Cubic critical, narrow range

Average Accuracy: 55.3% → 91.9%

Time: 30.9s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 0.34:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(a \cdot c\right)}^{4}}{a \cdot \left(-0.9481481481481482 \cdot {b}^{7}\right)}\right)\right)\right)\\ \end{array} \]

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 0.34)
   (/ (* -0.3333333333333333 (- b (sqrt (fma b b (* (* a c) -3.0))))) a)
   (fma
    -0.5
    (/ c b)
    (fma
     -0.375
     (* a (/ c (/ (pow b 3.0) c)))
     (fma
      -0.5625
      (/ (* a a) (* (/ 1.0 (* c c)) (/ (pow b 5.0) c)))
      (/ (pow (* a c) 4.0) (* a (* -0.9481481481481482 (pow b 7.0)))))))))

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) {
	double tmp;
	if (b <= 0.34) {
		tmp = (-0.3333333333333333 * (b - sqrt(fma(b, b, ((a * c) * -3.0))))) / a;
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.375, (a * (c / (pow(b, 3.0) / c))), fma(-0.5625, ((a * a) / ((1.0 / (c * c)) * (pow(b, 5.0) / c))), (pow((a * c), 4.0) / (a * (-0.9481481481481482 * pow(b, 7.0)))))));
	}
	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 <= 0.34)
		tmp = Float64(Float64(-0.3333333333333333 * Float64(b - sqrt(fma(b, b, Float64(Float64(a * c) * -3.0))))) / a);
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.375, Float64(a * Float64(c / Float64((b ^ 3.0) / c))), fma(-0.5625, Float64(Float64(a * a) / Float64(Float64(1.0 / Float64(c * c)) * Float64((b ^ 5.0) / c))), Float64((Float64(a * c) ^ 4.0) / Float64(a * Float64(-0.9481481481481482 * (b ^ 7.0)))))));
	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, 0.34], N[(N[(-0.3333333333333333 * N[(b - N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[(1.0 / N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, 5.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(a * N[(-0.9481481481481482 * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.340000000000000024

   1. Initial program 87.5%

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

   2. Simplified 87.7%

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

      [Start] 87.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 87.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 87.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/r/ [=>] 87.5	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \left(--1\right)} \]
      metadata-eval [=>] 87.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{1} \]
      metadata-eval [<=] 87.5	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \cdot \color{blue}{\frac{-1}{-1}} \]
      times-frac [<=] 87.5	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\left(3 \cdot a\right) \cdot -1}} \]
      *-commutative [<=] 87.5	\[ \frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}{\color{blue}{-1 \cdot \left(3 \cdot a\right)}} \]
      times-frac [=>] 87.4	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      *-commutative [<=] 87.4	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}} \]
      associate-/r* [=>] 87.5	\[ \color{blue}{\frac{\frac{-1}{3}}{a}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \]
      associate-*l/ [=>] 87.6	\[ \color{blue}{\frac{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1}}{a}} \]

   if 0.340000000000000024 < b

   1. Initial program 52.4%

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

   2. Simplified 52.6%

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

      [Start] 52.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      /-rgt-identity [<=] 52.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\frac{3 \cdot a}{1}}} \]
      metadata-eval [<=] 52.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\frac{3 \cdot a}{\color{blue}{--1}}} \]
      associate-/l* [<=] 52.4	\[ \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(--1\right)}{3 \cdot a}} \]
      associate-*r/ [<=] 52.4	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \frac{--1}{3 \cdot a}} \]
      *-commutative [=>] 52.4	\[ \color{blue}{\frac{--1}{3 \cdot a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)} \]
      associate-*l/ [=>] 52.4	\[ \color{blue}{\frac{\left(--1\right) \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{3 \cdot a}} \]
      associate-*r/ [<=] 52.4	\[ \color{blue}{\left(--1\right) \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [=>] 52.4	\[ \color{blue}{1} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      metadata-eval [<=] 52.4	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 52.4	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      neg-mul-1 [<=] 52.4	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{-3 \cdot a}} \]
      distribute-rgt-neg-in [=>] 52.4	\[ \frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{\color{blue}{3 \cdot \left(-a\right)}} \]
      times-frac [=>] 52.4	\[ \color{blue}{\frac{-1}{3} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a}} \]
      metadata-eval [=>] 52.4	\[ \color{blue}{-0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-a} \]
      neg-mul-1 [=>] 52.4	\[ -0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-1 \cdot a}} \]

   3. Taylor expanded in b around inf 93.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}\right)\right)}}{a} \]

   4. Simplified 93.1%

      \[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \mathsf{fma}\left(1.125, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{{b}^{3}}, 0.5 \cdot \frac{{\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot -1.125\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}\right)\right)\right)}}{a} \]
      Step-by-step derivation

      [Start] 93.1	\[ -0.3333333333333333 \cdot \frac{1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}} + \left(1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}\right)\right)}{a} \]
      fma-def [=>] 93.1	\[ -0.3333333333333333 \cdot \frac{\color{blue}{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, 1.5 \cdot \frac{c \cdot a}{b} + \left(1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}\right)\right)}}{a} \]
      fma-def [=>] 93.1	\[ -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, \color{blue}{\mathsf{fma}\left(1.5, \frac{c \cdot a}{b}, 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}\right)}\right)}{a} \]
      associate-/l* [=>] 93.1	\[ -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.5, \color{blue}{\frac{c}{\frac{b}{a}}}, 1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}\right)\right)}{a} \]
      fma-def [=>] 93.1	\[ -0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.6875, \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, \color{blue}{\mathsf{fma}\left(1.125, \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}}, 0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}\right)}\right)\right)}{a} \]

   5. Taylor expanded in c around 0 93.5%

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

   6. Simplified 93.6%

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

      [Start] 93.5	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      +-commutative [=>] 93.5	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} \]
      associate-+r+ [=>] 93.5	\[ \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]
      associate-/r* [=>] 93.5	\[ \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + -0.16666666666666666 \cdot \color{blue}{\frac{\frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a}}{{b}^{7}}} \]
      associate-*r/ [=>] 93.5	\[ \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) + \color{blue}{\frac{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a}}{{b}^{7}}} \]

   7. Applied egg-rr 93.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\color{blue}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}\right)\right)\right) \]
      Step-by-step derivation

      [Start] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}\right)\right)\right) \]
      *-un-lft-identity [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{\color{blue}{1 \cdot {b}^{5}}}{{c}^{3}}}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}\right)\right)\right) \]
      unpow3 [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1 \cdot {b}^{5}}{\color{blue}{\left(c \cdot c\right) \cdot c}}}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}\right)\right)\right) \]
      times-frac [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\color{blue}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}\right)\right)\right) \]

   8. Applied egg-rr 93.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{\left(a \cdot 0.1580246913580247\right) \cdot \left({b}^{7} \cdot -6\right)}}\right)\right)\right) \]
      Step-by-step derivation

      [Start] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{6.328125}} \cdot \frac{-0.16666666666666666}{{b}^{7}}\right)\right)\right) \]
      clear-num [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{\frac{a}{6.328125}} \cdot \color{blue}{\frac{1}{\frac{{b}^{7}}{-0.16666666666666666}}}\right)\right)\right) \]
      frac-times [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \color{blue}{\frac{{\left(c \cdot a\right)}^{4} \cdot 1}{\frac{a}{6.328125} \cdot \frac{{b}^{7}}{-0.16666666666666666}}}\right)\right)\right) \]
      *-commutative [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\color{blue}{1 \cdot {\left(c \cdot a\right)}^{4}}}{\frac{a}{6.328125} \cdot \frac{{b}^{7}}{-0.16666666666666666}}\right)\right)\right) \]
      *-un-lft-identity [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\color{blue}{{\left(c \cdot a\right)}^{4}}}{\frac{a}{6.328125} \cdot \frac{{b}^{7}}{-0.16666666666666666}}\right)\right)\right) \]
      div-inv [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{\color{blue}{\left(a \cdot \frac{1}{6.328125}\right)} \cdot \frac{{b}^{7}}{-0.16666666666666666}}\right)\right)\right) \]
      metadata-eval [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{\left(a \cdot \color{blue}{0.1580246913580247}\right) \cdot \frac{{b}^{7}}{-0.16666666666666666}}\right)\right)\right) \]
      div-inv [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{\left(a \cdot 0.1580246913580247\right) \cdot \color{blue}{\left({b}^{7} \cdot \frac{1}{-0.16666666666666666}\right)}}\right)\right)\right) \]
      metadata-eval [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{\left(a \cdot 0.1580246913580247\right) \cdot \left({b}^{7} \cdot \color{blue}{-6}\right)}\right)\right)\right) \]

   9. Simplified 93.6%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{a \cdot \left(-0.9481481481481482 \cdot {b}^{7}\right)}}\right)\right)\right) \]
      Step-by-step derivation

      [Start] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{\left(a \cdot 0.1580246913580247\right) \cdot \left({b}^{7} \cdot -6\right)}\right)\right)\right) \]
      associate-/l/ [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \color{blue}{\frac{\frac{{\left(c \cdot a\right)}^{4}}{{b}^{7} \cdot -6}}{a \cdot 0.1580246913580247}}\right)\right)\right) \]
      *-commutative [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\color{blue}{-6 \cdot {b}^{7}}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      metadata-eval [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\color{blue}{\frac{-1}{0.16666666666666666}} \cdot {b}^{7}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      metadata-eval [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\frac{\color{blue}{-1}}{0.16666666666666666} \cdot {b}^{7}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      associate-/r/ [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{{\left(c \cdot a\right)}^{4}}{\color{blue}{\frac{-1}{\frac{0.16666666666666666}{{b}^{7}}}}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      associate-/r/ [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{-1} \cdot \frac{0.16666666666666666}{{b}^{7}}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      times-frac [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\color{blue}{\frac{{\left(c \cdot a\right)}^{4} \cdot 0.16666666666666666}{\left(-1\right) \cdot {b}^{7}}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      metadata-eval [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{\left(--0.16666666666666666\right)}}{\left(-1\right) \cdot {b}^{7}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      distribute-rgt-neg-in [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{\color{blue}{-{\left(c \cdot a\right)}^{4} \cdot -0.16666666666666666}}{\left(-1\right) \cdot {b}^{7}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      metadata-eval [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{-{\left(c \cdot a\right)}^{4} \cdot -0.16666666666666666}{\color{blue}{-1} \cdot {b}^{7}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      neg-mul-1 [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\frac{-{\left(c \cdot a\right)}^{4} \cdot -0.16666666666666666}{\color{blue}{-{b}^{7}}}}{a \cdot 0.1580246913580247}\right)\right)\right) \]
      associate-/l/ [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \color{blue}{\frac{-{\left(c \cdot a\right)}^{4} \cdot -0.16666666666666666}{\left(a \cdot 0.1580246913580247\right) \cdot \left(-{b}^{7}\right)}}\right)\right)\right) \]
      distribute-rgt-neg-in [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{\color{blue}{{\left(c \cdot a\right)}^{4} \cdot \left(--0.16666666666666666\right)}}{\left(a \cdot 0.1580246913580247\right) \cdot \left(-{b}^{7}\right)}\right)\right)\right) \]
      metadata-eval [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{0.16666666666666666}}{\left(a \cdot 0.1580246913580247\right) \cdot \left(-{b}^{7}\right)}\right)\right)\right) \]
      distribute-rgt-neg-in [<=] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(c \cdot a\right)}^{4} \cdot 0.16666666666666666}{\color{blue}{-\left(a \cdot 0.1580246913580247\right) \cdot {b}^{7}}}\right)\right)\right) \]
      associate-/l* [=>] 93.6	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \color{blue}{\frac{{\left(c \cdot a\right)}^{4}}{\frac{-\left(a \cdot 0.1580246913580247\right) \cdot {b}^{7}}{0.16666666666666666}}}\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.34:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{1}{c \cdot c} \cdot \frac{{b}^{5}}{c}}, \frac{{\left(a \cdot c\right)}^{4}}{a \cdot \left(-0.9481481481481482 \cdot {b}^{7}\right)}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.0%
Cost	34372

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \mathsf{fma}\left(a \cdot c, -3, \mathsf{fma}\left(a \cdot c, -3, \left(a \cdot c\right) \cdot 3\right)\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 2
Accuracy	89.5%
Cost	33668

\[\begin{array}{l} \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5}{\frac{b}{c}}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	89.6%
Cost	33668

\[\begin{array}{l} \mathbf{if}\;b \leq 0.34:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	84.9%
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, \frac{-0.5}{\frac{b}{c}}\right)\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c \cdot c}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 6
Accuracy	84.8%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 7
Accuracy	84.8%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 8
Accuracy	84.8%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(b - \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -3\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 9
Accuracy	84.7%
Cost	15428

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.0005:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(\frac{c}{b} \cdot \frac{c}{b \cdot b}\right), -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{a \cdot 3}\\ \end{array} \]

Alternative 10
Accuracy	75.7%
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 11
Accuracy	64.5%
Cost	320

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

Alternative 12
Accuracy	64.6%
Cost	320

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

Reproduce

herbie shell --seed 2023158 
(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)))