Cubic critical

Average Error: 35.0 → 10.6

Time: 21.6s

Precision: binary64

Cost: 7624

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

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 -1e+125)
   (/ b (* a -1.5))
   (if (<= b 2.15e-26)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) (+ -0.5 (* (/ c (/ b a)) (/ -0.375 b)))))))

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 <= -1e+125) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.15e-26) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * (-0.5 + ((c / (b / a)) * (-0.375 / b)));
	}
	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 <= (-1d+125)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.15d-26) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * ((-0.5d0) + ((c / (b / a)) * ((-0.375d0) / b)))
    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 <= -1e+125) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.15e-26) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * (-0.5 + ((c / (b / a)) * (-0.375 / b)));
	}
	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 <= -1e+125:
		tmp = b / (a * -1.5)
	elif b <= 2.15e-26:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * (-0.5 + ((c / (b / a)) * (-0.375 / b)))
	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 <= -1e+125)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.15e-26)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * Float64(-0.5 + Float64(Float64(c / Float64(b / a)) * Float64(-0.375 / b))));
	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 <= -1e+125)
		tmp = b / (a * -1.5);
	elseif (b <= 2.15e-26)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * (-0.5 + ((c / (b / a)) * (-0.375 / b)));
	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, -1e+125], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.15e-26], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * N[(-0.5 + N[(N[(c / N[(b / a), $MachinePrecision]), $MachinePrecision] * N[(-0.375 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.9999999999999992e124

   1. Initial program 54.1

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

   2. Simplified 54.2

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

      [Start] 54.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      remove-double-neg [<=] 54.1	\[ \frac{\left(-b\right) + \color{blue}{\left(-\left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)\right)}}{3 \cdot a} \]
      sub-neg [<=] 54.1	\[ \frac{\color{blue}{\left(-b\right) - \left(-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a} \]
      div-sub [=>] 54.1	\[ \color{blue}{\frac{-b}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      neg-mul-1 [=>] 54.1	\[ \frac{\color{blue}{-1 \cdot b}}{3 \cdot a} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      associate-*l/ [<=] 54.1	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot b} - \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      distribute-frac-neg [=>] 54.1	\[ \frac{-1}{3 \cdot a} \cdot b - \color{blue}{\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      fma-neg [=>] 54.1	\[ \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot a}, b, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right)} \]
      /-rgt-identity [<=] 54.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b}{1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      metadata-eval [<=] 54.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{b}{\color{blue}{\frac{-1}{-1}}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      associate-/l* [<=] 54.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \color{blue}{\frac{b \cdot -1}{-1}}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      *-commutative [<=] 54.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-1 \cdot b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      neg-mul-1 [<=] 54.1	\[ \mathsf{fma}\left(\frac{-1}{3 \cdot a}, \frac{\color{blue}{-b}}{-1}, -\left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)\right) \]
      fma-neg [<=] 54.1	\[ \color{blue}{\frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \left(-\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\right)} \]
      neg-mul-1 [=>] 54.1	\[ \frac{-1}{3 \cdot a} \cdot \frac{-b}{-1} - \color{blue}{-1 \cdot \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   3. Taylor expanded in b around -inf 3.7

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   4. Simplified 3.7

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
      Proof

      [Start] 3.7	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      *-commutative [=>] 3.7	\[ \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Taylor expanded in b around 0 3.7

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   6. Simplified 3.6

      \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]
      Proof

      [Start] 3.7	\[ -0.6666666666666666 \cdot \frac{b}{a} \]
      *-commutative [<=] 3.7	\[ \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
      /-rgt-identity [<=] 3.7	\[ \color{blue}{\frac{\frac{b}{a} \cdot -0.6666666666666666}{1}} \]
      associate-/l* [=>] 3.6	\[ \color{blue}{\frac{\frac{b}{a}}{\frac{1}{-0.6666666666666666}}} \]
      associate-/r* [<=] 3.6	\[ \color{blue}{\frac{b}{a \cdot \frac{1}{-0.6666666666666666}}} \]
      metadata-eval [=>] 3.6	\[ \frac{b}{a \cdot \color{blue}{-1.5}} \]

   if -9.9999999999999992e124 < b < 2.14999999999999994e-26

   1. Initial program 15.0

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

   if 2.14999999999999994e-26 < b

   1. Initial program 55.1

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

   2. Simplified 55.1

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

      [Start] 55.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      *-lft-identity [<=] 55.1	\[ \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]
      metadata-eval [<=] 55.1	\[ \color{blue}{\frac{-1}{-1}} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      times-frac [<=] 55.1	\[ \color{blue}{\frac{-1 \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}{-1 \cdot \left(3 \cdot a\right)}} \]
      *-commutative [<=] 55.1	\[ \frac{\color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot -1}}{-1 \cdot \left(3 \cdot a\right)} \]
      times-frac [=>] 55.1	\[ \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot \frac{-1}{3 \cdot a}} \]
      associate-*r/ [=>] 55.1	\[ \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{-1} \cdot -1}{3 \cdot a}} \]

   3. Taylor expanded in b around inf 19.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   4. Simplified 15.0

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

      [Start] 19.0	\[ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      fma-def [=>] 19.0	\[ \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      associate-*r/ [=>] 19.0	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\frac{-0.375 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}}\right) \]
      associate-/l* [=>] 19.0	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\frac{-0.375}{\frac{{b}^{3}}{{c}^{2} \cdot a}}}\right) \]
      unpow2 [=>] 19.0	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{{b}^{3}}{\color{blue}{\left(c \cdot c\right)} \cdot a}}\right) \]
      associate-*l* [=>] 15.0	\[ \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{{b}^{3}}{\color{blue}{c \cdot \left(c \cdot a\right)}}}\right) \]

   5. Taylor expanded in c around 0 19.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}} \]

   6. Simplified 7.2

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

      [Start] 19.0	\[ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      *-commutative [=>] 19.0	\[ \color{blue}{\frac{c}{b} \cdot -0.5} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} \]
      associate-*r/ [=>] 19.0	\[ \frac{c}{b} \cdot -0.5 + \color{blue}{\frac{-0.375 \cdot \left({c}^{2} \cdot a\right)}{{b}^{3}}} \]
      unpow2 [=>] 19.0	\[ \frac{c}{b} \cdot -0.5 + \frac{-0.375 \cdot \left(\color{blue}{\left(c \cdot c\right)} \cdot a\right)}{{b}^{3}} \]
      associate-*r* [<=] 15.0	\[ \frac{c}{b} \cdot -0.5 + \frac{-0.375 \cdot \color{blue}{\left(c \cdot \left(c \cdot a\right)\right)}}{{b}^{3}} \]
      *-commutative [=>] 15.0	\[ \frac{c}{b} \cdot -0.5 + \frac{\color{blue}{\left(c \cdot \left(c \cdot a\right)\right) \cdot -0.375}}{{b}^{3}} \]
      unpow3 [=>] 15.0	\[ \frac{c}{b} \cdot -0.5 + \frac{\left(c \cdot \left(c \cdot a\right)\right) \cdot -0.375}{\color{blue}{\left(b \cdot b\right) \cdot b}} \]
      times-frac [=>] 15.0	\[ \frac{c}{b} \cdot -0.5 + \color{blue}{\frac{c \cdot \left(c \cdot a\right)}{b \cdot b} \cdot \frac{-0.375}{b}} \]
      times-frac [=>] 9.7	\[ \frac{c}{b} \cdot -0.5 + \color{blue}{\left(\frac{c}{b} \cdot \frac{c \cdot a}{b}\right)} \cdot \frac{-0.375}{b} \]
      associate-*l* [=>] 9.7	\[ \frac{c}{b} \cdot -0.5 + \color{blue}{\frac{c}{b} \cdot \left(\frac{c \cdot a}{b} \cdot \frac{-0.375}{b}\right)} \]
      distribute-lft-out [=>] 9.7	\[ \color{blue}{\frac{c}{b} \cdot \left(-0.5 + \frac{c \cdot a}{b} \cdot \frac{-0.375}{b}\right)} \]
      associate-/l* [=>] 7.2	\[ \frac{c}{b} \cdot \left(-0.5 + \color{blue}{\frac{c}{\frac{b}{a}}} \cdot \frac{-0.375}{b}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.6
Cost	7624

\[\begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+125}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

Alternative 2
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-128}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot 2\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7368

\[\begin{array}{l} \mathbf{if}\;b \leq -1.65 \cdot 10^{-128}:\\ \;\;\;\;\left(-1.5 \cdot \frac{c}{b} + \frac{b}{a} \cdot 2\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

Alternative 4
Error	23.1
Cost	1092

\[\begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{-207}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \left(-0.5 + \frac{c}{\frac{b}{a}} \cdot \frac{-0.375}{b}\right)\\ \end{array} \]

Alternative 5
Error	39.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	39.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7
Error	23.1
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-248}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 8
Error	23.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{-249}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]

Alternative 9
Error	22.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-257}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10
Error	56.1
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023045 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))