ 2.0))) - Float64(c / (b_2 ^ 2.0))))));
	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 = c * -a;
	t_1 = sqrt(t_0);
	t_2 = (-0.5 * c) / b_2;
	tmp = 0.0;
	if (b_2 <= -53000.0)
		tmp = t_2;
	elseif (b_2 <= -2.2e-67)
		tmp = (t_0 / (b_2 - hypot(b_2, t_1))) / a;
	elseif (b_2 <= -3e-81)
		tmp = t_2;
	elseif (b_2 <= 5e-152)
		tmp = (-b_2 - hypot(t_1, b_2)) / a;
	elseif (b_2 <= 4.5e+142)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (-0.5 * (b_2 * ((2.0 * ((c - c) / (b_2 ^ 2.0))) - (c / (b_2 ^ 2.0)))));
	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[(c * (-a)), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]}, If[LessEqual[b$95$2, -53000.0], t$95$2, If[LessEqual[b$95$2, -2.2e-67], N[(N[(t$95$0 / N[(b$95$2 - N[Sqrt[b$95$2 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -3e-81], t$95$2, If[LessEqual[b$95$2, 5e-152], N[(N[((-b$95$2) - N[Sqrt[t$95$1 ^ 2 + b$95$2 ^ 2], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 4.5e+142], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(b$95$2 * N[(N[(2.0 * N[(N[(c - c), $MachinePrecision] / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c / N[Power[b$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -53000 or -2.2000000000000001e-67 < b_2 < -2.9999999999999999e-81

   1. Initial program 18.2%

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

   2. Taylor expanded in b_2 around -inf 94.2%

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

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]
      Step-by-step derivation

      [Start] 94.2	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 94.2	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -53000 < b_2 < -2.2000000000000001e-67

   1. Initial program 59.6%

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

   2. Applied egg-rr 60.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
      Step-by-step derivation

      [Start] 59.6	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-sqr-sqrt [=>] 60.4	\[ \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      pow2 [=>] 60.4	\[ \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]
      pow1/2 [=>] 60.4	\[ \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]
      sqrt-pow1 [=>] 60.6	\[ \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]
      metadata-eval [=>] 60.6	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 59.1%

      \[\leadsto \frac{\color{blue}{\frac{-\left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}}{a} \]
      Step-by-step derivation

      [Start] 60.6	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{a} \]
      flip-- [=>] 60.6	\[ \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}}{a} \]
      frac-2neg [=>] 60.6	\[ \frac{\color{blue}{\frac{-\left(\left(-b_2\right) \cdot \left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}\right)}{-\left(\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}\right)}}}{a} \]

   4. Simplified 83.0%

      \[\leadsto \frac{\color{blue}{\frac{c \cdot \left(-a\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}}{a} \]
      Step-by-step derivation

      [Start] 59.1	\[ \frac{\frac{-\left(b_2 \cdot b_2 - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      unpow2 [<=] 59.1	\[ \frac{\frac{-\left(\color{blue}{{b_2}^{2}} - \left(b_2 \cdot b_2 - a \cdot c\right)\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      unpow2 [<=] 59.1	\[ \frac{\frac{-\left({b_2}^{2} - \left(\color{blue}{{b_2}^{2}} - a \cdot c\right)\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      associate--r- [=>] 83.0	\[ \frac{\frac{-\color{blue}{\left(\left({b_2}^{2} - {b_2}^{2}\right) + a \cdot c\right)}}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      +-inverses [=>] 83.0	\[ \frac{\frac{-\left(\color{blue}{0} + a \cdot c\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      *-commutative [<=] 83.0	\[ \frac{\frac{-\left(0 + \color{blue}{c \cdot a}\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      distribute-neg-in [=>] 83.0	\[ \frac{\frac{\color{blue}{\left(-0\right) + \left(-c \cdot a\right)}}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      metadata-eval [=>] 83.0	\[ \frac{\frac{\color{blue}{0} + \left(-c \cdot a\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      sub-neg [<=] 83.0	\[ \frac{\frac{\color{blue}{0 - c \cdot a}}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      neg-sub0 [<=] 83.0	\[ \frac{\frac{\color{blue}{-c \cdot a}}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      distribute-rgt-neg-in [=>] 83.0	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-a \cdot c}\right)}}{a} \]
      *-commutative [<=] 83.0	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right)}}{a} \]
      distribute-rgt-neg-in [=>] 83.0	\[ \frac{\frac{c \cdot \left(-a\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right)}}{a} \]

   if -2.9999999999999999e-81 < b_2 < 4.9999999999999997e-152

   1. Initial program 77.4%

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

   2. Applied egg-rr 85.1%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-a\right)}, b_2\right)}}{a} \]
      Step-by-step derivation

      [Start] 77.4	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      sub-neg [=>] 77.4	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{b_2 \cdot b_2 + \left(-a \cdot c\right)}}}{a} \]
      +-commutative [=>] 77.4	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a \cdot c\right) + b_2 \cdot b_2}}}{a} \]
      add-sqr-sqrt [=>] 77.4	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}} + b_2 \cdot b_2}}{a} \]
      hypot-def [=>] 85.1	\[ \frac{\left(-b_2\right) - \color{blue}{\mathsf{hypot}\left(\sqrt{-a \cdot c}, b_2\right)}}{a} \]
      *-commutative [=>] 85.1	\[ \frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{-\color{blue}{c \cdot a}}, b_2\right)}{a} \]
      distribute-rgt-neg-in [=>] 85.1	\[ \frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{\color{blue}{c \cdot \left(-a\right)}}, b_2\right)}{a} \]

   if 4.9999999999999997e-152 < b_2 < 4.4999999999999999e142

   1. Initial program 88.6%

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

   if 4.4999999999999999e142 < b_2

   1. Initial program 38.1%

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

   2. Applied egg-rr 38.0%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
      Step-by-step derivation

      [Start] 38.1	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-sqr-sqrt [=>] 38.0	\[ \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      pow2 [=>] 38.0	\[ \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]
      pow1/2 [=>] 38.0	\[ \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]
      sqrt-pow1 [=>] 38.0	\[ \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]
      metadata-eval [=>] 38.0	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 37.4%

      \[\leadsto \frac{\left(-b_2\right) - {\left({\color{blue}{\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)\right)}}^{0.25}\right)}^{2}}{a} \]
      Step-by-step derivation

      [Start] 38.0	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{a} \]
      prod-diff [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}^{0.25}\right)}^{2}}{a} \]
      *-commutative [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}^{0.25}\right)}^{2}}{a} \]
      fma-neg [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}^{0.25}\right)}^{2}}{a} \]
      prod-diff [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}^{0.25}\right)}^{2}}{a} \]
      *-commutative [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}^{0.25}\right)}^{2}}{a} \]
      fma-neg [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}^{0.25}\right)}^{2}}{a} \]
      associate-+l+ [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\color{blue}{\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)\right)}}^{0.25}\right)}^{2}}{a} \]
      *-commutative [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      fma-udef [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      distribute-lft-neg-in [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      *-commutative [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      distribute-rgt-neg-in [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      fma-def [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      *-commutative [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      fma-udef [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)\right)}^{0.25}\right)}^{2}}{a} \]
      distribute-lft-neg-in [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      *-commutative [<=] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      distribute-rgt-neg-in [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      fma-def [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)\right)}^{0.25}\right)}^{2}}{a} \]

   4. Simplified 37.4%

      \[\leadsto \frac{\left(-b_2\right) - {\left({\color{blue}{\left(\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)\right)}}^{0.25}\right)}^{2}}{a} \]
      Step-by-step derivation

      [Start] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      *-commutative [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)\right)}^{0.25}\right)}^{2}}{a} \]
      count-2 [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}\right)}^{0.25}\right)}^{2}}{a} \]
      *-commutative [=>] 37.4	\[ \frac{\left(-b_2\right) - {\left({\left(\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)\right)}^{0.25}\right)}^{2}}{a} \]

   5. Taylor expanded in a around 0 90.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + -0.5 \cdot \left(b_2 \cdot \left(2 \cdot \frac{c + -1 \cdot c}{{b_2}^{2}} - \frac{c}{{b_2}^{2}}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -53000:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-a\right)}{b_2 - \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -3 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{c \cdot \left(-a\right)}, b_2\right)}{a}\\ \mathbf{elif}\;b_2 \leq 4.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + -0.5 \cdot \left(b_2 \cdot \left(2 \cdot \frac{c - c}{{b_2}^{2}} - \frac{c}{{b_2}^{2}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.4%
Cost	13968

\[\begin{array}{l} t_0 := c \cdot \left(-a\right)\\ t_1 := \sqrt{t_0}\\ t_2 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -235000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b_2 \leq -9.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{t_0}{b_2 - \mathsf{hypot}\left(b_2, t_1\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-81}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(-b_2\right) - \mathsf{hypot}\left(t_1, b_2\right)}{a}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 2
Accuracy	86.9%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.65 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\left(-b_2\right) - \mathsf{hypot}\left(\sqrt{c \cdot \left(-a\right)}, b_2\right)}{a}\\ \mathbf{elif}\;b_2 \leq 5.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 3
Accuracy	86.4%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 4
Accuracy	81.2%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.9 \cdot 10^{-81}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.45 \cdot 10^{-46}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Accuracy	68.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Accuracy	68.7%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{0.5}{\frac{b_2}{c}}\\ \end{array} \]

Alternative 7
Accuracy	43.9%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.55 \cdot 10^{-221}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 8
Accuracy	68.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5}{\frac{b_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 9
Accuracy	68.5%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 10
Accuracy	23.4%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -3.05 \cdot 10^{-221}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b_2}{a}\\ \end{array} \]

Alternative 11
Accuracy	10.9%
Cost	192

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

Reproduce

herbie shell --seed 2023161 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))