Cubic critical, medium range

Average Accuracy: 31.7% → 95.2%

Time: 21.9s

Precision: binary64

Cost: 47680

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

Math

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

↓

\[\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c}{b} \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)\right) \]

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
  -0.16666666666666666
  (* (/ (pow a 3.0) b) (* (* (* c c) (* (* c c) (pow b -6.0))) 6.328125))
  (fma
   -0.5625
   (* (/ (pow c 3.0) (pow b 5.0)) (* a a))
   (fma -0.5 (/ c b) (* -0.375 (* (/ c b) (/ (* a c) (* 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 fma(-0.16666666666666666, ((pow(a, 3.0) / b) * (((c * c) * ((c * c) * pow(b, -6.0))) * 6.328125)), fma(-0.5625, ((pow(c, 3.0) / pow(b, 5.0)) * (a * a)), fma(-0.5, (c / b), (-0.375 * ((c / b) * ((a * c) / (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 fma(-0.16666666666666666, Float64(Float64((a ^ 3.0) / b) * Float64(Float64(Float64(c * c) * Float64(Float64(c * c) * (b ^ -6.0))) * 6.328125)), fma(-0.5625, Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a)), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(c / b) * Float64(Float64(a * c) / 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[(-0.16666666666666666 * N[(N[(N[Power[a, 3.0], $MachinePrecision] / b), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 6.328125), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(c / b), $MachinePrecision] * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

2. Simplified 31.8%

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

   [Start] 31.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   neg-sub0 [=>] 31.7	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   associate-+l- [=>] 31.7	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   sub0-neg [=>] 31.7	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   neg-mul-1 [=>] 31.7	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   associate-*r/ [<=] 31.7	\[ \color{blue}{-1 \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
   *-commutative [=>] 31.7	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot -1} \]
   metadata-eval [<=] 31.7	\[ \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{1}{-1}} \]
   metadata-eval [<=] 31.7	\[ \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \frac{\color{blue}{--1}}{-1} \]
   times-frac [<=] 31.7	\[ \color{blue}{\frac{\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{\left(3 \cdot a\right) \cdot -1}} \]
   *-commutative [<=] 31.7	\[ \frac{\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
   times-frac [=>] 31.7	\[ \color{blue}{\frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{--1}{3 \cdot a}} \]

3. Applied egg-rr 14.8%

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

   [Start] 31.8	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a} \]
   add-log-exp [=>] 14.8	\[ \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \color{blue}{\log \left(e^{\frac{0.3333333333333333}{a}}\right)} \]

4. Taylor expanded in a around 0 95.2%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

5. Simplified 95.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right)} \]
   Proof

   [Start] 95.2	\[ -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   fma-def [=>] 95.2	\[ \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b}, -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

6. Applied egg-rr 95.2%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \color{blue}{\left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)}\right)\right)\right) \]
   Proof

   [Start] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right) \]
   *-commutative [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{\color{blue}{\left(a \cdot c\right) \cdot c}}{{b}^{3}}\right)\right)\right) \]
   unpow3 [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{\left(a \cdot c\right) \cdot c}{\color{blue}{\left(b \cdot b\right) \cdot b}}\right)\right)\right) \]
   times-frac [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \color{blue}{\left(\frac{a \cdot c}{b \cdot b} \cdot \frac{c}{b}\right)}\right)\right)\right) \]
   *-commutative [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{\color{blue}{c \cdot a}}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]

7. Applied egg-rr 95.2%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right)} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   Proof

   [Start] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\frac{{c}^{4}}{{b}^{6}} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   div-inv [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left({c}^{4} \cdot \frac{1}{{b}^{6}}\right)} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   metadata-eval [<=] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left({c}^{\color{blue}{\left(2 + 2\right)}} \cdot \frac{1}{{b}^{6}}\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   metadata-eval [<=] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left({c}^{\left(\color{blue}{\sqrt{4}} + 2\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   metadata-eval [<=] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left({c}^{\left(\sqrt{4} + \color{blue}{\sqrt{4}}\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   pow-prod-up [<=] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\color{blue}{\left({c}^{\left(\sqrt{4}\right)} \cdot {c}^{\left(\sqrt{4}\right)}\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   pow-prod-down [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\color{blue}{{\left(c \cdot c\right)}^{\left(\sqrt{4}\right)}} \cdot \frac{1}{{b}^{6}}\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   metadata-eval [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left({\left(c \cdot c\right)}^{\color{blue}{2}} \cdot \frac{1}{{b}^{6}}\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   pow2 [<=] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)} \cdot \frac{1}{{b}^{6}}\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   associate-*l* [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\color{blue}{\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \frac{1}{{b}^{6}}\right)\right)} \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   pow-flip [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot \color{blue}{{b}^{\left(-6\right)}}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]
   metadata-eval [=>] 95.2	\[ \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{\color{blue}{-6}}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c \cdot a}{b \cdot b} \cdot \frac{c}{b}\right)\right)\right)\right) \]

8. Final simplification 95.2%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \left(\left(\left(c \cdot c\right) \cdot \left(\left(c \cdot c\right) \cdot {b}^{-6}\right)\right) \cdot 6.328125\right), \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \left(\frac{c}{b} \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	93.7%
Cost	33536

\[\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right) \]

Alternative 2
Accuracy	93.7%
Cost	33536

\[\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{{c}^{3} \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)}{{b}^{5}}\right)\right) \]

Alternative 3
Accuracy	93.2%
Cost	27648

\[\frac{\mathsf{fma}\left(-1.125, \frac{c \cdot c}{\frac{{b}^{3}}{a \cdot a}}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right) + {\left(a \cdot c\right)}^{3} \cdot \left({b}^{-5} \cdot -1.6875\right)\right)}{a \cdot 3} \]

Alternative 4
Accuracy	90.3%
Cost	21188

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.3:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \left(\frac{1}{a} \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + -0.375 \cdot \left(c \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	90.3%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.3:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + -0.375 \cdot \left(c \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	90.3%
Cost	14852

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -1.3:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b} + -0.375 \cdot \left(c \cdot \left(c \cdot \left(a \cdot {b}^{-3}\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	90.6%
Cost	7424

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

Alternative 8
Accuracy	80.8%
Cost	320

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

Alternative 9
Accuracy	81.0%
Cost	320

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

Reproduce

herbie shell --seed 2023152 
(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)))