The quadratic formula (r2)

Percentage Accurate: 52.4% → 88.2%

Time: 20.8s

Precision: binary64

Cost: 13960

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

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \frac{-0.5 \cdot c}{\frac{\frac{b}{a}}{-4}}\right)}\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-264}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\left(c \cdot -4\right) \cdot a}, b\right) - b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+74}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.22e-49)
   (* -0.5 (/ (* c -4.0) (fma b -2.0 (/ (* -0.5 c) (/ (/ b a) -4.0)))))
   (if (<= b -8.5e-264)
     (* -0.5 (/ (* c -4.0) (- (hypot (sqrt (* (* c -4.0) a)) b) b)))
     (if (<= b 4e+74)
       (* -0.5 (/ (+ b (sqrt (+ (* -4.0 (* c a)) (* 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 tmp;
	if (b <= -1.22e-49) {
		tmp = -0.5 * ((c * -4.0) / fma(b, -2.0, ((-0.5 * c) / ((b / a) / -4.0))));
	} else if (b <= -8.5e-264) {
		tmp = -0.5 * ((c * -4.0) / (hypot(sqrt(((c * -4.0) * a)), b) - b));
	} else if (b <= 4e+74) {
		tmp = -0.5 * ((b + sqrt(((-4.0 * (c * a)) + (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)
	tmp = 0.0
	if (b <= -1.22e-49)
		tmp = Float64(-0.5 * Float64(Float64(c * -4.0) / fma(b, -2.0, Float64(Float64(-0.5 * c) / Float64(Float64(b / a) / -4.0)))));
	elseif (b <= -8.5e-264)
		tmp = Float64(-0.5 * Float64(Float64(c * -4.0) / Float64(hypot(sqrt(Float64(Float64(c * -4.0) * a)), b) - b)));
	elseif (b <= 4e+74)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(Float64(Float64(-4.0 * Float64(c * a)) + 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_] := If[LessEqual[b, -1.22e-49], N[(-0.5 * N[(N[(c * -4.0), $MachinePrecision] / N[(b * -2.0 + N[(N[(-0.5 * c), $MachinePrecision] / N[(N[(b / a), $MachinePrecision] / -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.5e-264], N[(-0.5 * N[(N[(c * -4.0), $MachinePrecision] / N[(N[Sqrt[N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e+74], N[(-0.5 * N[(N[(b + N[Sqrt[N[(N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	52.4%
Target	71.0%
Herbie	88.2%

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

Derivation

1. Split input into 4 regimes

2. if b < -1.2199999999999999e-49

   1. Initial program 17.8%

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

   2. Simplified 17.9%

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

      [Start] 17.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 17.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 17.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 17.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 17.8	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 17.8	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 17.8	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 17.8	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 17.8	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 17.8	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 17.8	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 17.8	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 17.8	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 17.8	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 17.8	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 17.8	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 16.3	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 12.6%

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

      [Start] 17.9	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      flip-+ [=>] 16.9	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      add-sqr-sqrt [<=] 16.9	\[ -0.5 \cdot \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      div-sub [=>] 16.9	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      fma-udef [=>] 16.9	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      +-commutative [=>] 16.9	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      add-sqr-sqrt [=>] 12.6	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      hypot-def [=>] 12.6	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      associate-*r* [=>] 12.6	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]

   4. Simplified 46.6%

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

      [Start] 12.6	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      div-sub [<=] 12.6	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      *-lft-identity [<=] 12.6	\[ -0.5 \cdot \frac{\color{blue}{1 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      metadata-eval [<=] 12.6	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      times-frac [<=] 12.6	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 12.6	\[ -0.5 \cdot \frac{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{\color{blue}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 12.6	\[ -0.5 \cdot \frac{\frac{\color{blue}{-\left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}{a} \]

   5. Applied egg-rr 38.2%

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

      [Start] 46.6	\[ -0.5 \cdot \frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a} \]
      add-log-exp [=>] 19.0	\[ -0.5 \cdot \color{blue}{\log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)} \]
      *-un-lft-identity [=>] 19.0	\[ -0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)} \]
      log-prod [=>] 19.0	\[ -0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)\right)} \]
      metadata-eval [=>] 19.0	\[ -0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)\right) \]
      add-log-exp [<=] 46.6	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right) \]
      associate-/l/ [=>] 45.5	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right)}}\right) \]
      times-frac [=>] 38.2	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c}{a} \cdot \frac{a \cdot -4}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}\right) \]
      hypot-udef [=>] 31.5	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\color{blue}{\sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}}} - b}\right) \]
      add-sqr-sqrt [<=] 55.7	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -4\right)}} - b}\right) \]
      +-commutative [=>] 55.7	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}} - b}\right) \]
      add-sqr-sqrt [=>] 31.5	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\sqrt{\color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}} + b \cdot b} - b}\right) \]
      hypot-def [=>] 38.2	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\color{blue}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)} - b}\right) \]

   6. Simplified 50.8%

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

      [Start] 38.2	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}\right) \]
      +-lft-identity [=>] 38.2	\[ -0.5 \cdot \color{blue}{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}\right)} \]
      associate-*r/ [=>] 41.7	\[ -0.5 \cdot \color{blue}{\frac{\frac{c}{a} \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}} \]
      associate-*l/ [=>] 49.3	\[ -0.5 \cdot \frac{\color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{a}}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-*r* [=>] 49.3	\[ -0.5 \cdot \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -4}}{a}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-*l/ [<=] 49.3	\[ -0.5 \cdot \frac{\color{blue}{\frac{c \cdot a}{a} \cdot -4}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-/l* [=>] 50.8	\[ -0.5 \cdot \frac{\color{blue}{\frac{c}{\frac{a}{a}}} \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      *-inverses [=>] 50.8	\[ -0.5 \cdot \frac{\frac{c}{\color{blue}{1}} \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      /-rgt-identity [=>] 50.8	\[ -0.5 \cdot \frac{\color{blue}{c} \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-*r* [=>] 50.8	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}, b\right) - b} \]
      *-commutative [=>] 50.8	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}, b\right) - b} \]
      associate-*l* [=>] 50.8	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}, b\right) - b} \]

   7. Taylor expanded in b around -inf 0.0%

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

   8. Simplified 87.8%

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

      [Start] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{-2 \cdot b + -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}} \]
      *-commutative [=>] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{\color{blue}{b \cdot -2} + -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}} \]
      fma-def [=>] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{\color{blue}{\mathsf{fma}\left(b, -2, -0.5 \cdot \frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b}\right)}} \]
      *-commutative [=>] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c \cdot \left(a \cdot {\left(\sqrt{-4}\right)}^{2}\right)}{b} \cdot -0.5}\right)} \]
      associate-/l* [=>] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c}{\frac{b}{a \cdot {\left(\sqrt{-4}\right)}^{2}}}} \cdot -0.5\right)} \]
      associate-*l/ [=>] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \color{blue}{\frac{c \cdot -0.5}{\frac{b}{a \cdot {\left(\sqrt{-4}\right)}^{2}}}}\right)} \]
      associate-/r* [=>] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \frac{c \cdot -0.5}{\color{blue}{\frac{\frac{b}{a}}{{\left(\sqrt{-4}\right)}^{2}}}}\right)} \]
      unpow2 [=>] 0.0	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \frac{c \cdot -0.5}{\frac{\frac{b}{a}}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}}}\right)} \]
      rem-square-sqrt [=>] 87.8	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \frac{c \cdot -0.5}{\frac{\frac{b}{a}}{\color{blue}{-4}}}\right)} \]

   if -1.2199999999999999e-49 < b < -8.5000000000000001e-264

   1. Initial program 61.8%

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

   2. Simplified 61.8%

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

      [Start] 61.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 61.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 61.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 61.7	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 61.6	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 61.6	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 61.6	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 61.6	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 61.6	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 61.6	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 61.8	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 61.8	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 61.8	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 61.8	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 61.8	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 61.8	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 61.8	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 60.9%

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

      [Start] 61.8	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      flip-+ [=>] 61.1	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      add-sqr-sqrt [<=] 61.0	\[ -0.5 \cdot \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      div-sub [=>] 61.0	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}}{a} \]
      fma-udef [=>] 61.0	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      +-commutative [=>] 61.0	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -4\right)}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      add-sqr-sqrt [=>] 60.9	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} \cdot \sqrt{a \cdot \left(c \cdot -4\right)}}}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      hypot-def [=>] 60.9	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]
      associate-*r* [=>] 60.9	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}\right)} - \frac{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}{b - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]

   4. Simplified 66.0%

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

      [Start] 60.9	\[ -0.5 \cdot \frac{\frac{b \cdot b}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)} - \frac{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      div-sub [<=] 60.9	\[ -0.5 \cdot \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      *-lft-identity [<=] 60.9	\[ -0.5 \cdot \frac{\color{blue}{1 \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}}{a} \]
      metadata-eval [<=] 60.9	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1}{-1}} \cdot \frac{b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)}{b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)}}{a} \]
      times-frac [<=] 60.9	\[ -0.5 \cdot \frac{\color{blue}{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{-1 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 60.9	\[ -0.5 \cdot \frac{\frac{-1 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}{\color{blue}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}}{a} \]
      neg-mul-1 [<=] 60.9	\[ -0.5 \cdot \frac{\frac{\color{blue}{-\left(b \cdot b - \mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)\right)}}{-\left(b - \mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -4}\right)\right)}}{a} \]

   5. Applied egg-rr 61.7%

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

      [Start] 66.0	\[ -0.5 \cdot \frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a} \]
      add-log-exp [=>] 9.4	\[ -0.5 \cdot \color{blue}{\log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)} \]
      *-un-lft-identity [=>] 9.4	\[ -0.5 \cdot \log \color{blue}{\left(1 \cdot e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)} \]
      log-prod [=>] 9.4	\[ -0.5 \cdot \color{blue}{\left(\log 1 + \log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)\right)} \]
      metadata-eval [=>] 9.4	\[ -0.5 \cdot \left(\color{blue}{0} + \log \left(e^{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right)\right) \]
      add-log-exp [<=] 66.0	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{\frac{c \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}{a}}\right) \]
      associate-/l/ [=>] 58.0	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{a \cdot \left(\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b\right)}}\right) \]
      times-frac [=>] 61.7	\[ -0.5 \cdot \left(0 + \color{blue}{\frac{c}{a} \cdot \frac{a \cdot -4}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -4\right)}\right) - b}}\right) \]
      hypot-udef [=>] 57.5	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\color{blue}{\sqrt{b \cdot b + \sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}}} - b}\right) \]
      add-sqr-sqrt [<=] 60.1	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\sqrt{b \cdot b + \color{blue}{c \cdot \left(a \cdot -4\right)}} - b}\right) \]
      +-commutative [=>] 60.1	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\sqrt{\color{blue}{c \cdot \left(a \cdot -4\right) + b \cdot b}} - b}\right) \]
      add-sqr-sqrt [=>] 57.5	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\sqrt{\color{blue}{\sqrt{c \cdot \left(a \cdot -4\right)} \cdot \sqrt{c \cdot \left(a \cdot -4\right)}} + b \cdot b} - b}\right) \]
      hypot-def [=>] 61.7	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\color{blue}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right)} - b}\right) \]

   6. Simplified 76.9%

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

      [Start] 61.7	\[ -0.5 \cdot \left(0 + \frac{c}{a} \cdot \frac{a \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}\right) \]
      +-lft-identity [=>] 61.7	\[ -0.5 \cdot \color{blue}{\left(\frac{c}{a} \cdot \frac{a \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}\right)} \]
      associate-*r/ [=>] 61.9	\[ -0.5 \cdot \color{blue}{\frac{\frac{c}{a} \cdot \left(a \cdot -4\right)}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b}} \]
      associate-*l/ [=>] 66.1	\[ -0.5 \cdot \frac{\color{blue}{\frac{c \cdot \left(a \cdot -4\right)}{a}}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-*r* [=>] 66.1	\[ -0.5 \cdot \frac{\frac{\color{blue}{\left(c \cdot a\right) \cdot -4}}{a}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-*l/ [<=] 66.1	\[ -0.5 \cdot \frac{\color{blue}{\frac{c \cdot a}{a} \cdot -4}}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-/l* [=>] 76.9	\[ -0.5 \cdot \frac{\color{blue}{\frac{c}{\frac{a}{a}}} \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      *-inverses [=>] 76.9	\[ -0.5 \cdot \frac{\frac{c}{\color{blue}{1}} \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      /-rgt-identity [=>] 76.9	\[ -0.5 \cdot \frac{\color{blue}{c} \cdot -4}{\mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -4\right)}, b\right) - b} \]
      associate-*r* [=>] 76.9	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\color{blue}{\left(c \cdot a\right) \cdot -4}}, b\right) - b} \]
      *-commutative [=>] 76.9	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot c\right)} \cdot -4}, b\right) - b} \]
      associate-*l* [=>] 76.9	\[ -0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}, b\right) - b} \]

   if -8.5000000000000001e-264 < b < 3.99999999999999981e74

   1. Initial program 83.8%

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

   2. Simplified 83.8%

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

      [Start] 83.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
      /-rgt-identity [<=] 83.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2 \cdot a}{1}}} \]
      metadata-eval [<=] 83.8	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{2 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [=>] 83.6	\[ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\color{blue}{\frac{2}{\frac{--1}{a}}}} \]
      associate-/r/ [=>] 83.6	\[ \color{blue}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \cdot \frac{--1}{a}} \]
      *-commutative [<=] 83.6	\[ \color{blue}{\frac{--1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}} \]
      metadata-eval [=>] 83.6	\[ \frac{\color{blue}{1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      metadata-eval [<=] 83.6	\[ \frac{\color{blue}{-1 \cdot -1}}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-*l/ [<=] 83.6	\[ \color{blue}{\left(\frac{-1}{a} \cdot -1\right)} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      associate-/r/ [<=] 83.6	\[ \color{blue}{\frac{-1}{\frac{a}{-1}}} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} \]
      times-frac [<=] 83.8	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\frac{a}{-1} \cdot 2}} \]
      *-commutative [<=] 83.8	\[ \frac{-1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}{\color{blue}{2 \cdot \frac{a}{-1}}} \]
      times-frac [=>] 83.8	\[ \color{blue}{\frac{-1}{2} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}}} \]
      metadata-eval [=>] 83.8	\[ \color{blue}{-0.5} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\frac{a}{-1}} \]
      associate-/r/ [=>] 83.8	\[ -0.5 \cdot \color{blue}{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a} \cdot -1\right)} \]
      *-commutative [<=] 83.8	\[ -0.5 \cdot \color{blue}{\left(-1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)} \]
      div-sub [=>] 83.8	\[ -0.5 \cdot \left(-1 \cdot \color{blue}{\left(\frac{-b}{a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}\right)}\right) \]

   3. Applied egg-rr 83.8%

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

      [Start] 83.8	\[ -0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a} \]
      fma-udef [=>] 83.8	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right) + b \cdot b}}}{a} \]
      associate-*r* [=>] 83.8	\[ -0.5 \cdot \frac{b + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4} + b \cdot b}}{a} \]

   if 3.99999999999999981e74 < b

   1. Initial program 62.0%

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

   2. Taylor expanded in b around inf 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]
      Step-by-step derivation

      [Start] 95.9	\[ \frac{c}{b} + -1 \cdot \frac{b}{a} \]
      mul-1-neg [=>] 95.9	\[ \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]
      unsub-neg [=>] 95.9	\[ \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \frac{-0.5 \cdot c}{\frac{\frac{b}{a}}{-4}}\right)}\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-264}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot -4}{\mathsf{hypot}\left(\sqrt{\left(c \cdot -4\right) \cdot a}, b\right) - b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+74}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 6.1 \cdot 10^{+72}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 2
Accuracy	85.8%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{-108}:\\ \;\;\;\;-0.5 \cdot \frac{c \cdot -4}{\mathsf{fma}\left(b, -2, \frac{-0.5 \cdot c}{\frac{\frac{b}{a}}{-4}}\right)}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{+74}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 3
Accuracy	80.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-106}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-83}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{c \cdot \left(-4 \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 4
Accuracy	67.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array} \]

Alternative 5
Accuracy	42.8%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+37}:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 6
Accuracy	67.5%
Cost	388

\[\begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-298}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Alternative 7
Accuracy	12.4%
Cost	324

\[\begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot c\\ \end{array} \]

Alternative 8
Accuracy	4.5%
Cost	192

\[b \cdot c \]

Reproduce

herbie shell --seed 2023160 
(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)))