The quadratic formula (r2)

Average Accuracy: 48.0% → 87.1%

Time: 18.2s

Precision: binary64

Cost: 13960

Math

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

↓

\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \mathbf{if}\;b \leq -1 \cdot 10^{-15}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, \frac{-0.5 \cdot c}{\frac{b}{-4 \cdot a}}\right)}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-221}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{t_0}, b\right) - b}\right)\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{t_0 + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

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 (* -4.0 (* c a))))
   (if (<= b -1e-15)
     (* -0.5 (* -4.0 (/ c (fma b -2.0 (/ (* -0.5 c) (/ b (* -4.0 a)))))))
     (if (<= b 1.25e-221)
       (* -0.5 (* -4.0 (/ c (- (hypot (sqrt t_0) b) b))))
       (if (<= b 1.12e+135)
         (* -0.5 (/ (+ b (sqrt (+ t_0 (* b b)))) a))
         (- (/ 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 = -4.0 * (c * a);
	double tmp;
	if (b <= -1e-15) {
		tmp = -0.5 * (-4.0 * (c / fma(b, -2.0, ((-0.5 * c) / (b / (-4.0 * a))))));
	} else if (b <= 1.25e-221) {
		tmp = -0.5 * (-4.0 * (c / (hypot(sqrt(t_0), b) - b)));
	} else if (b <= 1.12e+135) {
		tmp = -0.5 * ((b + sqrt((t_0 + (b * b)))) / a);
	} 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(-4.0 * Float64(c * a))
	tmp = 0.0
	if (b <= -1e-15)
		tmp = Float64(-0.5 * Float64(-4.0 * Float64(c / fma(b, -2.0, Float64(Float64(-0.5 * c) / Float64(b / Float64(-4.0 * a)))))));
	elseif (b <= 1.25e-221)
		tmp = Float64(-0.5 * Float64(-4.0 * Float64(c / Float64(hypot(sqrt(t_0), b) - b))));
	elseif (b <= 1.12e+135)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(t_0 + Float64(b * b)))) / a));
	else
		tmp = Float64(Float64(c / b) - Float64(b / a));
	end
	return 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[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1e-15], N[(-0.5 * N[(-4.0 * N[(c / N[(b * -2.0 + N[(N[(-0.5 * c), $MachinePrecision] / N[(b / N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.25e-221], N[(-0.5 * N[(-4.0 * N[(c / N[(N[Sqrt[N[Sqrt[t$95$0], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.12e+135], N[(-0.5 * N[(N[(b + N[Sqrt[N[(t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	48.0%
Target	68.0%
Herbie	87.1%

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

Derivation

1. Split input into 4 regimes

2. if b < -1.0000000000000001e-15

   1. Initial program 15.1%

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

   2. Simplified 15.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]
      Proof

      [Start] 15.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 15.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 15.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 15.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 15.1	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 15.1	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 15.1	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 15.1	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 15.1	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 15.1	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 15.1	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 15.1	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 15.1	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 15.1	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 15.1	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 15.1	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 13.8	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 11.7%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      Proof

      [Start] 15.1	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      flip-+ [=>] 15.1	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      add-sqr-sqrt [<=] 15.1	\[ -0.5 \cdot \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      div-sub [=>] 15.1	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      fma-udef [=>] 15.1	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      +-commutative [=>] 15.1	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      add-sqr-sqrt [=>] 11.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      hypot-def [=>] 11.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      associate-*r* [=>] 11.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]

   4. Simplified 47.3%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}}{a} \]
      Proof

      [Start] 11.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      div-sub [<=] 11.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      *-lft-identity [<=] 11.7	\[ -0.5 \cdot \frac{\color{blue}{1 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      metadata-eval [<=] 11.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      times-frac [<=] 11.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 11.7	\[ -0.5 \cdot \frac{\frac{\color{blue}{-\left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}{a} \]

   5. Applied egg-rr 41.1%

      \[\leadsto -0.5 \cdot \color{blue}{\left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right)} \]
      Proof

      [Start] 47.3	\[ -0.5 \cdot \frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a} \]
      add-log-exp [=>] 19.7	\[ -0.5 \cdot \color{blue}{\log \left(e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)} \]
      *-un-lft-identity [=>] 19.7	\[ -0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)} \]
      log-prod [=>] 19.7	\[ -0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)\right)} \]
      metadata-eval [=>] 19.7	\[ -0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)\right) \]
      add-log-exp [<=] 47.3	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right) \]
      associate-/l/ [=>] 45.9	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c \cdot \left(-4 \cdot a\right)}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b\right)}}\right) \]
      times-frac [=>] 41.1	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}\right) \]
      hypot-udef [=>] 35.4	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\color{blue}{\sqrt{b \cdot b + \sqrt{c \cdot \left(-4 \cdot a\right)} \cdot \sqrt{c \cdot \left(-4 \cdot a\right)}}} - b}\right) \]
      add-sqr-sqrt [<=] 54.4	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\sqrt{b \cdot b + \color{blue}{c \cdot \left(-4 \cdot a\right)}} - b}\right) \]
      +-commutative [=>] 54.4	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right) + b \cdot b}} - b}\right) \]
      add-sqr-sqrt [=>] 35.4	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\sqrt{\color{blue}{\sqrt{c \cdot \left(-4 \cdot a\right)} \cdot \sqrt{c \cdot \left(-4 \cdot a\right)}} + b \cdot b} - b}\right) \]
      hypot-def [=>] 41.1	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\color{blue}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right)} - b}\right) \]

   6. Simplified 55.4%

      \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{-4 \cdot \left(c \cdot a\right)}, b\right) - b}\right)} \]
      Proof

      [Start] 41.1	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      +-lft-identity [=>] 41.1	\[ -0.5 \cdot \color{blue}{\left(\frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right)} \]
      associate-*r/ [=>] 45.8	\[ -0.5 \cdot \color{blue}{\frac{\frac{c}{a} \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}} \]
      associate-*l/ [=>] 51.1	\[ -0.5 \cdot \frac{\color{blue}{\frac{c \cdot \left(-4 \cdot a\right)}{a}}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b} \]
      *-commutative [=>] 51.1	\[ -0.5 \cdot \frac{\frac{\color{blue}{\left(-4 \cdot a\right) \cdot c}}{a}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b} \]
      associate-*l/ [<=] 55.3	\[ -0.5 \cdot \frac{\color{blue}{\frac{-4 \cdot a}{a} \cdot c}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b} \]
      associate-*r/ [<=] 55.3	\[ -0.5 \cdot \color{blue}{\left(\frac{-4 \cdot a}{a} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right)} \]
      associate-/l* [=>] 55.4	\[ -0.5 \cdot \left(\color{blue}{\frac{-4}{\frac{a}{a}}} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      *-inverses [=>] 55.4	\[ -0.5 \cdot \left(\frac{-4}{\color{blue}{1}} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      metadata-eval [=>] 55.4	\[ -0.5 \cdot \left(\color{blue}{-4} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      *-commutative [=>] 55.4	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}, b\right) - b}\right) \]
      associate-*l* [=>] 55.4	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}, b\right) - b}\right) \]
      *-commutative [<=] 55.4	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)}}, b\right) - b}\right) \]

   7. Taylor expanded in b around -inf 0.0%

      \[\leadsto -0.5 \cdot \left(-4 \cdot \frac{c}{\color{blue}{-2 \cdot b + -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}}}\right) \]

   8. Simplified 89.7%

      \[\leadsto -0.5 \cdot \left(-4 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(b, -2, \frac{c \cdot -0.5}{\frac{b}{a \cdot -4}}\right)}}\right) \]
      Proof

      [Start] 0.0	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{-2 \cdot b + -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}}\right) \]
      *-commutative [=>] 0.0	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\color{blue}{b \cdot -2} + -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}}\right) \]
      fma-def [=>] 0.0	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(b, -2, -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}\right)}}\right) \]
      associate-/l* [=>] 0.0	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, -0.5 \cdot \color{blue}{\frac{c}{\frac{b}{a \cdot {\left(\sqrt{-4}\right)}^{2}}}}\right)}\right) \]
      associate-*r/ [=>] 0.0	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, \color{blue}{\frac{-0.5 \cdot c}{\frac{b}{a \cdot {\left(\sqrt{-4}\right)}^{2}}}}\right)}\right) \]
      *-commutative [=>] 0.0	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, \frac{\color{blue}{c \cdot -0.5}}{\frac{b}{a \cdot {\left(\sqrt{-4}\right)}^{2}}}\right)}\right) \]
      unpow2 [=>] 0.0	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, \frac{c \cdot -0.5}{\frac{b}{a \cdot \color{blue}{\left(\sqrt{-4} \cdot \sqrt{-4}\right)}}}\right)}\right) \]
      rem-square-sqrt [=>] 89.7	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, \frac{c \cdot -0.5}{\frac{b}{a \cdot \color{blue}{-4}}}\right)}\right) \]

   if -1.0000000000000001e-15 < b < 1.24999999999999999e-221

   1. Initial program 64.5%

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

   2. Simplified 64.6%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]
      Proof

      [Start] 64.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 64.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 64.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 64.5	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 64.5	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 64.5	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 64.5	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 64.5	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 64.5	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 64.5	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 64.5	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 64.5	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 64.6	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 64.6	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 64.6	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 64.6	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 64.6	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 63.7%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      Proof

      [Start] 64.6	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      flip-+ [=>] 64.4	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      add-sqr-sqrt [<=] 64.3	\[ -0.5 \cdot \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      div-sub [=>] 64.3	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      fma-udef [=>] 64.3	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      +-commutative [=>] 64.3	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      add-sqr-sqrt [=>] 63.8	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      hypot-def [=>] 63.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      associate-*r* [=>] 63.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]

   4. Simplified 68.0%

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}}{a} \]
      Proof

      [Start] 63.7	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      div-sub [<=] 63.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      *-lft-identity [<=] 63.7	\[ -0.5 \cdot \frac{\color{blue}{1 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      metadata-eval [<=] 63.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      times-frac [<=] 63.7	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 63.7	\[ -0.5 \cdot \frac{\frac{\color{blue}{-\left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}{a} \]

   5. Applied egg-rr 56.5%

      \[\leadsto -0.5 \cdot \color{blue}{\left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right)} \]
      Proof

      [Start] 68.0	\[ -0.5 \cdot \frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a} \]
      add-log-exp [=>] 3.6	\[ -0.5 \cdot \color{blue}{\log \left(e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)} \]
      *-un-lft-identity [=>] 3.6	\[ -0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)} \]
      log-prod [=>] 3.6	\[ -0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)\right)} \]
      metadata-eval [=>] 3.6	\[ -0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right)\right) \]
      add-log-exp [<=] 68.0	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}{a}}\right) \]
      associate-/l/ [=>] 57.6	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c \cdot \left(-4 \cdot a\right)}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b\right)}}\right) \]
      times-frac [=>] 56.5	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(-4 \cdot a\right)}\right) - b}}\right) \]
      hypot-udef [=>] 54.6	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\color{blue}{\sqrt{b \cdot b + \sqrt{c \cdot \left(-4 \cdot a\right)} \cdot \sqrt{c \cdot \left(-4 \cdot a\right)}}} - b}\right) \]
      add-sqr-sqrt [<=] 58.7	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\sqrt{b \cdot b + \color{blue}{c \cdot \left(-4 \cdot a\right)}} - b}\right) \]
      +-commutative [=>] 58.7	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\sqrt{\color{blue}{c \cdot \left(-4 \cdot a\right) + b \cdot b}} - b}\right) \]
      add-sqr-sqrt [=>] 54.6	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\sqrt{\color{blue}{\sqrt{c \cdot \left(-4 \cdot a\right)} \cdot \sqrt{c \cdot \left(-4 \cdot a\right)}} + b \cdot b} - b}\right) \]
      hypot-def [=>] 56.5	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\color{blue}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right)} - b}\right) \]

   6. Simplified 78.4%

      \[\leadsto -0.5 \cdot \color{blue}{\left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{-4 \cdot \left(c \cdot a\right)}, b\right) - b}\right)} \]
      Proof

      [Start] 56.5	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      +-lft-identity [=>] 56.5	\[ -0.5 \cdot \color{blue}{\left(\frac{c}{a} \cdot \frac{-4 \cdot a}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right)} \]
      associate-*r/ [=>] 56.6	\[ -0.5 \cdot \color{blue}{\frac{\frac{c}{a} \cdot \left(-4 \cdot a\right)}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}} \]
      associate-*l/ [=>] 68.2	\[ -0.5 \cdot \frac{\color{blue}{\frac{c \cdot \left(-4 \cdot a\right)}{a}}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b} \]
      *-commutative [=>] 68.2	\[ -0.5 \cdot \frac{\frac{\color{blue}{\left(-4 \cdot a\right) \cdot c}}{a}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b} \]
      associate-*l/ [<=] 78.4	\[ -0.5 \cdot \frac{\color{blue}{\frac{-4 \cdot a}{a} \cdot c}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b} \]
      associate-*r/ [<=] 78.4	\[ -0.5 \cdot \color{blue}{\left(\frac{-4 \cdot a}{a} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right)} \]
      associate-/l* [=>] 78.4	\[ -0.5 \cdot \left(\color{blue}{\frac{-4}{\frac{a}{a}}} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      *-inverses [=>] 78.4	\[ -0.5 \cdot \left(\frac{-4}{\color{blue}{1}} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      metadata-eval [=>] 78.4	\[ -0.5 \cdot \left(\color{blue}{-4} \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{c \cdot \left(-4 \cdot a\right)}, b\right) - b}\right) \]
      *-commutative [=>] 78.4	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{\color{blue}{\left(-4 \cdot a\right) \cdot c}}, b\right) - b}\right) \]
      associate-*l* [=>] 78.4	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}, b\right) - b}\right) \]
      *-commutative [<=] 78.4	\[ -0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)}}, b\right) - b}\right) \]

   if 1.24999999999999999e-221 < b < 1.1199999999999999e135

   1. Initial program 88.1%

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

   2. Simplified 88.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]
      Proof

      [Start] 88.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 88.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 88.1	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 87.9	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 87.8	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 87.8	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 87.8	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 87.8	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 87.8	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 87.8	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 88.1	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 88.1	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 88.1	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 88.1	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 88.1	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 88.1	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 88.1	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 88.1%

      \[\leadsto -0.5 \cdot \frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4 + b \cdot b}}}{a} \]
      Proof

      [Start] 88.1	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      fma-udef [=>] 88.1	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}}{a} \]
      associate-*r* [=>] 88.1	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4} + b \cdot b}}{a} \]

   if 1.1199999999999999e135 < b

   1. Initial program 11.9%

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

   2. Taylor expanded in b around inf 95.2%

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

   3. Simplified 95.2%

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

      [Start] 95.2	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 95.2	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 95.2	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-15}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, \frac{-0.5 \cdot c}{\frac{b}{-4 \cdot a}}\right)}\right)\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-221}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{hypot}\left(\sqrt{-4 \cdot \left(c \cdot a\right)}, b\right) - b}\right)\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{+135}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.6%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 10^{+135}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	82.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-159}:\\ \;\;\;\;-0.5 \cdot \left(-4 \cdot \frac{c}{\mathsf{fma}\left(b, -2, \frac{-0.5 \cdot c}{\frac{b}{-4 \cdot a}}\right)}\right)\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+134}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Accuracy	76.4%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 0.31:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(-4 \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 4
Accuracy	63.6%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-207}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 5
Accuracy	29.7%
Cost	256

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

Reproduce

herbie shell --seed 2023152 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))