Cubic critical, narrow range

Average Error: 28.7 → 5.0

Time: 23.5s

Precision: binary64

Cost: 53700

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 0.0095:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + {c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)\right)\right)\\ \end{array} \]

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 0.0095)
   (*
    (+ (- b) (sqrt (- (* b b) (* c (/ a 0.3333333333333333)))))
    (/ 0.3333333333333333 a))
   (+
    (* -0.5 (/ c b))
    (+
     (* a (* (/ (pow c 2.0) (pow b 3.0)) -0.375))
     (+
      (* (pow a 2.0) (* -0.5625 (/ (pow c 3.0) (pow b 5.0))))
      (* (pow c 4.0) (* (/ (pow a 3.0) (pow b 7.0)) -1.0546875)))))))

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 <= 0.0095) {
		tmp = (-b + sqrt(((b * b) - (c * (a / 0.3333333333333333))))) * (0.3333333333333333 / a);
	} else {
		tmp = (-0.5 * (c / b)) + ((a * ((pow(c, 2.0) / pow(b, 3.0)) * -0.375)) + ((pow(a, 2.0) * (-0.5625 * (pow(c, 3.0) / pow(b, 5.0)))) + (pow(c, 4.0) * ((pow(a, 3.0) / pow(b, 7.0)) * -1.0546875))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.0095d0) then
        tmp = (-b + sqrt(((b * b) - (c * (a / 0.3333333333333333d0))))) * (0.3333333333333333d0 / a)
    else
        tmp = ((-0.5d0) * (c / b)) + ((a * (((c ** 2.0d0) / (b ** 3.0d0)) * (-0.375d0))) + (((a ** 2.0d0) * ((-0.5625d0) * ((c ** 3.0d0) / (b ** 5.0d0)))) + ((c ** 4.0d0) * (((a ** 3.0d0) / (b ** 7.0d0)) * (-1.0546875d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0095) {
		tmp = (-b + Math.sqrt(((b * b) - (c * (a / 0.3333333333333333))))) * (0.3333333333333333 / a);
	} else {
		tmp = (-0.5 * (c / b)) + ((a * ((Math.pow(c, 2.0) / Math.pow(b, 3.0)) * -0.375)) + ((Math.pow(a, 2.0) * (-0.5625 * (Math.pow(c, 3.0) / Math.pow(b, 5.0)))) + (Math.pow(c, 4.0) * ((Math.pow(a, 3.0) / Math.pow(b, 7.0)) * -1.0546875))));
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

↓

def code(a, b, c):
	tmp = 0
	if b <= 0.0095:
		tmp = (-b + math.sqrt(((b * b) - (c * (a / 0.3333333333333333))))) * (0.3333333333333333 / a)
	else:
		tmp = (-0.5 * (c / b)) + ((a * ((math.pow(c, 2.0) / math.pow(b, 3.0)) * -0.375)) + ((math.pow(a, 2.0) * (-0.5625 * (math.pow(c, 3.0) / math.pow(b, 5.0)))) + (math.pow(c, 4.0) * ((math.pow(a, 3.0) / math.pow(b, 7.0)) * -1.0546875))))
	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 <= 0.0095)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(c * Float64(a / 0.3333333333333333))))) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(a * Float64(Float64((c ^ 2.0) / (b ^ 3.0)) * -0.375)) + Float64(Float64((a ^ 2.0) * Float64(-0.5625 * Float64((c ^ 3.0) / (b ^ 5.0)))) + Float64((c ^ 4.0) * Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * -1.0546875)))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.0095)
		tmp = (-b + sqrt(((b * b) - (c * (a / 0.3333333333333333))))) * (0.3333333333333333 / a);
	else
		tmp = (-0.5 * (c / b)) + ((a * (((c ^ 2.0) / (b ^ 3.0)) * -0.375)) + (((a ^ 2.0) * (-0.5625 * ((c ^ 3.0) / (b ^ 5.0)))) + ((c ^ 4.0) * (((a ^ 3.0) / (b ^ 7.0)) * -1.0546875))));
	end
	tmp_2 = 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, 0.0095], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00949999999999999976

   1. Initial program 8.3

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

   2. Applied egg-rr 8.3

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

   3. Applied egg-rr 9.5

      \[\leadsto \color{blue}{-1 + \left(1 - \frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right)} \]

   4. Simplified 8.3

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

      [Start] 9.5	\[ -1 + \left(1 - \frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) \]
      rational.json-simplify-1 [=>] 9.5	\[ \color{blue}{\left(1 - \frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) + -1} \]
      rational.json-simplify-15 [=>] 9.5	\[ \color{blue}{\left(1 - \frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)\right) - 1} \]
      rational.json-simplify-42 [=>] 8.3	\[ \color{blue}{\left(1 - 1\right) - \frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)} \]
      metadata-eval [=>] 8.3	\[ \color{blue}{0} - \frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \]
      rational.json-simplify-12 [<=] 8.3	\[ \color{blue}{-\frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)} \]
      rational.json-simplify-10 [=>] 8.3	\[ \color{blue}{\frac{\frac{-0.3333333333333333}{a} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right)}{-1}} \]
      rational.json-simplify-49 [=>] 8.3	\[ \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\frac{-0.3333333333333333}{a}}{-1}} \]
      rational.json-simplify-44 [=>] 8.3	\[ \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\frac{\frac{-0.3333333333333333}{-1}}{a}} \]
      metadata-eval [=>] 8.3	\[ \left(\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{\color{blue}{0.3333333333333333}}{a} \]

   if 0.00949999999999999976 < b

   1. Initial program 30.0

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

   2. Taylor expanded in a around 0 4.8

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

   3. Simplified 4.8

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + 1.265625 \cdot {\left(\frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right) \cdot \left(-0.16666666666666666 \cdot \frac{{a}^{3}}{b}\right)\right)\right)} \]
      Proof

      [Start] 4.8	\[ -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      rational.json-simplify-1 [=>] 4.8	\[ -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \color{blue}{\left(\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
      rational.json-simplify-41 [=>] 4.8	\[ \color{blue}{\left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right) + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b}\right)} \]
      rational.json-simplify-1 [=>] 4.8	\[ \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b}\right) + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      rational.json-simplify-41 [=>] 4.8	\[ \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b}\right)\right)} \]

   4. Taylor expanded in c around 0 4.8

      \[\leadsto -0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \color{blue}{-1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}}\right)\right) \]

   5. Simplified 4.8

      \[\leadsto -0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \color{blue}{{c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)}\right)\right) \]
      Proof

      [Start] 4.8	\[ -0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + -1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right)\right) \]
      rational.json-simplify-2 [=>] 4.8	\[ -0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + -1.0546875 \cdot \frac{\color{blue}{{a}^{3} \cdot {c}^{4}}}{{b}^{7}}\right)\right) \]
      rational.json-simplify-49 [=>] 4.8	\[ -0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + -1.0546875 \cdot \color{blue}{\left({c}^{4} \cdot \frac{{a}^{3}}{{b}^{7}}\right)}\right)\right) \]
      rational.json-simplify-43 [=>] 4.8	\[ -0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + \color{blue}{{c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0095:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + {c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.0
Cost	47492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.011:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \left(-0.5 \cdot \frac{c}{b} + \left({a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 2
Error	5.0
Cost	47492

\[\begin{array}{l} \mathbf{if}\;b \leq 0.0092:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \left({c}^{2} \cdot \frac{a}{{b}^{3}}\right) + -0.5625 \cdot \left({c}^{3} \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right)\right) + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\\ \end{array} \]

Alternative 3
Error	5.3
Cost	41284

\[\begin{array}{l} \mathbf{if}\;b \leq 0.01:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(\frac{a}{b} \cdot -1.5\right) + \left(-0.5 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)\right)\right) \cdot \frac{0.3333333333333333}{a}\\ \end{array} \]

Alternative 4
Error	5.2
Cost	41284

\[\begin{array}{l} \mathbf{if}\;b \leq 0.011:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-1.5 \cdot \frac{c}{b}\right) + \left(-0.5 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a}\\ \end{array} \]

Alternative 5
Error	6.7
Cost	33796

\[\begin{array}{l} \mathbf{if}\;b \leq 0.011:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6
Error	6.7
Cost	33796

\[\begin{array}{l} \mathbf{if}\;b \leq 0.011:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + {a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right)\\ \end{array} \]

Alternative 7
Error	7.0
Cost	27716

\[\begin{array}{l} \mathbf{if}\;b \leq 0.011:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{\frac{b}{a}} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)\right) \cdot \frac{0.3333333333333333}{a}\\ \end{array} \]

Alternative 8
Error	7.0
Cost	27716

\[\begin{array}{l} \mathbf{if}\;b \leq 0.011:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{a} \cdot \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)\right)\\ \end{array} \]

Alternative 9
Error	7.0
Cost	27716

\[\begin{array}{l} \mathbf{if}\;b \leq 0.011:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{a}{\frac{b}{c}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{a} \cdot 0.3333333333333333\\ \end{array} \]

Alternative 10
Error	6.9
Cost	27716

\[\begin{array}{l} \mathbf{if}\;b \leq 0.0105:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \left(a \cdot \frac{c}{b}\right) + \left(-1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}} + -1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}\right)}{3 \cdot a}\\ \end{array} \]

Alternative 11
Error	9.7
Cost	13892

\[\begin{array}{l} \mathbf{if}\;b \leq 106:\\ \;\;\;\;\frac{2}{\frac{a \cdot 6}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right)\\ \end{array} \]

Alternative 12
Error	16.9
Cost	7684

\[\begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;\frac{2}{\frac{a \cdot 6}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13
Error	16.9
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 14
Error	16.9
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 15
Error	16.9
Cost	7556

\[\begin{array}{l} \mathbf{if}\;b \leq 1800:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \frac{a}{0.3333333333333333}}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 16
Error	22.6
Cost	320

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

Reproduce

herbie shell --seed 2023067 
(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)))