quad2m (problem 3.2.1, negative)

Percentage Accurate: 57.4% → 88.2%

Time: 30.8s

Precision: binary64

Cost: 20808

• could not determine a ground truth

  1. a = 3.5075299573203835e+146
  2. b_2 = -1.2188738977806838e-214
  3. c = -1.1298104663403366e-125

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.75 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + c \cdot a}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -1.35 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 0.0002:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

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 (/ (* -0.5 c) b_2)))
   (if (<= b_2 -1.85e+28)
     t_0
     (if (<= b_2 -1.75e-37)
       (/
        (/
         (+ (- (* b_2 b_2) (* b_2 b_2)) (* c a))
         (fma -1.0 b_2 (hypot b_2 (sqrt (* c (- a))))))
        a)
       (if (<= b_2 -1.35e-119)
         t_0
         (if (<= b_2 0.0002)
           (/
            (-
             (- b_2)
             (sqrt (+ (- (* b_2 b_2) (* c a)) (* 2.0 (fma a (- c) (* c a))))))
            a)
           (* -2.0 (/ b_2 a))))))))

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 = (-0.5 * c) / b_2;
	double tmp;
	if (b_2 <= -1.85e+28) {
		tmp = t_0;
	} else if (b_2 <= -1.75e-37) {
		tmp = ((((b_2 * b_2) - (b_2 * b_2)) + (c * a)) / fma(-1.0, b_2, hypot(b_2, sqrt((c * -a))))) / a;
	} else if (b_2 <= -1.35e-119) {
		tmp = t_0;
	} else if (b_2 <= 0.0002) {
		tmp = (-b_2 - sqrt((((b_2 * b_2) - (c * a)) + (2.0 * fma(a, -c, (c * a)))))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	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(-0.5 * c) / b_2)
	tmp = 0.0
	if (b_2 <= -1.85e+28)
		tmp = t_0;
	elseif (b_2 <= -1.75e-37)
		tmp = Float64(Float64(Float64(Float64(Float64(b_2 * b_2) - Float64(b_2 * b_2)) + Float64(c * a)) / fma(-1.0, b_2, hypot(b_2, sqrt(Float64(c * Float64(-a)))))) / a);
	elseif (b_2 <= -1.35e-119)
		tmp = t_0;
	elseif (b_2 <= 0.0002)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(Float64(b_2 * b_2) - Float64(c * a)) + Float64(2.0 * fma(a, Float64(-c), Float64(c * a)))))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return 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[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]}, If[LessEqual[b$95$2, -1.85e+28], t$95$0, If[LessEqual[b$95$2, -1.75e-37], N[(N[(N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * b$95$2 + N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, -1.35e-119], t$95$0, If[LessEqual[b$95$2, 0.0002], N[(N[((-b$95$2) - N[Sqrt[N[(N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(a * (-c) + N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.85e28 or -1.7500000000000001e-37 < b_2 < -1.35000000000000013e-119

   1. Initial program 20.5%

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

   2. Taylor expanded in b_2 around -inf 89.9%

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

   3. Simplified 89.9%

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

      [Start] 89.9%	\[ -0.5 \cdot \frac{c}{b_2} \]
      associate-*r/ [=>] 89.9%	\[ \color{blue}{\frac{-0.5 \cdot c}{b_2}} \]

   if -1.85e28 < b_2 < -1.7500000000000001e-37

   1. Initial program 73.9%

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

   2. Applied egg-rr 51.9%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}}{a} \]
      Step-by-step derivation

      [Start] 73.9%	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      add-cbrt-cube [=>] 55.9%	\[ \frac{\left(-b_2\right) - \color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]
      pow3 [=>] 55.3%	\[ \frac{\left(-b_2\right) - \sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}}}{a} \]
      pow1/3 [=>] 51.9%	\[ \frac{\left(-b_2\right) - \color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}}}{a} \]
      sqrt-pow2 [=>] 51.9%	\[ \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333}}{a} \]
      metadata-eval [=>] 51.9%	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}}{a} \]

   3. Simplified 55.6%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      Step-by-step derivation

      [Start] 51.9%	\[ \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{1.5}\right)}^{0.3333333333333333}}{a} \]
      unpow1/3 [=>] 55.6%	\[ \frac{\left(-b_2\right) - \color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{1.5}}}}{a} \]
      *-commutative [=>] 55.6%	\[ \frac{\left(-b_2\right) - \sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}}}{a} \]

   4. Applied egg-rr 73.5%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}}{a} \]
      Step-by-step derivation

      [Start] 55.6%	\[ \frac{\left(-b_2\right) - \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}{a} \]
      flip-- [=>] 54.7%	\[ \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} \cdot \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}}{a} \]
      pow2 [=>] 54.7%	\[ \frac{\frac{\color{blue}{{\left(-b_2\right)}^{2}} - \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} \cdot \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      pow1/3 [=>] 51.2%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - \color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      pow1/3 [=>] 50.3%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - {\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333} \cdot \color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      pow-prod-down [=>] 42.5%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - \color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      pow-prod-up [=>] 42.5%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - {\color{blue}{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 + 1.5\right)}\right)}}^{0.3333333333333333}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      metadata-eval [=>] 42.5%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - {\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{3}}\right)}^{0.3333333333333333}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      pow3 [<=] 42.5%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - {\color{blue}{\left(\left(\left(b_2 \cdot b_2 - c \cdot a\right) \cdot \left(b_2 \cdot b_2 - c \cdot a\right)\right) \cdot \left(b_2 \cdot b_2 - c \cdot a\right)\right)}}^{0.3333333333333333}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      pow1/3 [<=] 45.6%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - \color{blue}{\sqrt[3]{\left(\left(b_2 \cdot b_2 - c \cdot a\right) \cdot \left(b_2 \cdot b_2 - c \cdot a\right)\right) \cdot \left(b_2 \cdot b_2 - c \cdot a\right)}}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      add-cbrt-cube [<=] 55.9%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - \color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)}}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]
      *-commutative [=>] 55.9%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - \color{blue}{a \cdot c}\right)}{\left(-b_2\right) + \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}}}{a} \]

   5. Simplified 91.1%

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

      [Start] 73.5%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a} \]
      unpow2 [<=] 73.5%	\[ \frac{\frac{{\left(-b_2\right)}^{2} - \left(\color{blue}{{b_2}^{2}} - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a} \]
      associate--r- [=>] 91.1%	\[ \frac{\frac{\color{blue}{\left({\left(-b_2\right)}^{2} - {b_2}^{2}\right) + a \cdot c}}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a} \]
      unpow2 [=>] 91.1%	\[ \frac{\frac{\left(\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)} - {b_2}^{2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a} \]
      sqr-neg [=>] 91.1%	\[ \frac{\frac{\left(\color{blue}{b_2 \cdot b_2} - {b_2}^{2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a} \]
      unpow2 [=>] 91.1%	\[ \frac{\frac{\left(b_2 \cdot b_2 - \color{blue}{b_2 \cdot b_2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a} \]
      distribute-rgt-neg-out [=>] 91.1%	\[ \frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{\color{blue}{-c \cdot a}}\right)\right)}}{a} \]
      *-commutative [<=] 91.1%	\[ \frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{a \cdot c}}\right)\right)}}{a} \]
      distribute-rgt-neg-out [<=] 91.1%	\[ \frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{\color{blue}{a \cdot \left(-c\right)}}\right)\right)}}{a} \]

   if -1.35000000000000013e-119 < b_2 < 2.0000000000000001e-4

   1. Initial program 81.0%

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

   2. Applied egg-rr 81.0%

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

      [Start] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
      prod-diff [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}}{a} \]
      *-commutative [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]
      fma-neg [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]
      prod-diff [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]
      *-commutative [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]
      fma-neg [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}}{a} \]
      associate-+l+ [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}}{a} \]
      *-commutative [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]
      fma-udef [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]
      distribute-lft-neg-in [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]
      *-commutative [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]
      distribute-rgt-neg-in [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]
      fma-def [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}}{a} \]
      *-commutative [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)}}{a} \]
      fma-udef [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\left(\left(-c\right) \cdot a + a \cdot c\right)}\right)}}{a} \]
      distribute-lft-neg-in [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)\right)}}{a} \]
      *-commutative [<=] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)\right)}}{a} \]
      distribute-rgt-neg-in [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \left(\color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)\right)}}{a} \]
      fma-def [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)}}{a} \]

   3. Simplified 81.0%

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

      [Start] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - a \cdot c\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]
      *-commutative [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right) + \left(\mathsf{fma}\left(a, -c, a \cdot c\right) + \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}}{a} \]
      count-2 [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + \color{blue}{2 \cdot \mathsf{fma}\left(a, -c, a \cdot c\right)}}}{a} \]
      *-commutative [=>] 81.0%	\[ \frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, \color{blue}{c \cdot a}\right)}}{a} \]

   if 2.0000000000000001e-4 < b_2

   1. Initial program 86.6%

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

   2. Taylor expanded in b_2 around inf 100.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.75 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + c \cdot a}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -1.35 \cdot 10^{-119}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 0.0002:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	88.2%
Cost	20808

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.75 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{\left(b_2 \cdot b_2 - b_2 \cdot b_2\right) + c \cdot a}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}{a}\\ \mathbf{elif}\;b_2 \leq -1.35 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 0.0002:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\left(b_2 \cdot b_2 - c \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 2
Accuracy	86.6%
Cost	14672

\[\begin{array}{l} t_0 := b_2 \cdot b_2 - c \cdot a\\ t_1 := \frac{-0.5 \cdot c}{b_2}\\ \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -1.82 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{t_0}}{a}\\ \mathbf{elif}\;b_2 \leq -2.3 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq 0.0002:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{t_0 + 2 \cdot \mathsf{fma}\left(a, -c, c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 3
Accuracy	86.5%
Cost	7696

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot c}{b_2}\\ t_1 := \frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -1.82 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b_2 \leq -2.25 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 0.0002:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 4
Accuracy	85.9%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{-120}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.5 \cdot 10^{-179}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 5
Accuracy	85.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{-\sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 6
Accuracy	77.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Alternative 7
Accuracy	47.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 8
Accuracy	76.8%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -8 \cdot 10^{-304}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 9
Accuracy	28.6%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a}\\ \end{array} \]

Alternative 10
Accuracy	10.0%
Cost	192

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

Reproduce

herbie shell --seed 2023277 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))