Cubic critical

Average Error: 34.2 → 6.7

Time: 26.6s

Precision: binary64

Cost: 7948

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{+88}:\\ \;\;\;\;\frac{\frac{c}{\frac{a}{-a}}}{b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

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 -2e+154)
   (/ (fma 1.5 (* (/ c b) a) (* b -2.0)) (* a 3.0))
   (if (<= b -1.7e-283)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (if (<= b 1e+88)
       (/ (/ c (/ a (- a))) (+ b (sqrt (+ (* b b) (* (* c a) -3.0)))))
       (* (/ c b) -0.5)))))

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 <= -2e+154) {
		tmp = fma(1.5, ((c / b) * a), (b * -2.0)) / (a * 3.0);
	} else if (b <= -1.7e-283) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else if (b <= 1e+88) {
		tmp = (c / (a / -a)) / (b + sqrt(((b * b) + ((c * a) * -3.0))));
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= -2e+154)
		tmp = Float64(fma(1.5, Float64(Float64(c / b) * a), Float64(b * -2.0)) / Float64(a * 3.0));
	elseif (b <= -1.7e-283)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	elseif (b <= 1e+88)
		tmp = Float64(Float64(c / Float64(a / Float64(-a))) / Float64(b + sqrt(Float64(Float64(b * b) + Float64(Float64(c * a) * -3.0)))));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return 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, -2e+154], N[(N[(1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.7e-283], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+88], N[(N[(c / N[(a / (-a)), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.00000000000000007e154

   1. Initial program 64.0

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

   2. Taylor expanded in b around -inf 12.9

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{3 \cdot a} \]

   3. Simplified 3.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, b \cdot -2\right)}}{3 \cdot a} \]
      Proof

      [Start] 12.9	\[ \frac{1.5 \cdot \frac{c \cdot a}{b} + -2 \cdot b}{3 \cdot a} \]
      fma-def [=>] 12.9	\[ \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, -2 \cdot b\right)}}{3 \cdot a} \]
      associate-/l* [=>] 3.6	\[ \frac{\mathsf{fma}\left(1.5, \color{blue}{\frac{c}{\frac{b}{a}}}, -2 \cdot b\right)}{3 \cdot a} \]
      associate-/r/ [=>] 3.6	\[ \frac{\mathsf{fma}\left(1.5, \color{blue}{\frac{c}{b} \cdot a}, -2 \cdot b\right)}{3 \cdot a} \]
      *-commutative [=>] 3.6	\[ \frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, \color{blue}{b \cdot -2}\right)}{3 \cdot a} \]

   if -2.00000000000000007e154 < b < -1.6999999999999999e-283

   1. Initial program 8.1

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

   if -1.6999999999999999e-283 < b < 9.99999999999999959e87

   1. Initial program 31.7

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

   2. Simplified 31.8

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

      [Start] 31.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      *-lft-identity [<=] 31.7	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 31.7	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 31.7	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      neg-mul-1 [<=] 31.7	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
      distribute-rgt-neg-in [=>] 31.7	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
      times-frac [=>] 31.8	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
      *-commutative [=>] 31.8	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

   3. Applied egg-rr 36.3

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

   4. Simplified 17.2

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

      [Start] 36.3	\[ \frac{\left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) \cdot -0.3333333333333333}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
      *-commutative [=>] 36.3	\[ \frac{\color{blue}{-0.3333333333333333 \cdot \left(b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
      times-frac [=>] 31.8	\[ \color{blue}{\frac{-0.3333333333333333}{a} \cdot \frac{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      fma-udef [=>] 31.8	\[ \frac{-0.3333333333333333}{a} \cdot \frac{b \cdot b - \color{blue}{\left(a \cdot \left(c \cdot -3\right) + b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      associate-*l* [<=] 31.8	\[ \frac{-0.3333333333333333}{a} \cdot \frac{b \cdot b - \left(\color{blue}{\left(a \cdot c\right) \cdot -3} + b \cdot b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      +-commutative [=>] 31.8	\[ \frac{-0.3333333333333333}{a} \cdot \frac{b \cdot b - \color{blue}{\left(b \cdot b + \left(a \cdot c\right) \cdot -3\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      associate--r+ [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(a \cdot c\right) \cdot -3}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      distribute-lft-out-- [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{\color{blue}{b \cdot \left(b - b\right)} - \left(a \cdot c\right) \cdot -3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      *-commutative [<=] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{b \cdot \left(b - b\right) - \color{blue}{\left(c \cdot a\right)} \cdot -3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      associate-*l* [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{b \cdot \left(b - b\right) - \color{blue}{c \cdot \left(a \cdot -3\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]

   5. Applied egg-rr 17.2

      \[\leadsto \frac{-0.3333333333333333}{a} \cdot \color{blue}{\left(\frac{b \cdot \left(b - b\right) - c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} + \frac{0}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \]

   6. Simplified 17.2

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

      [Start] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \left(\frac{b \cdot \left(b - b\right) - c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} + \frac{0}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right) \]
      div0 [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \left(\frac{b \cdot \left(b - b\right) - c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} + \color{blue}{0}\right) \]
      +-rgt-identity [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \color{blue}{\frac{b \cdot \left(b - b\right) - c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      +-inverses [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{b \cdot \color{blue}{0} - c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      mul0-rgt [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{\color{blue}{0} - c \cdot \left(a \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      neg-sub0 [<=] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{\color{blue}{-c \cdot \left(a \cdot -3\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      associate-*r* [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{-\color{blue}{\left(c \cdot a\right) \cdot -3}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      distribute-rgt-neg-in [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{\color{blue}{\left(c \cdot a\right) \cdot \left(--3\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      metadata-eval [=>] 17.2	\[ \frac{-0.3333333333333333}{a} \cdot \frac{\left(c \cdot a\right) \cdot \color{blue}{3}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]

   7. Applied egg-rr 50.4

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{a} \cdot \frac{\left(c \cdot a\right) \cdot 3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} - 1} \]

   8. Simplified 9.5

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

      [Start] 50.4	\[ e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{a} \cdot \frac{\left(c \cdot a\right) \cdot 3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} - 1 \]
      expm1-def [=>] 28.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{a} \cdot \frac{\left(c \cdot a\right) \cdot 3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)\right)} \]
      expm1-log1p [=>] 17.2	\[ \color{blue}{\frac{-0.3333333333333333}{a} \cdot \frac{\left(c \cdot a\right) \cdot 3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      associate-*l/ [=>] 17.2	\[ \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\left(c \cdot a\right) \cdot 3}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}} \]
      *-commutative [=>] 17.2	\[ \frac{-0.3333333333333333 \cdot \frac{\color{blue}{3 \cdot \left(c \cdot a\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a} \]
      associate-/l* [=>] 17.4	\[ \frac{-0.3333333333333333 \cdot \color{blue}{\frac{3}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{c \cdot a}}}}{a} \]
      associate-*r/ [=>] 17.3	\[ \frac{\color{blue}{\frac{-0.3333333333333333 \cdot 3}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{c \cdot a}}}}{a} \]
      metadata-eval [=>] 17.3	\[ \frac{\frac{\color{blue}{-1}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{c \cdot a}}}{a} \]
      associate-/l* [<=] 17.1	\[ \frac{\color{blue}{\frac{-1 \cdot \left(c \cdot a\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{a} \]
      associate-/r* [<=] 21.8	\[ \color{blue}{\frac{-1 \cdot \left(c \cdot a\right)}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot a}} \]
      associate-/l/ [<=] 16.4	\[ \color{blue}{\frac{\frac{-1 \cdot \left(c \cdot a\right)}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
      mul-1-neg [=>] 16.4	\[ \frac{\frac{\color{blue}{-c \cdot a}}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      distribute-rgt-neg-in [=>] 16.4	\[ \frac{\frac{\color{blue}{c \cdot \left(-a\right)}}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
      associate-/l* [=>] 9.5	\[ \frac{\color{blue}{\frac{c}{\frac{a}{-a}}}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]

   9. Applied egg-rr 9.5

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

   if 9.99999999999999959e87 < b

   1. Initial program 58.5

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

   2. Simplified 58.5

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

      [Start] 58.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 58.5	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 58.5	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 59.2	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 59.2	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 60.1	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 60.1	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 62.7	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 62.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 62.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 62.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 62.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 62.7	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 60.1	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 60.1	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 3.1

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq -1.7 \cdot 10^{-283}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{+88}:\\ \;\;\;\;\frac{\frac{c}{\frac{a}{-a}}}{b + \sqrt{b \cdot b + \left(c \cdot a\right) \cdot -3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-110}:\\ \;\;\;\;\left(b - \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 2
Error	10.2
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -0.000115:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-112}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 4
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-7}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-114}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 5
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-7}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{-110}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{c \cdot \left(a \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 6
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-7}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 7
Error	14.1
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, \frac{c}{b} \cdot a, b \cdot -2\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 8
Error	22.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-248}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 9
Error	22.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-248}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 10
Error	39.9
Cost	320

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

Reproduce

herbie shell --seed 2023187 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))