Cubic critical

Average Error: 34.8 → 10.3

Time: 17.2s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}{a} \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.06e+126)
   (* (/ (fma -1.5 (/ c (/ b a)) (* b 2.0)) a) -0.3333333333333333)
   (if (<= b 4.4e-57)
     (/ (- (sqrt (+ (* b b) (* c (* a -3.0)))) 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 tmp;
	if (b <= -1.06e+126) {
		tmp = (fma(-1.5, (c / (b / a)), (b * 2.0)) / a) * -0.3333333333333333;
	} else if (b <= 4.4e-57) {
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - 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)
	tmp = 0.0
	if (b <= -1.06e+126)
		tmp = Float64(Float64(fma(-1.5, Float64(c / Float64(b / a)), Float64(b * 2.0)) / a) * -0.3333333333333333);
	elseif (b <= 4.4e-57)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	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, -1.06e+126], N[(N[(N[(-1.5 * N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], If[LessEqual[b, 4.4e-57], 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], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0600000000000001e126

   1. Initial program 55.0

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

   2. Simplified 55.0

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333} \]
      Proof
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -3))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 3)))))) a) -1/3): 4 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 3)))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 a c))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 a) c)))))) a) -1/3): 6 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) a) -1/3): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) 1)) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= metadata-eval (/.f64 -1 -1))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1)) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 -1 a))) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= neg-mul-1_binary64 (neg.f64 a))) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 a)) (Rewrite<= metadata-eval (/.f64 -1 3))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 (neg.f64 a) 3))): 19 points increase in error, 32 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 a 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 a) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 -1 -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite=> metadata-eval 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite<= metadata-eval (neg.f64 -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around -inf 9.8

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}{a} \cdot -0.3333333333333333 \]

   4. Simplified 3.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}{a} \cdot -0.3333333333333333 \]
      Proof
      (fma.f64 -3/2 (/.f64 c (/.f64 b a)) (*.f64 b 2)): 0 points increase in error, 0 points decrease in error
      (fma.f64 -3/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 c a) b)) (*.f64 b 2)): 15 points increase in error, 25 points decrease in error
      (fma.f64 -3/2 (/.f64 (*.f64 c a) b) (Rewrite<= *-commutative_binary64 (*.f64 2 b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -3/2 (/.f64 (*.f64 c a) b)) (*.f64 2 b))): 0 points increase in error, 9 points decrease in error

   if -1.0600000000000001e126 < b < 4.39999999999999997e-57

   1. Initial program 14.2

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

   if 4.39999999999999997e-57 < b

   1. Initial program 54.0

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

   2. Simplified 54.0

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a} \cdot -0.3333333333333333} \]
      Proof
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c -3))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (*.f64 a (*.f64 c (Rewrite<= metadata-eval (neg.f64 3))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a c) (neg.f64 3)))))) a) -1/3): 4 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 a c) 3)))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 (*.f64 a c))))))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 3 a) c)))))) a) -1/3): 6 points increase in error, 4 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) a) -1/3): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) 1)) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= metadata-eval (/.f64 -1 -1))) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1)) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) a) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 -1 a))) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite<= neg-mul-1_binary64 (neg.f64 a))) -1/3): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 a)) (Rewrite<= metadata-eval (/.f64 -1 3))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 (neg.f64 a) 3))): 19 points increase in error, 32 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 a 3)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 3 a)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 3 a) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 -1 -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite=> metadata-eval 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (Rewrite<= metadata-eval (neg.f64 -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (neg.f64 -1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 (*.f64 3 a) (Rewrite=> metadata-eval 1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (Rewrite=> /-rgt-identity_binary64 (*.f64 3 a))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in b around inf 8.1

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}{a} \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.5, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}{a} \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-56}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 2
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{c \cdot 0.5}{b}\right)\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-57}:\\ \;\;\;\;\left(\sqrt{-3 \cdot \left(c \cdot a\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 3
Error	13.8
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{c \cdot 0.5}{b}\right)\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{b - \sqrt{-3 \cdot \left(c \cdot a\right)}}{\frac{a}{-0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4
Error	13.6
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-111}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{c \cdot 0.5}{b}\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5
Error	22.2
Cost	7108

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{c \cdot 0.5}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6
Error	22.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-304}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666 + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7
Error	22.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{-206}:\\ \;\;\;\;\frac{b}{a \cdot 3} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8
Error	36.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-211}:\\ \;\;\;\;b \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{-206}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10
Error	22.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 11
Error	22.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-214}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 12
Error	39.6
Cost	320

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

Reproduce

herbie shell --seed 2022332 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))