Cubic critical, medium range

Average Accuracy: 30.9% → 99.0%

Time: 18.5s

Precision: binary64

Cost: 28096

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

\[\frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \mathsf{fma}\left(-6, b \cdot b, \left(c \cdot a\right) \cdot 9\right)\right)} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/
  (fma 6.0 (* c (* a (* b b))) (* -9.0 (* (* c c) (* a a))))
  (*
   a
   (*
    (+ b (sqrt (fma a (* c -3.0) (* b b))))
    (fma -6.0 (* b b) (* (* c a) 9.0))))))

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) {
	return fma(6.0, (c * (a * (b * b))), (-9.0 * ((c * c) * (a * a)))) / (a * ((b + sqrt(fma(a, (c * -3.0), (b * b)))) * fma(-6.0, (b * b), ((c * a) * 9.0))));
}

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)
	return Float64(fma(6.0, Float64(c * Float64(a * Float64(b * b))), Float64(-9.0 * Float64(Float64(c * c) * Float64(a * a)))) / Float64(a * Float64(Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))) * fma(-6.0, Float64(b * b), Float64(Float64(c * a) * 9.0)))))
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_] := N[(N[(6.0 * N[(c * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-9.0 * N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-6.0 * N[(b * b), $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Simplified 30.9%

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

   [Start] 30.9	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 30.9	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 30.9	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 30.5	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 30.5	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 30.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 30.7	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 32.3	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 32.3	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 32.3	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 32.3	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 32.3	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 32.3	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 30.7	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 30.7	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 32.4%

   \[\leadsto \color{blue}{\frac{{b}^{4} - {\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}^{2}}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}} \]

4. Taylor expanded in b around 0 99.0%

   \[\leadsto \frac{\color{blue}{6 \cdot \left(c \cdot \left(a \cdot {b}^{2}\right)\right) + -9 \cdot \left({c}^{2} \cdot {a}^{2}\right)}}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]

5. Simplified 99.0%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]
   Proof

   [Start] 99.0	\[ \frac{6 \cdot \left(c \cdot \left(a \cdot {b}^{2}\right)\right) + -9 \cdot \left({c}^{2} \cdot {a}^{2}\right)}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]
   fma-def [=>] 99.0	\[ \frac{\color{blue}{\mathsf{fma}\left(6, c \cdot \left(a \cdot {b}^{2}\right), -9 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]
   unpow2 [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \color{blue}{\left(b \cdot b\right)}\right), -9 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]
   unpow2 [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot {a}^{2}\right)\right)}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]
   unpow2 [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]

6. Applied egg-rr 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{\left(a \cdot -3\right) \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(b \cdot b\right)\right) + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) \cdot \left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)}} \]

7. Simplified 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(-3 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)\right)\right)}} \]
   Proof

   [Start] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(a \cdot -3\right) \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(b \cdot b\right)\right) + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) \cdot \left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \]
   associate-*r* [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b\right)} + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) \cdot \left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \]
   *-commutative [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b\right) + \color{blue}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   distribute-lft-in [<=] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \left(b \cdot b + \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}} \]
   fma-udef [<=] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}} \]
   associate-*l* [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{\left(a \cdot \left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)\right)} \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)} \]
   associate-*r* [<=] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{a \cdot \left(\left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)\right)}} \]
   *-commutative [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\color{blue}{\left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot -3\right)} \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)\right)} \]
   associate-*l* [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \color{blue}{\left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(-3 \cdot \mathsf{fma}\left(b, b, \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)\right)\right)}} \]

8. Taylor expanded in b around 0 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\left(-6 \cdot {b}^{2} + 9 \cdot \left(c \cdot a\right)\right)}\right)} \]

9. Simplified 99.0%

   \[\leadsto \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-6, b \cdot b, \left(c \cdot a\right) \cdot 9\right)}\right)} \]
   Proof

   [Start] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(-6 \cdot {b}^{2} + 9 \cdot \left(c \cdot a\right)\right)\right)} \]
   fma-def [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(-6, {b}^{2}, 9 \cdot \left(c \cdot a\right)\right)}\right)} \]
   unpow2 [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \mathsf{fma}\left(-6, \color{blue}{b \cdot b}, 9 \cdot \left(c \cdot a\right)\right)\right)} \]
   *-commutative [=>] 99.0	\[ \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \mathsf{fma}\left(-6, b \cdot b, \color{blue}{\left(c \cdot a\right) \cdot 9}\right)\right)} \]

10. Final simplification 99.0%

    \[\leadsto \frac{\mathsf{fma}\left(6, c \cdot \left(a \cdot \left(b \cdot b\right)\right), -9 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}{a \cdot \left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \mathsf{fma}\left(-6, b \cdot b, \left(c \cdot a\right) \cdot 9\right)\right)} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	14144

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

Alternative 2
Accuracy	93.9%
Cost	8448

\[\frac{\frac{-0.3333333333333333}{a}}{\left(\frac{\left(c \cdot a\right) \cdot 0.375 + \left(c \cdot a\right) \cdot -0.75}{{b}^{3}} + 0.6666666666666666 \cdot \frac{b}{c \cdot a}\right) + 0.5 \cdot \frac{-1}{b}} \]

Alternative 3
Accuracy	90.9%
Cost	7232

\[\frac{\frac{-0.3333333333333333}{a}}{\mathsf{fma}\left(0.6666666666666666, \frac{b}{c \cdot a}, \frac{-0.5}{b}\right)} \]

Alternative 4
Accuracy	90.8%
Cost	1088

\[\frac{\frac{-0.3333333333333333}{a}}{0.6666666666666666 \cdot \frac{b}{c \cdot a} + 0.5 \cdot \frac{-1}{b}} \]

Alternative 5
Accuracy	90.9%
Cost	960

\[\frac{\frac{-0.3333333333333333}{a}}{\frac{b \cdot 0.6666666666666666}{c \cdot a} + \frac{-0.5}{b}} \]

Alternative 6
Accuracy	81.6%
Cost	320

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

Alternative 7
Accuracy	3.2%
Cost	192

\[\frac{0}{a} \]

Reproduce

herbie shell --seed 2023125 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))