Cubic critical, narrow range

Average Accuracy: 55.3% → 99.1%

Time: 21.2s

Precision: binary64

Cost: 15040

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

Math

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

↓

\[\frac{\frac{3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\frac{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot 9 - {b}^{4}}{c \cdot \left(a \cdot -3\right) - b \cdot b}}}}{3 \cdot a} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (/
  (/
   (* 3.0 (* c a))
   (-
    (- b)
    (sqrt
     (/
      (- (* (* (* c a) (* c a)) 9.0) (pow b 4.0))
      (- (* c (* a -3.0)) (* b 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) {
	return ((3.0 * (c * a)) / (-b - sqrt((((((c * a) * (c * a)) * 9.0) - pow(b, 4.0)) / ((c * (a * -3.0)) - (b * b)))))) / (3.0 * a);
}

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
    code = ((3.0d0 * (c * a)) / (-b - sqrt((((((c * a) * (c * a)) * 9.0d0) - (b ** 4.0d0)) / ((c * (a * (-3.0d0))) - (b * b)))))) / (3.0d0 * a)
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) {
	return ((3.0 * (c * a)) / (-b - Math.sqrt((((((c * a) * (c * a)) * 9.0) - Math.pow(b, 4.0)) / ((c * (a * -3.0)) - (b * b)))))) / (3.0 * a);
}

Python

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

↓

def code(a, b, c):
	return ((3.0 * (c * a)) / (-b - math.sqrt((((((c * a) * (c * a)) * 9.0) - math.pow(b, 4.0)) / ((c * (a * -3.0)) - (b * b)))))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * Float64(c * a)) / Float64(Float64(-b) - sqrt(Float64(Float64(Float64(Float64(Float64(c * a) * Float64(c * a)) * 9.0) - (b ^ 4.0)) / Float64(Float64(c * Float64(a * -3.0)) - Float64(b * b)))))) / Float64(3.0 * a))
end

MATLAB

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

↓

function tmp = code(a, b, c)
	tmp = ((3.0 * (c * a)) / (-b - sqrt((((((c * a) * (c * a)) * 9.0) - (b ^ 4.0)) / ((c * (a * -3.0)) - (b * b)))))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[(N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[((-b) - N[Sqrt[N[(N[(N[(N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision] * 9.0), $MachinePrecision] - N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

2. Applied egg-rr 56.5%

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

   [Start] 55.3	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   flip-+ [=>] 55.3	\[ \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]
   sub-neg [=>] 55.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\left(-b\right) + \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]
   add-sqr-sqrt [=>] 55.2	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-\color{blue}{\sqrt{b} \cdot \sqrt{b}}\right) + \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   sqrt-prod [<=] 55.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-\color{blue}{\sqrt{b \cdot b}}\right) + \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   sqr-neg [<=] 55.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}\right) + \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   sqrt-unprod [<=] 0.0	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}\right) + \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   add-sqr-sqrt [<=] 1.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-\color{blue}{\left(-b\right)}\right) + \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
   distribute-neg-in [<=] 1.3	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{-\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a} \]
   add-sqr-sqrt [=>] 0.0	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-\color{blue}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}{3 \cdot a} \]
   distribute-rgt-neg-in [=>] 0.0	\[ \frac{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} \cdot \left(-\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\right)}}}{3 \cdot a} \]

3. Simplified 56.8%

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

   [Start] 56.5	\[ \frac{\frac{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}{3 \cdot a} \]
   associate-/r* [<=] 56.5	\[ \frac{\color{blue}{\frac{b \cdot b - \mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}}{3 \cdot a} \]
   fma-def [<=] 56.8	\[ \frac{\frac{b \cdot b - \color{blue}{\left(b \cdot b + c \cdot \left(a \cdot -3\right)\right)}}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}{3 \cdot a} \]
   +-commutative [=>] 56.8	\[ \frac{\frac{b \cdot b - \color{blue}{\left(c \cdot \left(a \cdot -3\right) + b \cdot b\right)}}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}{3 \cdot a} \]
   fma-def [=>] 56.8	\[ \frac{\frac{b \cdot b - \color{blue}{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}{\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \left(-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}\right)}}{3 \cdot a} \]
   distribute-rgt-neg-in [<=] 56.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{\color{blue}{-\sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}} \cdot \sqrt{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}}{3 \cdot a} \]
   rem-square-sqrt [=>] 56.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}}}{3 \cdot a} \]
   fma-def [<=] 56.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{b \cdot b + c \cdot \left(a \cdot -3\right)}}\right)}}{3 \cdot a} \]
   +-commutative [=>] 56.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}\right)}}{3 \cdot a} \]
   fma-def [=>] 56.8	\[ \frac{\frac{b \cdot b - \mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}{-\left(b + \sqrt{\color{blue}{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}\right)}}{3 \cdot a} \]

4. Taylor expanded in b around 0 99.1%

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

5. Applied egg-rr 99.1%

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

   [Start] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}}{3 \cdot a} \]
   fma-udef [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right) + b \cdot b}}\right)}}{3 \cdot a} \]
   flip-+ [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\color{blue}{\frac{\left(c \cdot \left(a \cdot -3\right)\right) \cdot \left(c \cdot \left(a \cdot -3\right)\right) - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{c \cdot \left(a \cdot -3\right) - b \cdot b}}}\right)}}{3 \cdot a} \]
   associate-*r* [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\color{blue}{\left(\left(c \cdot a\right) \cdot -3\right)} \cdot \left(c \cdot \left(a \cdot -3\right)\right) - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{c \cdot \left(a \cdot -3\right) - b \cdot b}}\right)}}{3 \cdot a} \]
   associate-*r* [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \color{blue}{\left(\left(c \cdot a\right) \cdot -3\right)} - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{c \cdot \left(a \cdot -3\right) - b \cdot b}}\right)}}{3 \cdot a} \]
   pow2 [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right) - \color{blue}{{b}^{2}} \cdot \left(b \cdot b\right)}{c \cdot \left(a \cdot -3\right) - b \cdot b}}\right)}}{3 \cdot a} \]
   pow2 [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right) - {b}^{2} \cdot \color{blue}{{b}^{2}}}{c \cdot \left(a \cdot -3\right) - b \cdot b}}\right)}}{3 \cdot a} \]
   pow-sqr [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right) - \color{blue}{{b}^{\left(2 \cdot 2\right)}}}{c \cdot \left(a \cdot -3\right) - b \cdot b}}\right)}}{3 \cdot a} \]
   metadata-eval [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right) - {b}^{\color{blue}{4}}}{c \cdot \left(a \cdot -3\right) - b \cdot b}}\right)}}{3 \cdot a} \]
   associate-*r* [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right) - {b}^{4}}{\color{blue}{\left(c \cdot a\right) \cdot -3} - b \cdot b}}\right)}}{3 \cdot a} \]

6. Simplified 99.1%

   \[\leadsto \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\color{blue}{\frac{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot 9 - {b}^{4}}{c \cdot \left(a \cdot -3\right) - b \cdot b}}}\right)}}{3 \cdot a} \]
   Proof

   [Start] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot -3\right) \cdot \left(\left(c \cdot a\right) \cdot -3\right) - {b}^{4}}{\left(c \cdot a\right) \cdot -3 - b \cdot b}}\right)}}{3 \cdot a} \]
   swap-sqr [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\color{blue}{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \left(-3 \cdot -3\right)} - {b}^{4}}{\left(c \cdot a\right) \cdot -3 - b \cdot b}}\right)}}{3 \cdot a} \]
   metadata-eval [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot \color{blue}{9} - {b}^{4}}{\left(c \cdot a\right) \cdot -3 - b \cdot b}}\right)}}{3 \cdot a} \]
   associate-*l* [=>] 99.1	\[ \frac{\frac{3 \cdot \left(c \cdot a\right)}{-\left(b + \sqrt{\frac{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot 9 - {b}^{4}}{\color{blue}{c \cdot \left(a \cdot -3\right)} - b \cdot b}}\right)}}{3 \cdot a} \]

7. Final simplification 99.1%

   \[\leadsto \frac{\frac{3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\frac{\left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right) \cdot 9 - {b}^{4}}{c \cdot \left(a \cdot -3\right) - b \cdot b}}}}{3 \cdot a} \]

Alternatives

Alternative 1
Accuracy	85.1%
Cost	14852

\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.007:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5 + -0.375 \cdot \left(a \cdot \frac{c \cdot c}{{b}^{3}}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	14080

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

Alternative 3
Accuracy	99.1%
Cost	7808

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

Alternative 4
Accuracy	84.8%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;\left(b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

Alternative 5
Accuracy	84.8%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;b \leq 105:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

Alternative 6
Accuracy	81.5%
Cost	960

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

Alternative 7
Accuracy	64.4%
Cost	320

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

Reproduce

herbie shell --seed 2023135 
(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)))