Cubic critical

Average Accuracy: 47.7% → 82.8%

Time: 29.2s

Precision: binary64

Cost: 20492

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{a \cdot -3} \cdot \sqrt{c}, b\right)}{-3}}}\\ \mathbf{elif}\;b \leq 410000000:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{-3 \cdot \left(a \cdot c\right)}\right)}{3 \cdot \left(a \cdot c\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.3e+103)
   (* (/ b a) -0.6666666666666666)
   (if (<= b -4.2e-162)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (if (<= b -1.32e-235)
       (/ 1.0 (/ a (/ (- b (hypot (* (sqrt (* a -3.0)) (sqrt c)) b)) -3.0)))
       (if (<= b 410000000.0)
         (/
          (/
           -0.3333333333333333
           (/ (+ b (hypot b (sqrt (* -3.0 (* a c))))) (* 3.0 (* a c))))
          a)
         (* -0.5 (/ c b)))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+103) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= -4.2e-162) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else if (b <= -1.32e-235) {
		tmp = 1.0 / (a / ((b - hypot((sqrt((a * -3.0)) * sqrt(c)), b)) / -3.0));
	} else if (b <= 410000000.0) {
		tmp = (-0.3333333333333333 / ((b + hypot(b, sqrt((-3.0 * (a * c))))) / (3.0 * (a * c)))) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.3e+103) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= -4.2e-162) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else if (b <= -1.32e-235) {
		tmp = 1.0 / (a / ((b - Math.hypot((Math.sqrt((a * -3.0)) * Math.sqrt(c)), b)) / -3.0));
	} else if (b <= 410000000.0) {
		tmp = (-0.3333333333333333 / ((b + Math.hypot(b, Math.sqrt((-3.0 * (a * c))))) / (3.0 * (a * c)))) / a;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= -2.3e+103:
		tmp = (b / a) * -0.6666666666666666
	elif b <= -4.2e-162:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	elif b <= -1.32e-235:
		tmp = 1.0 / (a / ((b - math.hypot((math.sqrt((a * -3.0)) * math.sqrt(c)), b)) / -3.0))
	elif b <= 410000000.0:
		tmp = (-0.3333333333333333 / ((b + math.hypot(b, math.sqrt((-3.0 * (a * c))))) / (3.0 * (a * c)))) / a
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.3e+103)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	elseif (b <= -4.2e-162)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	elseif (b <= -1.32e-235)
		tmp = Float64(1.0 / Float64(a / Float64(Float64(b - hypot(Float64(sqrt(Float64(a * -3.0)) * sqrt(c)), b)) / -3.0)));
	elseif (b <= 410000000.0)
		tmp = Float64(Float64(-0.3333333333333333 / Float64(Float64(b + hypot(b, sqrt(Float64(-3.0 * Float64(a * c))))) / Float64(3.0 * Float64(a * c)))) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.3e+103)
		tmp = (b / a) * -0.6666666666666666;
	elseif (b <= -4.2e-162)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	elseif (b <= -1.32e-235)
		tmp = 1.0 / (a / ((b - hypot((sqrt((a * -3.0)) * sqrt(c)), b)) / -3.0));
	elseif (b <= 410000000.0)
		tmp = (-0.3333333333333333 / ((b + hypot(b, sqrt((-3.0 * (a * c))))) / (3.0 * (a * c)))) / a;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := If[LessEqual[b, -2.3e+103], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], If[LessEqual[b, -4.2e-162], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.32e-235], N[(1.0 / N[(a / N[(N[(b - N[Sqrt[N[(N[Sqrt[N[(a * -3.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 410000000.0], N[(N[(-0.3333333333333333 / N[(N[(b + N[Sqrt[b ^ 2 + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.30000000000000008e103

   1. Initial program 28.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around -inf 93.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
      Proof

      [Start] 93.0	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      *-commutative [=>] 93.0	\[ \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   if -2.30000000000000008e103 < b < -4.2e-162

   1. Initial program 90.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   if -4.2e-162 < b < -1.32e-235

   1. Initial program 72.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 71.9%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}} \]
      Proof

      [Start] 72.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 72.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 72.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 72.1	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{3 \cdot a}} \]
      associate-*r/ [<=] 71.9	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{3 \cdot a}} \]
      *-commutative [=>] 71.9	\[ \color{blue}{\frac{--1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
      associate-*l/ [=>] 72.1	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot a}} \]
      associate-*r/ [<=] 72.1	\[ \color{blue}{\left(--1\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [=>] 72.1	\[ \color{blue}{1} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      metadata-eval [<=] 72.1	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 72.1	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      neg-mul-1 [<=] 72.1	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
      distribute-rgt-neg-in [=>] 72.1	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
      times-frac [=>] 71.8	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
      metadata-eval [=>] 71.8	\[ \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]
      neg-mul-1 [=>] 71.8	\[ -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}} \]
      Proof

      [Start] 71.9	\[ -0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \]
      associate-*r/ [=>] 71.9	\[ \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}{a}} \]
      clear-num [=>] 71.8	\[ \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}}} \]
      fma-udef [=>] 71.8	\[ \frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}}\right)}} \]
      add-sqr-sqrt [=>] 71.8	\[ \frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + \color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}\right)}} \]
      hypot-def [=>] 81.6	\[ \frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}\right)}} \]

   4. Applied egg-rr 81.6%

      \[\leadsto \frac{1}{\color{blue}{0 + \frac{a \cdot -3}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -3\right)}, b\right)}}} \]
      Proof

      [Start] 81.6	\[ \frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}} \]
      add-log-exp [=>] 4.8	\[ \frac{1}{\color{blue}{\log \left(e^{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}\right)}} \]
      *-un-lft-identity [=>] 4.8	\[ \frac{1}{\log \color{blue}{\left(1 \cdot e^{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}\right)}} \]
      log-prod [=>] 4.8	\[ \frac{1}{\color{blue}{\log 1 + \log \left(e^{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}\right)}} \]
      metadata-eval [=>] 4.8	\[ \frac{1}{\color{blue}{0} + \log \left(e^{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}\right)} \]
      add-log-exp [<=] 81.6	\[ \frac{1}{0 + \color{blue}{\frac{a}{-0.3333333333333333 \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right)}}} \]
      associate-/r* [=>] 81.6	\[ \frac{1}{0 + \color{blue}{\frac{\frac{a}{-0.3333333333333333}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}}} \]
      div-inv [=>] 81.6	\[ \frac{1}{0 + \frac{\color{blue}{a \cdot \frac{1}{-0.3333333333333333}}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}} \]
      metadata-eval [=>] 81.6	\[ \frac{1}{0 + \frac{a \cdot \color{blue}{-3}}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}} \]
      hypot-udef [=>] 71.9	\[ \frac{1}{0 + \frac{a \cdot -3}{b - \color{blue}{\sqrt{b \cdot b + \sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}}} \]
      add-sqr-sqrt [<=] 71.9	\[ \frac{1}{0 + \frac{a \cdot -3}{b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -3\right)}}}} \]
      +-commutative [=>] 71.9	\[ \frac{1}{0 + \frac{a \cdot -3}{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right) + b \cdot b}}}} \]
      add-sqr-sqrt [=>] 71.9	\[ \frac{1}{0 + \frac{a \cdot -3}{b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}} + b \cdot b}}} \]
      hypot-def [=>] 81.6	\[ \frac{1}{0 + \frac{a \cdot -3}{b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -3\right)}, b\right)}}} \]

   5. Simplified 81.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -3\right)}, b\right)}{-3}}}} \]
      Proof

      [Start] 81.6	\[ \frac{1}{0 + \frac{a \cdot -3}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -3\right)}, b\right)}} \]
      +-lft-identity [=>] 81.6	\[ \frac{1}{\color{blue}{\frac{a \cdot -3}{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -3\right)}, b\right)}}} \]
      associate-/l* [=>] 81.7	\[ \frac{1}{\color{blue}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -3\right)}, b\right)}{-3}}}} \]
      associate-*r* [=>] 81.6	\[ \frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}, b\right)}{-3}}} \]
      *-commutative [=>] 81.6	\[ \frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3}, b\right)}{-3}}} \]
      associate-*r* [<=] 81.7	\[ \frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}}, b\right)}{-3}}} \]

   6. Applied egg-rr 47.8%

      \[\leadsto \frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{c}}, b\right)}{-3}}} \]
      Proof

      [Start] 81.7	\[ \frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{c \cdot \left(a \cdot -3\right)}, b\right)}{-3}}} \]
      *-commutative [=>] 81.7	\[ \frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{\color{blue}{\left(a \cdot -3\right) \cdot c}}, b\right)}{-3}}} \]
      sqrt-prod [=>] 47.8	\[ \frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{c}}, b\right)}{-3}}} \]

   if -1.32e-235 < b < 4.1e8

   1. Initial program 62.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}} \]
      Proof

      [Start] 62.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 62.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 62.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 62.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 62.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 62.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 62.0	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 62.0	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 61.9	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      *-commutative [<=] 61.9	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-/r* [=>] 61.8	\[ \color{blue}{\frac{\frac{-1}{3}}{a}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \]
      associate-*l/ [=>] 61.8	\[ \color{blue}{\frac{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{a}} \]

   3. Applied egg-rr 60.9%

      \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}}}{a} \]
      Proof

      [Start] 61.8	\[ \frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a} \]
      flip-- [=>] 61.7	\[ \frac{-0.3333333333333333 \cdot \color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}}{a} \]
      associate-*r/ [=>] 61.6	\[ \frac{\color{blue}{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}}{a} \]
      add-sqr-sqrt [<=] 61.7	\[ \frac{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \color{blue}{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}{a} \]
      associate-*l* [=>] 61.6	\[ \frac{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)\right)}{b + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}}{a} \]
      fma-udef [=>] 61.6	\[ \frac{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}{b + \sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -3}}}}{a} \]
      add-sqr-sqrt [=>] 60.8	\[ \frac{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}{b + \sqrt{b \cdot b + \color{blue}{\sqrt{\left(a \cdot c\right) \cdot -3} \cdot \sqrt{\left(a \cdot c\right) \cdot -3}}}}}{a} \]
      hypot-def [=>] 60.8	\[ \frac{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}{b + \color{blue}{\mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -3}\right)}}}{a} \]
      associate-*l* [=>] 60.9	\[ \frac{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}\right)}}{a} \]

   4. Simplified 66.8%

      \[\leadsto \frac{\color{blue}{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{\left(c \cdot a\right) \cdot 3}}}}{a} \]
      Proof

      [Start] 60.9	\[ \frac{\frac{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)\right)}{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}}{a} \]
      associate-/l* [=>] 60.8	\[ \frac{\color{blue}{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}}{a} \]
      associate-*r* [=>] 60.8	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}\right)}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{a} \]
      *-commutative [=>] 60.8	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3}\right)}{b \cdot b - \mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{a} \]
      fma-udef [=>] 60.8	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{b \cdot b - \color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -3\right)\right)}}}}{a} \]
      associate--r+ [=>] 66.7	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{\color{blue}{\left(b \cdot b - b \cdot b\right) - a \cdot \left(c \cdot -3\right)}}}}{a} \]
      +-inverses [=>] 66.7	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{\color{blue}{0} - a \cdot \left(c \cdot -3\right)}}}{a} \]
      neg-sub0 [<=] 66.7	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{\color{blue}{-a \cdot \left(c \cdot -3\right)}}}}{a} \]
      associate-*r* [=>] 66.8	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{-\color{blue}{\left(a \cdot c\right) \cdot -3}}}}{a} \]
      *-commutative [=>] 66.8	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{-\color{blue}{\left(c \cdot a\right)} \cdot -3}}}{a} \]
      distribute-rgt-neg-in [=>] 66.8	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{\color{blue}{\left(c \cdot a\right) \cdot \left(--3\right)}}}}{a} \]
      metadata-eval [=>] 66.8	\[ \frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{\left(c \cdot a\right) \cdot -3}\right)}{\left(c \cdot a\right) \cdot \color{blue}{3}}}}{a} \]

   if 4.1e8 < b

   1. Initial program 12.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf 91.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq -4.2 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -1.32 \cdot 10^{-235}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{b - \mathsf{hypot}\left(\sqrt{a \cdot -3} \cdot \sqrt{c}, b\right)}{-3}}}\\ \mathbf{elif}\;b \leq 410000000:\\ \;\;\;\;\frac{\frac{-0.3333333333333333}{\frac{b + \mathsf{hypot}\left(b, \sqrt{-3 \cdot \left(a \cdot c\right)}\right)}{3 \cdot \left(a \cdot c\right)}}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.4%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2.55 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.46 \cdot 10^{-36}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Accuracy	83.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Accuracy	83.7%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Accuracy	78.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-38}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Accuracy	77.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{-3 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Accuracy	77.9%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-36}:\\ \;\;\;\;\left(b - \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Accuracy	78.0%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Accuracy	63.7%
Cost	644

\[\begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{-256}:\\ \;\;\;\;\frac{-\left(b + b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Accuracy	63.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-261}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b + b\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Accuracy	63.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.36 \cdot 10^{-260}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Accuracy	63.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-256}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Accuracy	37.7%
Cost	320

\[-0.5 \cdot \frac{c}{b} \]

Alternative 13
Accuracy	11.9%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023151 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))