Cubic critical, wide range

Average Error: 52.3 → 50.1

Time: 36.1s

Precision: binary64

Cost: 47364

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(a \cdot -3\right)\\ t_1 := \sqrt{b \cdot b + t_0} - b\\ t_2 := \mathsf{fma}\left(b, b, t_0\right)\\ \mathbf{if}\;\frac{t_1}{a \cdot 3} \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t_1 \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{\left({t_2}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{t_2}}}{a \cdot 3}\\ \end{array} \]

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 (* c (* a -3.0)))
        (t_1 (- (sqrt (+ (* b b) t_0)) b))
        (t_2 (fma b b t_0)))
   (if (<= (/ t_1 (* a 3.0)) -5e-33)
     (* t_1 (sqrt (/ 0.1111111111111111 (* a a))))
     (/
      (+ (- b) (sqrt (* (pow (pow t_2 2.0) 0.3333333333333333) (cbrt t_2))))
      (* a 3.0)))))

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 = c * (a * -3.0);
	double t_1 = sqrt(((b * b) + t_0)) - b;
	double t_2 = fma(b, b, t_0);
	double tmp;
	if ((t_1 / (a * 3.0)) <= -5e-33) {
		tmp = t_1 * sqrt((0.1111111111111111 / (a * a)));
	} else {
		tmp = (-b + sqrt((pow(pow(t_2, 2.0), 0.3333333333333333) * cbrt(t_2)))) / (a * 3.0);
	}
	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(c * Float64(a * -3.0))
	t_1 = Float64(sqrt(Float64(Float64(b * b) + t_0)) - b)
	t_2 = fma(b, b, t_0)
	tmp = 0.0
	if (Float64(t_1 / Float64(a * 3.0)) <= -5e-33)
		tmp = Float64(t_1 * sqrt(Float64(0.1111111111111111 / Float64(a * a))));
	else
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(((t_2 ^ 2.0) ^ 0.3333333333333333) * cbrt(t_2)))) / Float64(a * 3.0));
	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_] := Block[{t$95$0 = N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$2 = N[(b * b + t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -5e-33], N[(t$95$1 * N[Sqrt[N[(0.1111111111111111 / N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[((-b) + N[Sqrt[N[(N[Power[N[Power[t$95$2, 2.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -5.00000000000000028e-33

   1. Initial program 24.5

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

   2. Simplified 24.5

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]
      Proof
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a -3)))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (*.f64 a (Rewrite<= metadata-eval (neg.f64 3)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a 3)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (*.f64 c (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 a)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 c (*.f64 3 a)))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (fma.f64 b b (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 3 a) c))))) b) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 (sqrt.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) b) (/.f64 1/3 a)): 3 points increase in error, 6 points decrease in error
      (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))) (neg.f64 b))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) 1)) (/.f64 1/3 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)))) (Rewrite<= metadata-eval (*.f64 -1 -1))) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1)) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1)) -1) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite=> associate-/l*_binary64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (/.f64 -1 -1))) -1) (/.f64 1/3 a)): 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 1)) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite=> /-rgt-identity_binary64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) -1) (/.f64 1/3 a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (Rewrite<= metadata-eval (/.f64 1 3)) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (/.f64 (/.f64 (Rewrite<= metadata-eval (neg.f64 -1)) 3) a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 -1) (*.f64 3 a)))): 14 points increase in error, 13 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 -1)) (*.f64 -1 (*.f64 3 a)))): 6 points increase in error, 9 points decrease in error
      (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (neg.f64 -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 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 (neg.f64 -1) -1))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) (/.f64 (Rewrite=> metadata-eval 1) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (-.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
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) -1) (*.f64 3 a))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 b (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c))))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 b)) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 25.4

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25}, {\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{0.25}, -b\right)} \cdot \frac{0.3333333333333333}{a} \]

   4. Simplified 24.5

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - b\right)} \cdot \frac{0.3333333333333333}{a} \]
      Proof
      (-.f64 (sqrt.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b))) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= unpow1/2_binary64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/2)) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) (Rewrite<= metadata-eval (*.f64 2 1/4))) b): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/4) (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/4))) b): 63 points increase in error, 68 points decrease in error
      (Rewrite=> fma-neg_binary64 (fma.f64 (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/4) (pow.f64 (fma.f64 c (*.f64 a -3) (*.f64 b b)) 1/4) (neg.f64 b))): 113 points increase in error, 142 points decrease in error

   5. Applied egg-rr 24.5

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

   6. Applied egg-rr 24.5

      \[\leadsto \left(\sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b} - b\right) \cdot \color{blue}{\sqrt{\frac{0.1111111111111111}{a \cdot a}}} \]

   if -5.00000000000000028e-33 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 61.8

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

   2. Applied egg-rr 58.8

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

   3. Applied egg-rr 58.7

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right) \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{\left({\left(\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}{a \cdot 3}\\ \end{array} \]

Alternatives

Alternative 1
Error	50.6
Cost	91968

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ t_1 := \sqrt[3]{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\\ \mathsf{fma}\left(\left|t_1\right|, \sqrt{\left({t_0}^{0.16666666666666666} \cdot \sqrt[3]{\sqrt[3]{t_0}}\right) \cdot \sqrt{\sqrt[3]{t_1}}}, -b\right) \cdot \frac{0.3333333333333333}{a} \end{array} \]

Alternative 2
Error	50.7
Cost	85632

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ t_1 := {t_0}^{0.16666666666666666}\\ \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(\left|\sqrt[3]{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right|, \sqrt{\left(t_1 \cdot \sqrt[3]{\sqrt[3]{t_0}}\right) \cdot \sqrt[3]{t_1}}, -b\right) \end{array} \]

Alternative 3
Error	50.7
Cost	79232

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(\left|\sqrt[3]{t_1}\right|, \sqrt{\left({t_0}^{0.16666666666666666} \cdot \sqrt[3]{\sqrt[3]{t_0}}\right) \cdot {t_1}^{0.05555555555555555}}, -b\right) \end{array} \]

Alternative 4
Error	50.7
Cost	72320

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\\ \frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(\left|\sqrt[3]{t_0}\right|, \sqrt{\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)}} \cdot {\left({t_0}^{0.1111111111111111}\right)}^{2}}, -b\right) \end{array} \]

Alternative 5
Error	50.8
Cost	58944

\[\mathsf{fma}\left(\left|\sqrt[3]{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right|, \sqrt{{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{0.3333333333333333}}, -b\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.3333333333333333}{a}\right)\right) \]

Alternative 6
Error	50.8
Cost	46144

\[\frac{0.3333333333333333}{a} \cdot \mathsf{fma}\left(\left|\sqrt[3]{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right|, \sqrt{{\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{0.3333333333333333}}, -b\right) \]

Alternative 7
Error	50.1
Cost	34052

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\\ \mathbf{if}\;\frac{t_0}{a \cdot 3} \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{{\left({\left(\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)\right)}^{3}\right)}^{0.3333333333333333}}\right)\\ \end{array} \]

Alternative 8
Error	50.1
Cost	27652

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\\ \mathbf{if}\;\frac{t_0}{a \cdot 3} \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{a \cdot 3}\\ \end{array} \]

Alternative 9
Error	50.3
Cost	21316

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\\ \mathbf{if}\;\frac{t_0}{a \cdot 3} \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333 + \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{a \cdot -3}\\ \end{array} \]

Alternative 10
Error	52.3
Cost	13888

\[\left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right) \cdot \sqrt{\frac{0.1111111111111111}{a \cdot a}} \]

Alternative 11
Error	52.3
Cost	7360

\[\frac{0.3333333333333333}{a} \cdot \left(\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b\right) \]

Alternative 12
Error	52.3
Cost	7360

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

Reproduce

herbie shell --seed 2022334 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))