Cubic critical, narrow range

Average Accuracy: 55.7% → 92.2%

Time: 25.3s

Precision: binary64

Cost: 54724

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.5:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5625, c \cdot \left(\frac{a}{\frac{{b}^{5}}{a}} \cdot \left(c \cdot c\right)\right), \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma a (* c -3.0) (* b b))))
   (if (<= (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a)) -2.5)
     (/ (/ (- t_0 (* b b)) (+ b (sqrt t_0))) (* 3.0 a))
     (fma
      -0.5
      (/ c b)
      (fma
       -0.16666666666666666
       (* (/ (pow (* a c) 4.0) (pow b 7.0)) (/ 6.328125 a))
       (fma
        -0.5625
        (* c (* (/ a (/ (pow b 5.0) a)) (* c c)))
        (/ (* (* c c) -0.375) (/ (pow b 3.0) a))))))))

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 t_0 = fma(a, (c * -3.0), (b * b));
	double tmp;
	if (((sqrt(((b * b) - ((3.0 * a) * c))) - b) / (3.0 * a)) <= -2.5) {
		tmp = ((t_0 - (b * b)) / (b + sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(-0.5, (c / b), fma(-0.16666666666666666, ((pow((a * c), 4.0) / pow(b, 7.0)) * (6.328125 / a)), fma(-0.5625, (c * ((a / (pow(b, 5.0) / a)) * (c * c))), (((c * c) * -0.375) / (pow(b, 3.0) / a)))));
	}
	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)
	t_0 = fma(a, Float64(c * -3.0), Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c))) - b) / Float64(3.0 * a)) <= -2.5)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(b + sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(-0.5, Float64(c / b), fma(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / (b ^ 7.0)) * Float64(6.328125 / a)), fma(-0.5625, Float64(c * Float64(Float64(a / Float64((b ^ 5.0) / a)) * Float64(c * c))), Float64(Float64(Float64(c * c) * -0.375) / Float64((b ^ 3.0) / a)))));
	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_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.5], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(6.328125 / a), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(c * N[(N[(a / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -2.5

   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 84.2%

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

   3. Applied egg-rr 83.8%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}}}}{3 \cdot a} \]
      Proof

      [Start] 84.2	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a} \]
      sub-neg [=>] 84.2	\[ \frac{\left(-b\right) + \sqrt{\color{blue}{b \cdot b + \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      flip-+ [=>] 83.9	\[ \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}}{3 \cdot a} \]
      pow2 [=>] 83.9	\[ \frac{\left(-b\right) + \sqrt{\frac{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      pow2 [=>] 83.9	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{2} \cdot \color{blue}{{b}^{2}} - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      pow-prod-up [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{\color{blue}{{b}^{\left(2 + 2\right)}} - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      metadata-eval [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{4}} - \left(-3 \cdot \left(a \cdot c\right)\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      *-commutative [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(-\color{blue}{\left(a \cdot c\right) \cdot 3}\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      distribute-rgt-neg-in [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \color{blue}{\left(\left(a \cdot c\right) \cdot \left(-3\right)\right)} \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      metadata-eval [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot \color{blue}{-3}\right) \cdot \left(-3 \cdot \left(a \cdot c\right)\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      *-commutative [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(-\color{blue}{\left(a \cdot c\right) \cdot 3}\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      distribute-rgt-neg-in [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \color{blue}{\left(\left(a \cdot c\right) \cdot \left(-3\right)\right)}}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      metadata-eval [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot \color{blue}{-3}\right)}{b \cdot b - \left(-3 \cdot \left(a \cdot c\right)\right)}}}{3 \cdot a} \]
      *-commutative [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(-\color{blue}{\left(a \cdot c\right) \cdot 3}\right)}}}{3 \cdot a} \]
      distribute-rgt-neg-in [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \color{blue}{\left(a \cdot c\right) \cdot \left(-3\right)}}}}{3 \cdot a} \]
      metadata-eval [=>] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot \color{blue}{-3}}}}{3 \cdot a} \]

   4. Applied egg-rr 85.1%

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

      [Start] 83.8	\[ \frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}}}{3 \cdot a} \]
      +-commutative [=>] 83.8	\[ \frac{\color{blue}{\sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} + \left(-b\right)}}{3 \cdot a} \]
      flip-+ [=>] 83.4	\[ \frac{\color{blue}{\frac{\sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} \cdot \sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} - \left(-b\right) \cdot \left(-b\right)}{\sqrt{\frac{{b}^{4} - \left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right)}{b \cdot b - \left(a \cdot c\right) \cdot -3}} - \left(-b\right)}}}{3 \cdot a} \]

   5. Simplified 85.6%

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

      [Start] 85.1	\[ \frac{\frac{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      fma-def [<=] 85.5	\[ \frac{\frac{\color{blue}{\left(b \cdot b + a \cdot \left(c \cdot -3\right)\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      +-commutative [<=] 85.5	\[ \frac{\frac{\color{blue}{\left(a \cdot \left(c \cdot -3\right) + b \cdot b\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      fma-def [=>] 85.6	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b}{b + \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}{3 \cdot a} \]
      fma-def [<=] 85.5	\[ \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{b \cdot b + a \cdot \left(c \cdot -3\right)}}}}{3 \cdot a} \]
      +-commutative [<=] 85.5	\[ \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right) + b \cdot b}}}}{3 \cdot a} \]
      fma-def [=>] 85.6	\[ \frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}}{3 \cdot a} \]

   if -2.5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 51.9%

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

   2. Taylor expanded in b around inf 92.7%

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

   3. Simplified 92.7%

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

      [Start] 92.7	\[ \frac{-0.5 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}} + \left(-1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a} \]
      fma-def [=>] 92.7	\[ \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{{b}^{7}}, -1.125 \cdot \frac{{c}^{2} \cdot {a}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{c}^{3} \cdot {a}^{3}}{{b}^{5}}\right)\right)}}{3 \cdot a} \]

   4. Taylor expanded in c around 0 93.0%

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

   5. Simplified 93.1%

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

      [Start] 93.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      associate-+r+ [=>] 93.0	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(\left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + -0.5 \cdot \frac{c}{b}\right) + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      associate-+r+ [=>] 93.0	\[ \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + -0.5 \cdot \frac{c}{b}\right)\right) + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   6. Applied egg-rr 93.1%

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

      [Start] 93.1	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \frac{-0.375 \cdot \left(c \cdot c\right)}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
      associate-/r/ [=>] 93.1	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5625, \color{blue}{\frac{a \cdot a}{{b}^{5}} \cdot {c}^{3}}, \frac{-0.375 \cdot \left(c \cdot c\right)}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
      unpow3 [=>] 93.1	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{{b}^{5}} \cdot \color{blue}{\left(\left(c \cdot c\right) \cdot c\right)}, \frac{-0.375 \cdot \left(c \cdot c\right)}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
      associate-*r* [=>] 93.1	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5625, \color{blue}{\left(\frac{a \cdot a}{{b}^{5}} \cdot \left(c \cdot c\right)\right) \cdot c}, \frac{-0.375 \cdot \left(c \cdot c\right)}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
      associate-/l* [=>] 93.1	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5625, \left(\color{blue}{\frac{a}{\frac{{b}^{5}}{a}}} \cdot \left(c \cdot c\right)\right) \cdot c, \frac{-0.375 \cdot \left(c \cdot c\right)}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -2.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right) - b \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(a \cdot c\right)}^{4}}{{b}^{7}} \cdot \frac{6.328125}{a}, \mathsf{fma}\left(-0.5625, c \cdot \left(\frac{a}{\frac{{b}^{5}}{a}} \cdot \left(c \cdot c\right)\right), \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.8%
Cost	40964

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	89.8%
Cost	40964

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \frac{\left(c \cdot c\right) \cdot -0.375}{\frac{{b}^{3}}{a}}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	85.3%
Cost	28228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.35:\\ \;\;\;\;\frac{\frac{t_0 - b \cdot b}{b + \sqrt{t_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 4
Accuracy	85.1%
Cost	21188

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.35:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 5
Accuracy	85.1%
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.35:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 6
Accuracy	84.8%
Cost	21060

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.35:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot \frac{b}{c}}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}\\ \end{array} \]

Alternative 7
Accuracy	84.8%
Cost	15428

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.35:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\mathsf{fma}\left(1.5, \frac{c}{\frac{b}{a}}, 1.125 \cdot \left(\frac{c \cdot \left(a \cdot c\right)}{b \cdot b} \cdot \frac{a}{b}\right)\right)}{a}\\ \end{array} \]

Alternative 8
Accuracy	84.8%
Cost	15428

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -0.35:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot \frac{b}{c}}, -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}{3 \cdot a}\\ \end{array} \]

Alternative 9
Accuracy	76.7%
Cost	14788

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a} \leq -7.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10
Accuracy	64.0%
Cost	320

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

Alternative 11
Accuracy	64.1%
Cost	320

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

Reproduce

herbie shell --seed 2023147 
(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)))