Cubic critical, narrow range

Average Error: 29.2 → 4.8

Time: 41.9s

Precision: binary64

Cost: 97220

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\\ t_1 := \frac{\left(-b\right) + t_0}{3 \cdot a}\\ t_2 := t_1 \cdot t_1\\ t_3 := \frac{t_0 - b}{3 \cdot a}\\ t_4 := t_3 \cdot t_3\\ \mathbf{if}\;b \leq 0.0044:\\ \;\;\;\;\left(t_1 \cdot \left(t_1 \cdot t_2\right)\right) \cdot \frac{\frac{1}{\left(t_4 \cdot t_4\right) \cdot \frac{\frac{1}{t_3}}{t_4}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

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 (sqrt (- (* b b) (* 3.0 (* a c)))))
        (t_1 (/ (+ (- b) t_0) (* 3.0 a)))
        (t_2 (* t_1 t_1))
        (t_3 (/ (- t_0 b) (* 3.0 a)))
        (t_4 (* t_3 t_3)))
   (if (<= b 0.0044)
     (*
      (* t_1 (* t_1 t_2))
      (/ (/ 1.0 (* (* t_4 t_4) (/ (/ 1.0 t_3) t_4))) t_2))
     (+
      (* -0.5 (/ c b))
      (+
       (* -0.5625 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
       (+
        (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (*
         -0.16666666666666666
         (/ (* (pow (* c a) 4.0) 6.328125) (* a (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 t_0 = sqrt(((b * b) - (3.0 * (a * c))));
	double t_1 = (-b + t_0) / (3.0 * a);
	double t_2 = t_1 * t_1;
	double t_3 = (t_0 - b) / (3.0 * a);
	double t_4 = t_3 * t_3;
	double tmp;
	if (b <= 0.0044) {
		tmp = (t_1 * (t_1 * t_2)) * ((1.0 / ((t_4 * t_4) * ((1.0 / t_3) / t_4))) / t_2);
	} else {
		tmp = (-0.5 * (c / b)) + ((-0.5625 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((c * a), 4.0) * 6.328125) / (a * pow(b, 7.0))))));
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (3.0d0 * (a * c))))
    t_1 = (-b + t_0) / (3.0d0 * a)
    t_2 = t_1 * t_1
    t_3 = (t_0 - b) / (3.0d0 * a)
    t_4 = t_3 * t_3
    if (b <= 0.0044d0) then
        tmp = (t_1 * (t_1 * t_2)) * ((1.0d0 / ((t_4 * t_4) * ((1.0d0 / t_3) / t_4))) / t_2)
    else
        tmp = ((-0.5d0) * (c / b)) + (((-0.5625d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0))) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((c * a) ** 4.0d0) * 6.328125d0) / (a * (b ** 7.0d0))))))
    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 t_0 = Math.sqrt(((b * b) - (3.0 * (a * c))));
	double t_1 = (-b + t_0) / (3.0 * a);
	double t_2 = t_1 * t_1;
	double t_3 = (t_0 - b) / (3.0 * a);
	double t_4 = t_3 * t_3;
	double tmp;
	if (b <= 0.0044) {
		tmp = (t_1 * (t_1 * t_2)) * ((1.0 / ((t_4 * t_4) * ((1.0 / t_3) / t_4))) / t_2);
	} else {
		tmp = (-0.5 * (c / b)) + ((-0.5625 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((c * a), 4.0) * 6.328125) / (a * Math.pow(b, 7.0))))));
	}
	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):
	t_0 = math.sqrt(((b * b) - (3.0 * (a * c))))
	t_1 = (-b + t_0) / (3.0 * a)
	t_2 = t_1 * t_1
	t_3 = (t_0 - b) / (3.0 * a)
	t_4 = t_3 * t_3
	tmp = 0
	if b <= 0.0044:
		tmp = (t_1 * (t_1 * t_2)) * ((1.0 / ((t_4 * t_4) * ((1.0 / t_3) / t_4))) / t_2)
	else:
		tmp = (-0.5 * (c / b)) + ((-0.5625 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0))) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((c * a), 4.0) * 6.328125) / (a * math.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)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c))))
	t_1 = Float64(Float64(Float64(-b) + t_0) / Float64(3.0 * a))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(Float64(t_0 - b) / Float64(3.0 * a))
	t_4 = Float64(t_3 * t_3)
	tmp = 0.0
	if (b <= 0.0044)
		tmp = Float64(Float64(t_1 * Float64(t_1 * t_2)) * Float64(Float64(1.0 / Float64(Float64(t_4 * t_4) * Float64(Float64(1.0 / t_3) / t_4))) / t_2));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.5625 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) * 6.328125) / Float64(a * (b ^ 7.0)))))));
	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)
	t_0 = sqrt(((b * b) - (3.0 * (a * c))));
	t_1 = (-b + t_0) / (3.0 * a);
	t_2 = t_1 * t_1;
	t_3 = (t_0 - b) / (3.0 * a);
	t_4 = t_3 * t_3;
	tmp = 0.0;
	if (b <= 0.0044)
		tmp = (t_1 * (t_1 * t_2)) * ((1.0 / ((t_4 * t_4) * ((1.0 / t_3) / t_4))) / t_2);
	else
		tmp = (-0.5 * (c / b)) + ((-0.5625 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((c * a) ^ 4.0) * 6.328125) / (a * (b ^ 7.0))))));
	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_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[((-b) + t$95$0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * t$95$3), $MachinePrecision]}, If[LessEqual[b, 0.0044], N[(N[(t$95$1 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[(N[(1.0 / t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.00440000000000000027

   1. Initial program 8.1

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

   2. Simplified 8.1

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

      [Start] 8.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
      rational.json-simplify-2 [=>] 8.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{c \cdot \left(3 \cdot a\right)}}}{3 \cdot a} \]
      rational.json-simplify-43 [=>] 8.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   3. Applied egg-rr 31.6

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{\left(\left(0 - \left(-1 - 3 \cdot \left(a \cdot c\right)\right)\right) - 1\right)}}}{3 \cdot a} \]

   4. Applied egg-rr 8.2

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

   5. Applied egg-rr 8.2

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

   if 0.00440000000000000027 < b

   1. Initial program 30.4

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

   2. Taylor expanded in b around inf 4.7

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   3. Simplified 4.6

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)} \]
      Proof

      [Start] 4.7	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
      rational.json-simplify-41 [=>] 4.7	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]
      rational.json-simplify-41 [=>] 4.6	\[ \color{blue}{-0.5 \cdot \frac{c}{b} + \left(\left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]
      rational.json-simplify-1 [=>] 4.6	\[ -0.5 \cdot \frac{c}{b} + \color{blue}{\left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0044:\\ \;\;\;\;\left(\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a} \cdot \left(\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a} \cdot \left(\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}\right)\right)\right) \cdot \frac{\frac{1}{\left(\left(\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a} \cdot \frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\right) \cdot \left(\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a} \cdot \frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\right)\right) \cdot \frac{\frac{1}{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}}}{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a} \cdot \frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}}}}{\frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	4.8
Cost	67012

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\\ t_1 := \frac{\left(-b\right) + t_0}{3 \cdot a}\\ t_2 := t_0 - b\\ t_3 := t_1 \cdot t_1\\ \mathbf{if}\;b \leq 0.0044:\\ \;\;\;\;\left(t_1 \cdot \left(\frac{\frac{1}{t_2} \cdot \left(t_2 \cdot t_2\right)}{3 \cdot a} \cdot t_3\right)\right) \cdot \frac{\frac{1}{t_1}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 2
Error	4.8
Cost	52612

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{3 \cdot a}\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;b \leq 0.0039:\\ \;\;\;\;\left(t_0 \cdot \left(t_0 \cdot t_1\right)\right) \cdot \frac{\frac{1}{t_0}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 3
Error	4.9
Cost	47492

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;b \leq 0.0044:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 4
Error	4.8
Cost	47492

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;b \leq 0.0042:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a \cdot {b}^{7}}\right)\right)\\ \end{array} \]

Alternative 5
Error	5.1
Cost	41284

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;b \leq 0.0045:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + \left(-0.5 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)\right)}{3 \cdot a}\\ \end{array} \]

Alternative 6
Error	5.1
Cost	41284

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;b \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(\left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right) + -0.5 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{{b}^{7}}\right)}{3 \cdot a}\\ \end{array} \]

Alternative 7
Error	6.5
Cost	33796

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;b \leq 0.0039:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\\ \end{array} \]

Alternative 8
Error	9.3
Cost	29316

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\\ \end{array} \]

Alternative 9
Error	6.7
Cost	27716

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;b \leq 0.0046:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.5 \cdot \frac{c \cdot a}{b} + \left(-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{3 \cdot a}\\ \end{array} \]

Alternative 10
Error	6.7
Cost	27716

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b\\ \mathbf{if}\;b \leq 0.004:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1.125 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}} + \left(-1.5 \cdot \frac{c \cdot a}{b} + -1.6875 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}\right)}{3 \cdot a}\\ \end{array} \]

Alternative 11
Error	9.3
Cost	21252

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0028:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\\ \end{array} \]

Alternative 12
Error	15.1
Cost	14852

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13
Error	20.1
Cost	7492

\[\begin{array}{l} \mathbf{if}\;a \leq 5.2:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{3 \cdot a}\\ \end{array} \]

Alternative 14
Error	22.3
Cost	320

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

Reproduce

herbie shell --seed 2023077 
(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)))