Cubic critical, narrow range

Average Accuracy: 55.6% → 91.4%

Time: 24.5s

Precision: binary64

Cost: 47428

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(c \cdot \left(c \cdot a\right)\right)}{\frac{{b}^{5}}{c}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}, -0.375 \cdot \frac{c \cdot a}{\frac{{b}^{3}}{c}}\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.0)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (fma
    -0.5
    (/ c b)
    (fma
     -0.5625
     (/ (* a (* c (* c a))) (/ (pow b 5.0) c))
     (fma
      (/ -0.16666666666666666 a)
      (/ (* (pow (* c a) 4.0) 6.328125) (pow b 7.0))
      (* -0.375 (/ (* c a) (/ (pow b 3.0) 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) {
	double tmp;
	if (b <= 4.0) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.5625, ((a * (c * (c * a))) / (pow(b, 5.0) / c)), fma((-0.16666666666666666 / a), ((pow((c * a), 4.0) * 6.328125) / pow(b, 7.0)), (-0.375 * ((c * a) / (pow(b, 3.0) / c))))));
	}
	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)
	tmp = 0.0
	if (b <= 4.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.5625, Float64(Float64(a * Float64(c * Float64(c * a))) / Float64((b ^ 5.0) / c)), fma(Float64(-0.16666666666666666 / a), Float64(Float64((Float64(c * a) ^ 4.0) * 6.328125) / (b ^ 7.0)), Float64(-0.375 * Float64(Float64(c * a) / Float64((b ^ 3.0) / c))))));
	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_] := If[LessEqual[b, 4.0], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.5625 * N[(N[(a * N[(c * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 / a), $MachinePrecision] * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(c * a), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4

   1. Initial program 84.2%

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

   2. Simplified 84.5%

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

      [Start] 84.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      neg-sub0 [=>] 84.2	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-+l- [=>] 84.2	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      sub0-neg [=>] 84.2	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      neg-mul-1 [=>] 84.2	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      associate-*r/ [<=] 84.2	\[ \color{blue}{-1 \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 84.2	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      metadata-eval [<=] 84.2	\[ \frac{\color{blue}{--1}}{-1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 84.2	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [=>] 84.2	\[ \frac{\left(--1\right) \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{\left(3 \cdot a\right) \cdot -1}} \]
      times-frac [=>] 84.2	\[ \color{blue}{\frac{--1}{3 \cdot a} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-*l/ [=>] 84.2	\[ \color{blue}{\frac{\left(--1\right) \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{3 \cdot a}} \]

   if 4 < b

   1. Initial program 49.3%

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

   2. Simplified 49.5%

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

      [Start] 49.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      neg-sub0 [=>] 49.3	\[ \frac{\color{blue}{\left(0 - b\right)} + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-+l- [=>] 49.3	\[ \frac{\color{blue}{0 - \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      sub0-neg [=>] 49.3	\[ \frac{\color{blue}{-\left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      neg-mul-1 [=>] 49.3	\[ \frac{\color{blue}{-1 \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      associate-*r/ [<=] 49.3	\[ \color{blue}{-1 \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 49.3	\[ \color{blue}{\frac{1}{-1}} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      metadata-eval [<=] 49.3	\[ \frac{\color{blue}{--1}}{-1} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 49.3	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [=>] 49.3	\[ \frac{\left(--1\right) \cdot \left(b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{\left(3 \cdot a\right) \cdot -1}} \]
      times-frac [=>] 49.3	\[ \color{blue}{\frac{--1}{3 \cdot a} \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-*l/ [=>] 49.3	\[ \color{blue}{\frac{\left(--1\right) \cdot \frac{b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 94.0%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}} + \left(-1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}}{3 \cdot a} \]

   4. Simplified 94.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(5.0625, {c}^{4} \cdot {a}^{4}, \left({c}^{4} \cdot {a}^{4}\right) \cdot 1.265625\right)}{{b}^{7}}, \mathsf{fma}\left(-1.125, \frac{c \cdot c}{{b}^{3}} \cdot \left(a \cdot a\right), \mathsf{fma}\left(-1.6875, \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}, -1.5 \cdot \left(\frac{c}{b} \cdot a\right)\right)\right)\right)}}{3 \cdot a} \]
      Proof

      [Start] 94.0	\[ \frac{-0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}} + \left(-1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]
      fma-def [=>] 94.0	\[ \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}, -1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}}{3 \cdot a} \]

   5. Taylor expanded in c around 0 94.4%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   6. Simplified 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{\left(c \cdot a\right)}^{2}}{\frac{{b}^{5}}{c}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}, -0.375 \cdot \frac{c \cdot a}{\frac{{b}^{3}}{c}}\right)\right)\right)} \]
      Proof

      [Start] 94.4	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      +-commutative [=>] 94.4	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} \]
      associate-+r+ [=>] 94.4	\[ \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]
      associate-*r/ [=>] 94.4	\[ \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + \color{blue}{\frac{-0.16666666666666666 \cdot \left({c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)\right)}{a \cdot {b}^{7}}} \]
      times-frac [=>] 94.4	\[ \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + \color{blue}{\frac{-0.16666666666666666}{a} \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{{b}^{7}}} \]

   7. Applied egg-rr 94.5%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot c\right) \cdot a}}{\frac{{b}^{5}}{c}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}, -0.375 \cdot \frac{c \cdot a}{\frac{{b}^{3}}{c}}\right)\right)\right) \]
      Proof

      [Start] 94.5	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{\left(c \cdot a\right)}^{2}}{\frac{{b}^{5}}{c}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}, -0.375 \cdot \frac{c \cdot a}{\frac{{b}^{3}}{c}}\right)\right)\right) \]
      unpow2 [=>] 94.5	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{\color{blue}{\left(c \cdot a\right) \cdot \left(c \cdot a\right)}}{\frac{{b}^{5}}{c}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}, -0.375 \cdot \frac{c \cdot a}{\frac{{b}^{3}}{c}}\right)\right)\right) \]
      associate-*r* [=>] 94.5	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{\color{blue}{\left(\left(c \cdot a\right) \cdot c\right) \cdot a}}{\frac{{b}^{5}}{c}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}, -0.375 \cdot \frac{c \cdot a}{\frac{{b}^{3}}{c}}\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{a \cdot \left(c \cdot \left(c \cdot a\right)\right)}{\frac{{b}^{5}}{c}}, \mathsf{fma}\left(\frac{-0.16666666666666666}{a}, \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}, -0.375 \cdot \frac{c \cdot a}{\frac{{b}^{3}}{c}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.5%
Cost	46596

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{e^{\mathsf{log1p}\left(\frac{{b}^{3}}{a}\right)} + -1}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	89.5%
Cost	27460

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \left(c \cdot c\right) \cdot \left(c \cdot \frac{a \cdot a}{{b}^{5}}\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	75.9%
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4
Accuracy	85.2%
Cost	13828

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 6
Accuracy	85.0%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 7
Accuracy	85.0%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 8
Accuracy	84.9%
Cost	8132

\[\begin{array}{l} \mathbf{if}\;b \leq 4:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 9
Accuracy	64.2%
Cost	320

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

Alternative 10
Accuracy	64.2%
Cost	320

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

Reproduce

herbie shell --seed 2023153 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))