Cubic critical, medium range

Average Accuracy: 31.5% → 99.8%

Time: 19.4s

Precision: binary64

Cost: 13568

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/ (- c) (+ b (sqrt (fma a (* c -3.0) (* b 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) {
	return -c / (b + sqrt(fma(a, (c * -3.0), (b * b))));
}

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(Float64(-c) / Float64(b + sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))))
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[((-c) / N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

2. Simplified 31.5%

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

   [Start] 31.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   *-lft-identity [<=] 31.5	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   metadata-eval [<=] 31.5	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   times-frac [<=] 31.5	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
   neg-mul-1 [<=] 31.5	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
   distribute-rgt-neg-in [=>] 31.5	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
   times-frac [=>] 31.5	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
   *-commutative [=>] 31.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \cdot \frac{-1}{3}} \]

3. Applied egg-rr 31.1%

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

   [Start] 31.5	\[ \frac{b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot -0.3333333333333333 \]
   div-sub [=>] 31.1	\[ \color{blue}{\left(\frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)} \cdot -0.3333333333333333 \]
   flip-- [=>] 31.1	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \cdot -0.3333333333333333 \]
   associate-*l/ [=>] 31.1	\[ \color{blue}{\frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]

4. Simplified 32.2%

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

   [Start] 31.1	\[ \frac{\left(\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 31.1	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{-0.3333333333333333}}} \]
   associate-/r/ [=>] 31.1	\[ \color{blue}{\frac{\frac{b}{a} \cdot \frac{b}{a} - \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333} \]

5. Taylor expanded in b around 0 99.1%

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

6. Applied egg-rr 37.8%

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

   [Start] 99.1	\[ \frac{3 \cdot \frac{c}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333 \]
   expm1-log1p-u [=>] 84.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{3 \cdot \frac{c}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333\right)\right)} \]
   expm1-udef [=>] 37.7	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{3 \cdot \frac{c}{a}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot -0.3333333333333333\right)} - 1} \]
   associate-*l/ [=>] 37.7	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \frac{c}{a}\right) \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}\right)} - 1 \]
   *-commutative [=>] 37.7	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\frac{c}{a} \cdot 3\right)} \cdot -0.3333333333333333}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]
   associate-*l* [=>] 37.8	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{c}{a} \cdot \left(3 \cdot -0.3333333333333333\right)}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]
   metadata-eval [=>] 37.8	\[ e^{\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot \color{blue}{-1}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]

7. Simplified 99.4%

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

   [Start] 37.8	\[ e^{\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} - 1 \]
   expm1-def [=>] 84.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)\right)} \]
   expm1-log1p [=>] 99.4	\[ \color{blue}{\frac{\frac{c}{a} \cdot -1}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   /-rgt-identity [<=] 99.4	\[ \frac{\color{blue}{\frac{\frac{c}{a} \cdot -1}{1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l* [=>] 99.4	\[ \frac{\color{blue}{\frac{\frac{c}{a}}{\frac{1}{-1}}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   metadata-eval [=>] 99.4	\[ \frac{\frac{\frac{c}{a}}{\color{blue}{-1}}}{\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   associate-/l/ [=>] 99.4	\[ \color{blue}{\frac{\frac{c}{a}}{\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot -1}} \]
   *-commutative [<=] 99.4	\[ \frac{\frac{c}{a}}{\color{blue}{-1 \cdot \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   neg-mul-1 [<=] 99.4	\[ \frac{\frac{c}{a}}{\color{blue}{-\left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   *-lft-identity [<=] 99.4	\[ \frac{\frac{c}{a}}{-\color{blue}{1 \cdot \left(\frac{b}{a} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}} \]
   distribute-rgt-in [=>] 99.4	\[ \frac{\frac{c}{a}}{-\color{blue}{\left(\frac{b}{a} \cdot 1 + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)}} \]
   *-commutative [<=] 99.4	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{1 \cdot \frac{b}{a}} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)} \]
   associate-*r/ [=>] 99.4	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{\frac{1 \cdot b}{a}} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)} \]
   associate-*l/ [<=] 99.3	\[ \frac{\frac{c}{a}}{-\left(\color{blue}{\frac{1}{a} \cdot b} + \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot 1\right)} \]
   associate-*l/ [=>] 99.3	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot 1}{a}}\right)} \]
   associate-*r/ [<=] 99.3	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \frac{1}{a}}\right)} \]
   *-commutative [<=] 99.3	\[ \frac{\frac{c}{a}}{-\left(\frac{1}{a} \cdot b + \color{blue}{\frac{1}{a} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \]
   distribute-lft-in [<=] 99.3	\[ \frac{\frac{c}{a}}{-\color{blue}{\frac{1}{a} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]

8. Applied egg-rr 99.4%

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

   [Start] 99.4	\[ \frac{\frac{c}{a}}{-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   *-un-lft-identity [=>] 99.4	\[ \frac{\color{blue}{1 \cdot \frac{c}{a}}}{-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   neg-mul-1 [=>] 99.4	\[ \frac{1 \cdot \frac{c}{a}}{\color{blue}{-1 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   times-frac [=>] 99.4	\[ \color{blue}{\frac{1}{-1} \cdot \frac{\frac{c}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   metadata-eval [=>] 99.4	\[ \color{blue}{-1} \cdot \frac{\frac{c}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \]
   add-sqr-sqrt [=>] 99.0	\[ -1 \cdot \frac{\frac{c}{a}}{\color{blue}{\sqrt{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot \sqrt{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}} \]
   sqrt-unprod [=>] 99.4	\[ -1 \cdot \frac{\frac{c}{a}}{\color{blue}{\sqrt{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a} \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}} \]
   sqr-neg [<=] 99.4	\[ -1 \cdot \frac{\frac{c}{a}}{\sqrt{\color{blue}{\left(-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right) \cdot \left(-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}\right)}}} \]
   sqrt-unprod [<=] 0.0	\[ -1 \cdot \frac{\frac{c}{a}}{\color{blue}{\sqrt{-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}} \cdot \sqrt{-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}} \]
   add-sqr-sqrt [<=] 1.6	\[ -1 \cdot \frac{\frac{c}{a}}{\color{blue}{-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   div-inv [=>] 1.6	\[ -1 \cdot \color{blue}{\left(\frac{c}{a} \cdot \frac{1}{-\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}\right)} \]

9. Simplified 99.8%

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

   [Start] 99.4	\[ -1 \cdot \left(\frac{c}{a} \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right) \]
   mul-1-neg [=>] 99.4	\[ \color{blue}{-\frac{c}{a} \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-*l/ [=>] 99.3	\[ -\color{blue}{\frac{c \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{a}} \]
   *-rgt-identity [<=] 99.3	\[ -\frac{c \cdot \color{blue}{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot 1\right)}}{a} \]
   associate-*r/ [<=] 99.3	\[ -\color{blue}{c \cdot \frac{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot 1}{a}} \]
   associate-*r/ [<=] 99.2	\[ -c \cdot \color{blue}{\left(\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \cdot \frac{1}{a}\right)} \]
   *-commutative [<=] 99.2	\[ -c \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\right)} \]
   associate-*r/ [=>] 99.4	\[ -c \cdot \color{blue}{\frac{\frac{1}{a} \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   associate-/l* [=>] 99.3	\[ -c \cdot \color{blue}{\frac{\frac{1}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}} \]
   /-rgt-identity [<=] 99.3	\[ -\color{blue}{\frac{c \cdot \frac{\frac{1}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}{1}} \]
   associate-/l* [=>] 99.3	\[ -\color{blue}{\frac{c}{\frac{1}{\frac{\frac{1}{a}}{\frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{a}}}}} \]
   associate-/l* [<=] 99.5	\[ -\frac{c}{\frac{1}{\color{blue}{\frac{\frac{1}{a} \cdot a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}} \]

10. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	91.0%
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3} \leq -1650:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{a}}{\frac{c}{b} \cdot 1.5 + \frac{b}{a} \cdot -2}\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	7552

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

Alternative 3
Accuracy	90.7%
Cost	960

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

Alternative 4
Accuracy	90.8%
Cost	960

\[\frac{\frac{c}{a}}{\frac{c}{b} \cdot 1.5 + \frac{b}{a} \cdot -2} \]

Alternative 5
Accuracy	81.2%
Cost	320

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

Reproduce

herbie shell --seed 2023135 
(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)))