Cubic critical, medium range

Average Accuracy: 31.5% → 99.5%

Time: 27.3s

Precision: binary64

Cost: 13696

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

Math

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

↓

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

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (- (- b) (sqrt (fma a (* -3.0 c) (* b b)))) c)))

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 1.0 / ((-b - sqrt(fma(a, (-3.0 * c), (b * b)))) / c);
}

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

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

   [Start] 31.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   remove-double-neg [<=] 31.5	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
   sub-neg [<=] 31.5	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   div-sub [=>] 31.1	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   neg-mul-1 [=>] 31.1	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-*l/ [<=] 31.3	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   distribute-frac-neg [=>] 31.3	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   fma-neg [=>] 32.8	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
   /-rgt-identity [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   metadata-eval [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   associate-/l* [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   *-commutative [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   neg-mul-1 [<=] 32.8	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
   fma-neg [<=] 31.3	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
   neg-mul-1 [=>] 31.3	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

3. Applied egg-rr 32.4%

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

   [Start] 31.5	\[ \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \frac{-0.3333333333333333}{a} \]
   *-commutative [=>] 31.5	\[ \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)} \]
   clear-num [=>] 31.5	\[ \color{blue}{\frac{1}{\frac{a}{-0.3333333333333333}}} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \]
   flip-- [=>] 31.5	\[ \frac{1}{\frac{a}{-0.3333333333333333}} \cdot \color{blue}{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   frac-times [=>] 31.5	\[ \color{blue}{\frac{1 \cdot \left(b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\frac{a}{-0.3333333333333333} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}} \]
   associate-/l* [=>] 31.5	\[ \color{blue}{\frac{1}{\frac{\frac{a}{-0.3333333333333333} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}} \]
   div-inv [=>] 31.5	\[ \frac{1}{\frac{\color{blue}{\left(a \cdot \frac{1}{-0.3333333333333333}\right)} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   metadata-eval [=>] 31.5	\[ \frac{1}{\frac{\left(a \cdot \color{blue}{-3}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]
   add-sqr-sqrt [<=] 32.4	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}} \]

4. Applied egg-rr 99.1%

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

   [Start] 32.4	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{b \cdot b - \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}} \]
   sub-neg [=>] 32.4	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\color{blue}{b \cdot b + \left(-\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}} \]
   +-commutative [=>] 32.4	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\color{blue}{\left(-\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right) + b \cdot b}}} \]
   fma-udef [=>] 32.4	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\left(a \cdot \left(c \cdot -3\right) + b \cdot b\right)}\right) + b \cdot b}} \]
   distribute-neg-in [=>] 32.4	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\color{blue}{\left(\left(-a \cdot \left(c \cdot -3\right)\right) + \left(-b \cdot b\right)\right)} + b \cdot b}} \]
   associate-+l+ [=>] 99.1	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\frac{0 - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}{a \cdot \left(c \cdot 3\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   Proof

   [Start] 99.1	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-a \cdot \left(c \cdot -3\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   sqrt-unprod [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\sqrt{\left(a \cdot \left(c \cdot -3\right)\right) \cdot \left(a \cdot \left(c \cdot -3\right)\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   sqr-neg [<=] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\sqrt{\color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right) \cdot \left(-a \cdot \left(c \cdot -3\right)\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   sqrt-unprod [<=] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\sqrt{-a \cdot \left(c \cdot -3\right)} \cdot \sqrt{-a \cdot \left(c \cdot -3\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   add-sqr-sqrt [<=] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   neg-sub0 [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\left(0 - a \cdot \left(c \cdot -3\right)\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   metadata-eval [<=] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\left(\color{blue}{\log 1} - a \cdot \left(c \cdot -3\right)\right)\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   flip-- [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\frac{\log 1 \cdot \log 1 - \left(a \cdot \left(c \cdot -3\right)\right) \cdot \left(a \cdot \left(c \cdot -3\right)\right)}{\log 1 + a \cdot \left(c \cdot -3\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   metadata-eval [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{\color{blue}{0} \cdot \log 1 - \left(a \cdot \left(c \cdot -3\right)\right) \cdot \left(a \cdot \left(c \cdot -3\right)\right)}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   metadata-eval [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 \cdot \color{blue}{0} - \left(a \cdot \left(c \cdot -3\right)\right) \cdot \left(a \cdot \left(c \cdot -3\right)\right)}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   metadata-eval [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{\color{blue}{0} - \left(a \cdot \left(c \cdot -3\right)\right) \cdot \left(a \cdot \left(c \cdot -3\right)\right)}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   sqr-neg [<=] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - \color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right) \cdot \left(-a \cdot \left(c \cdot -3\right)\right)}}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   pow2 [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - \color{blue}{{\left(-a \cdot \left(c \cdot -3\right)\right)}^{2}}}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   distribute-rgt-neg-in [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - {\color{blue}{\left(a \cdot \left(-c \cdot -3\right)\right)}}^{2}}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   distribute-rgt-neg-in [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - {\left(a \cdot \color{blue}{\left(c \cdot \left(--3\right)\right)}\right)}^{2}}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   metadata-eval [=>] 1.6	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - {\left(a \cdot \left(c \cdot \color{blue}{3}\right)\right)}^{2}}{\log 1 + a \cdot \left(c \cdot -3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}{\log 1 + \color{blue}{\sqrt{a \cdot \left(c \cdot -3\right)} \cdot \sqrt{a \cdot \left(c \cdot -3\right)}}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   sqrt-unprod [=>] 99.0	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}{\log 1 + \color{blue}{\sqrt{\left(a \cdot \left(c \cdot -3\right)\right) \cdot \left(a \cdot \left(c \cdot -3\right)\right)}}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   sqr-neg [<=] 99.0	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}{\log 1 + \sqrt{\color{blue}{\left(-a \cdot \left(c \cdot -3\right)\right) \cdot \left(-a \cdot \left(c \cdot -3\right)\right)}}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]

6. Simplified 99.2%

   \[\leadsto \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\color{blue}{\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   Proof

   [Start] 99.0	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{0 - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}{a \cdot \left(c \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   sub0-neg [=>] 99.0	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{\color{blue}{-{\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{a \cdot \left(c \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   associate-*r* [=>] 98.7	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{-{\color{blue}{\left(\left(a \cdot c\right) \cdot 3\right)}}^{2}}{a \cdot \left(c \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   *-commutative [<=] 98.7	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{-{\left(\color{blue}{\left(c \cdot a\right)} \cdot 3\right)}^{2}}{a \cdot \left(c \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   associate-*l* [=>] 98.9	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{-{\color{blue}{\left(c \cdot \left(a \cdot 3\right)\right)}}^{2}}{a \cdot \left(c \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   associate-*r* [=>] 98.9	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{\color{blue}{\left(a \cdot c\right) \cdot 3}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   *-commutative [<=] 98.9	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{\color{blue}{\left(c \cdot a\right)} \cdot 3}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   associate-*l* [=>] 99.2	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{\color{blue}{c \cdot \left(a \cdot 3\right)}}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]

7. Applied egg-rr 99.0%

   \[\leadsto \frac{1}{\color{blue}{a \cdot \left(\left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}\right)}} \]
   Proof

   [Start] 99.2	\[ \frac{1}{\frac{\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   div-inv [=>] 99.1	\[ \frac{1}{\color{blue}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \frac{1}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}}} \]
   associate-*l* [=>] 99.0	\[ \frac{1}{\color{blue}{\left(a \cdot \left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)\right)} \cdot \frac{1}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}} \]
   associate-*l* [=>] 99.0	\[ \frac{1}{\color{blue}{a \cdot \left(\left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \frac{1}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)}\right) + \left(\left(-b \cdot b\right) + b \cdot b\right)}\right)}} \]
   +-commutative [=>] 99.0	\[ \frac{1}{a \cdot \left(\left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \frac{1}{\left(-\frac{-{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)}\right) + \color{blue}{\left(b \cdot b + \left(-b \cdot b\right)\right)}}\right)} \]

8. Simplified 99.5%

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

   [Start] 99.0	\[ \frac{1}{a \cdot \left(\left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}\right)} \]
   associate-*r* [=>] 99.0	\[ \frac{1}{\color{blue}{\left(a \cdot \left(-3 \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)\right) \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}}} \]
   associate-*r* [=>] 99.1	\[ \frac{1}{\color{blue}{\left(\left(a \cdot -3\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right)} \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}} \]
   metadata-eval [<=] 99.1	\[ \frac{1}{\left(\left(a \cdot \color{blue}{\left(-3\right)}\right) \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}} \]
   distribute-rgt-neg-in [<=] 99.1	\[ \frac{1}{\left(\color{blue}{\left(-a \cdot 3\right)} \cdot \left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\right) \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}} \]
   *-commutative [<=] 99.1	\[ \frac{1}{\color{blue}{\left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(-a \cdot 3\right)\right)} \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}} \]
   distribute-rgt-neg-out [=>] 99.1	\[ \frac{1}{\color{blue}{\left(-\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)\right)} \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}} \]
   distribute-lft-neg-out [=>] 99.1	\[ \frac{1}{\color{blue}{-\left(\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)\right) \cdot \frac{1}{\frac{{\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}{c \cdot \left(a \cdot 3\right)} + \left(b \cdot b\right) \cdot 0}}} \]

9. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	90.8%
Cost	832

\[\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \]

Alternative 2
Accuracy	81.1%
Cost	320

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

Alternative 3
Accuracy	3.2%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023136 
(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)))