Cubic critical

Percentage Accurate: 52.4% → 86.0%

Time: 23.9s

Precision: binary64

Cost: 13964

Math

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

↓

\[\begin{array}{l} t_0 := a \cdot \left(-3 \cdot c\right)\\ \mathbf{if}\;b \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;{\left(-3 \cdot \frac{a \cdot 0.5}{b}\right)}^{-1}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + t_0}\right)}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{b - \mathsf{hypot}\left(\sqrt{t_0}, b\right)}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* -3.0 c))))
   (if (<= b -1.35e+130)
     (pow (* -3.0 (/ (* a 0.5) b)) -1.0)
     (if (<= b -5.5e-125)
       (/ (* -0.3333333333333333 (- b (sqrt (+ (* b b) t_0)))) a)
       (if (<= b 1.6e-93)
         (/ (/ (- b (hypot (sqrt t_0) b)) a) -3.0)
         (* -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 t_0 = a * (-3.0 * c);
	double tmp;
	if (b <= -1.35e+130) {
		tmp = pow((-3.0 * ((a * 0.5) / b)), -1.0);
	} else if (b <= -5.5e-125) {
		tmp = (-0.3333333333333333 * (b - sqrt(((b * b) + t_0)))) / a;
	} else if (b <= 1.6e-93) {
		tmp = ((b - hypot(sqrt(t_0), b)) / a) / -3.0;
	} 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 t_0 = a * (-3.0 * c);
	double tmp;
	if (b <= -1.35e+130) {
		tmp = Math.pow((-3.0 * ((a * 0.5) / b)), -1.0);
	} else if (b <= -5.5e-125) {
		tmp = (-0.3333333333333333 * (b - Math.sqrt(((b * b) + t_0)))) / a;
	} else if (b <= 1.6e-93) {
		tmp = ((b - Math.hypot(Math.sqrt(t_0), b)) / a) / -3.0;
	} 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):
	t_0 = a * (-3.0 * c)
	tmp = 0
	if b <= -1.35e+130:
		tmp = math.pow((-3.0 * ((a * 0.5) / b)), -1.0)
	elif b <= -5.5e-125:
		tmp = (-0.3333333333333333 * (b - math.sqrt(((b * b) + t_0)))) / a
	elif b <= 1.6e-93:
		tmp = ((b - math.hypot(math.sqrt(t_0), b)) / a) / -3.0
	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)
	t_0 = Float64(a * Float64(-3.0 * c))
	tmp = 0.0
	if (b <= -1.35e+130)
		tmp = Float64(-3.0 * Float64(Float64(a * 0.5) / b)) ^ -1.0;
	elseif (b <= -5.5e-125)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(Float64(Float64(b * b) + t_0)))) / a);
	elseif (b <= 1.6e-93)
		tmp = Float64(Float64(Float64(b - hypot(sqrt(t_0), b)) / a) / -3.0);
	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)
	t_0 = a * (-3.0 * c);
	tmp = 0.0;
	if (b <= -1.35e+130)
		tmp = (-3.0 * ((a * 0.5) / b)) ^ -1.0;
	elseif (b <= -5.5e-125)
		tmp = (-0.3333333333333333 * (b - sqrt(((b * b) + t_0)))) / a;
	elseif (b <= 1.6e-93)
		tmp = ((b - hypot(sqrt(t_0), b)) / a) / -3.0;
	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_] := Block[{t$95$0 = N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.35e+130], N[Power[N[(-3.0 * N[(N[(a * 0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[b, -5.5e-125], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.6e-93], N[(N[(N[(b - N[Sqrt[N[Sqrt[t$95$0], $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.3499999999999999e130

   1. Initial program 45.1%

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

   2. Simplified 45.0%

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

      [Start] 45.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 45.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 45.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 45.1	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 45.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 45.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 45.1	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 45.1	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 45.0	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      *-commutative [<=] 45.0	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-/r* [=>] 45.0	\[ \color{blue}{\frac{\frac{-1}{3}}{a}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \]
      associate-*l/ [=>] 45.0	\[ \color{blue}{\frac{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{a}} \]

   3. Applied egg-rr 44.1%

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

      [Start] 45.0	\[ \frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a} \]
      clear-num [=>] 45.1	\[ \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}}} \]
      inv-pow [=>] 45.1	\[ \color{blue}{{\left(\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}\right)}^{-1}} \]
      *-un-lft-identity [=>] 45.1	\[ {\left(\frac{\color{blue}{1 \cdot a}}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}\right)}^{-1} \]
      times-frac [=>] 45.1	\[ {\color{blue}{\left(\frac{1}{-0.3333333333333333} \cdot \frac{a}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}\right)}}^{-1} \]
      metadata-eval [=>] 45.1	\[ {\left(\color{blue}{-3} \cdot \frac{a}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}\right)}^{-1} \]
      fma-udef [=>] 45.1	\[ {\left(-3 \cdot \frac{a}{b - \sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -3}}}\right)}^{-1} \]
      add-sqr-sqrt [=>] 21.3	\[ {\left(-3 \cdot \frac{a}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{\left(a \cdot c\right) \cdot -3} \cdot \sqrt{\left(a \cdot c\right) \cdot -3}}}}\right)}^{-1} \]
      hypot-def [=>] 44.1	\[ {\left(-3 \cdot \frac{a}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -3}\right)}}\right)}^{-1} \]
      associate-*l* [=>] 44.1	\[ {\left(-3 \cdot \frac{a}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}\right)}\right)}^{-1} \]

   4. Taylor expanded in b around -inf 98.2%

      \[\leadsto {\left(-3 \cdot \color{blue}{\left(0.5 \cdot \frac{a}{b}\right)}\right)}^{-1} \]

   5. Simplified 98.2%

      \[\leadsto {\left(-3 \cdot \color{blue}{\frac{a \cdot 0.5}{b}}\right)}^{-1} \]
      Step-by-step derivation

      [Start] 98.2	\[ {\left(-3 \cdot \left(0.5 \cdot \frac{a}{b}\right)\right)}^{-1} \]
      *-commutative [=>] 98.2	\[ {\left(-3 \cdot \color{blue}{\left(\frac{a}{b} \cdot 0.5\right)}\right)}^{-1} \]
      associate-*l/ [=>] 98.2	\[ {\left(-3 \cdot \color{blue}{\frac{a \cdot 0.5}{b}}\right)}^{-1} \]

   if -1.3499999999999999e130 < b < -5.4999999999999997e-125

   1. Initial program 95.8%

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

   2. Simplified 96.0%

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

      [Start] 95.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 95.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 95.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 95.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 95.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 95.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 95.8	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 95.8	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 95.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      *-commutative [<=] 95.8	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-/r* [=>] 95.9	\[ \color{blue}{\frac{\frac{-1}{3}}{a}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \]
      associate-*l/ [=>] 96.0	\[ \color{blue}{\frac{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{a}} \]

   3. Applied egg-rr 96.0%

      \[\leadsto \frac{-0.3333333333333333 \cdot \left(b - \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}}\right)}{a} \]
      Step-by-step derivation

      [Start] 96.0	\[ \frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a} \]
      fma-udef [=>] 96.0	\[ \frac{-0.3333333333333333 \cdot \left(b - \sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -3}}\right)}{a} \]
      associate-*l* [=>] 96.0	\[ \frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -3\right)}}\right)}{a} \]

   if -5.4999999999999997e-125 < b < 1.5999999999999999e-93

   1. Initial program 71.5%

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

   2. Simplified 71.5%

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

      [Start] 71.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 71.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 71.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 71.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 71.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 71.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 71.5	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 71.5	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 71.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      *-commutative [<=] 71.5	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-/r* [=>] 71.4	\[ \color{blue}{\frac{\frac{-1}{3}}{a}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \]
      associate-*l/ [=>] 71.5	\[ \color{blue}{\frac{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{a}} \]

   3. Applied egg-rr 79.4%

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

      [Start] 71.5	\[ \frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a} \]
      clear-num [=>] 71.4	\[ \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}}} \]
      inv-pow [=>] 71.4	\[ \color{blue}{{\left(\frac{a}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}\right)}^{-1}} \]
      *-un-lft-identity [=>] 71.4	\[ {\left(\frac{\color{blue}{1 \cdot a}}{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}\right)}^{-1} \]
      times-frac [=>] 71.4	\[ {\color{blue}{\left(\frac{1}{-0.3333333333333333} \cdot \frac{a}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}\right)}}^{-1} \]
      metadata-eval [=>] 71.4	\[ {\left(\color{blue}{-3} \cdot \frac{a}{b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}}\right)}^{-1} \]
      fma-udef [=>] 71.4	\[ {\left(-3 \cdot \frac{a}{b - \sqrt{\color{blue}{b \cdot b + \left(a \cdot c\right) \cdot -3}}}\right)}^{-1} \]
      add-sqr-sqrt [=>] 71.4	\[ {\left(-3 \cdot \frac{a}{b - \sqrt{b \cdot b + \color{blue}{\sqrt{\left(a \cdot c\right) \cdot -3} \cdot \sqrt{\left(a \cdot c\right) \cdot -3}}}}\right)}^{-1} \]
      hypot-def [=>] 79.3	\[ {\left(-3 \cdot \frac{a}{b - \color{blue}{\mathsf{hypot}\left(b, \sqrt{\left(a \cdot c\right) \cdot -3}\right)}}\right)}^{-1} \]
      associate-*l* [=>] 79.4	\[ {\left(-3 \cdot \frac{a}{b - \mathsf{hypot}\left(b, \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}\right)}\right)}^{-1} \]

   4. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{\frac{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -3\right)}, b\right)}{a}}{-3}} \]
      Step-by-step derivation

      [Start] 79.4	\[ {\left(-3 \cdot \frac{a}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}\right)}^{-1} \]
      unpow-1 [=>] 79.4	\[ \color{blue}{\frac{1}{-3 \cdot \frac{a}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}}} \]
      *-commutative [=>] 79.4	\[ \frac{1}{\color{blue}{\frac{a}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)} \cdot -3}} \]
      associate-/r* [=>] 79.3	\[ \color{blue}{\frac{\frac{1}{\frac{a}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}}}{-3}} \]
      clear-num [<=] 79.4	\[ \frac{\color{blue}{\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}}{-3} \]
      hypot-udef [=>] 71.5	\[ \frac{\frac{b - \color{blue}{\sqrt{b \cdot b + \sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}}{a}}{-3} \]
      add-sqr-sqrt [<=] 71.5	\[ \frac{\frac{b - \sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -3\right)}}}{a}}{-3} \]
      +-commutative [=>] 71.5	\[ \frac{\frac{b - \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right) + b \cdot b}}}{a}}{-3} \]
      add-sqr-sqrt [=>] 71.5	\[ \frac{\frac{b - \sqrt{\color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}} + b \cdot b}}{a}}{-3} \]
      hypot-def [=>] 79.4	\[ \frac{\frac{b - \color{blue}{\mathsf{hypot}\left(\sqrt{a \cdot \left(c \cdot -3\right)}, b\right)}}{a}}{-3} \]

   if 1.5999999999999999e-93 < b

   1. Initial program 20.4%

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

   2. Taylor expanded in b around inf 88.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;{\left(-3 \cdot \frac{a \cdot 0.5}{b}\right)}^{-1}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{b - \mathsf{hypot}\left(\sqrt{a \cdot \left(-3 \cdot c\right)}, b\right)}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.0%
Cost	13964

\[\begin{array}{l} t_0 := a \cdot \left(-3 \cdot c\right)\\ \mathbf{if}\;b \leq -1.75 \cdot 10^{+130}:\\ \;\;\;\;{\left(-3 \cdot \frac{a \cdot 0.5}{b}\right)}^{-1}\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-125}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + t_0}\right)}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{a}{b - \mathsf{hypot}\left(\sqrt{t_0}, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Accuracy	86.0%
Cost	13964

\[\begin{array}{l} t_0 := a \cdot \left(-3 \cdot c\right)\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{+129}:\\ \;\;\;\;{\left(-3 \cdot \frac{a \cdot 0.5}{b}\right)}^{-1}\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + t_0}\right)}{a}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{b - \mathsf{hypot}\left(b, \sqrt{t_0}\right)}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Accuracy	85.6%
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{+126}:\\ \;\;\;\;{\left(-3 \cdot \frac{a \cdot 0.5}{b}\right)}^{-1}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{b \cdot b + a \cdot \left(-3 \cdot c\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Accuracy	80.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-124}:\\ \;\;\;\;\frac{0.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-93}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(-3 \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Accuracy	80.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{0.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(-3 \cdot c\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Accuracy	80.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{-124}:\\ \;\;\;\;\frac{0.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-93}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(-3 \cdot a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Accuracy	80.5%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{0.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-3 \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Accuracy	68.1%
Cost	964

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{c}{b} \cdot -1.5 + 2 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Accuracy	68.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Accuracy	68.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Accuracy	47.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Accuracy	68.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13
Accuracy	68.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14
Accuracy	68.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 15
Accuracy	34.4%
Cost	320

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

Reproduce

herbie shell --seed 2023162 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))