Quadratic roots, narrow range

Average Error: 28.1 → 5.2

Time: 28.6s

Precision: binary64

Cost: 54660

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

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

↓

\[\begin{array}{l} t_0 := b \cdot b - \left(4 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{t_0}}{2 \cdot a} \leq -5.4:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* 4.0 a) c))))
   (if (<= (/ (+ (- b) (sqrt t_0)) (* 2.0 a)) -5.4)
     (/ (+ (- b) (sqrt (* (/ 1.0 t_0) (* t_0 t_0)))) (* a 2.0))
     (+
      (- (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0))))
      (+
       (* -2.0 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))
       (* -0.25 (/ (* (pow (* c a) 4.0) 20.0) (* a (pow b 7.0)))))))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((4.0 * a) * c);
	double tmp;
	if (((-b + sqrt(t_0)) / (2.0 * a)) <= -5.4) {
		tmp = (-b + sqrt(((1.0 / t_0) * (t_0 * t_0)))) / (a * 2.0);
	} else {
		tmp = -((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0))) + ((-2.0 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0))) + (-0.25 * ((pow((c * a), 4.0) * 20.0) / (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) - ((4.0d0 * a) * c)))) / (2.0d0 * 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) :: tmp
    t_0 = (b * b) - ((4.0d0 * a) * c)
    if (((-b + sqrt(t_0)) / (2.0d0 * a)) <= (-5.4d0)) then
        tmp = (-b + sqrt(((1.0d0 / t_0) * (t_0 * t_0)))) / (a * 2.0d0)
    else
        tmp = -((c / b) + ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + (((-2.0d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0))) + ((-0.25d0) * ((((c * a) ** 4.0d0) * 20.0d0) / (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) - ((4.0 * a) * c)))) / (2.0 * a);
}

↓

public static double code(double a, double b, double c) {
	double t_0 = (b * b) - ((4.0 * a) * c);
	double tmp;
	if (((-b + Math.sqrt(t_0)) / (2.0 * a)) <= -5.4) {
		tmp = (-b + Math.sqrt(((1.0 / t_0) * (t_0 * t_0)))) / (a * 2.0);
	} else {
		tmp = -((c / b) + ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + ((-2.0 * ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) / Math.pow(b, 5.0))) + (-0.25 * ((Math.pow((c * a), 4.0) * 20.0) / (a * Math.pow(b, 7.0)))));
	}
	return tmp;
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

↓

def code(a, b, c):
	t_0 = (b * b) - ((4.0 * a) * c)
	tmp = 0
	if ((-b + math.sqrt(t_0)) / (2.0 * a)) <= -5.4:
		tmp = (-b + math.sqrt(((1.0 / t_0) * (t_0 * t_0)))) / (a * 2.0)
	else:
		tmp = -((c / b) + ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + ((-2.0 * ((math.pow(c, 3.0) * math.pow(a, 2.0)) / math.pow(b, 5.0))) + (-0.25 * ((math.pow((c * a), 4.0) * 20.0) / (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(4.0 * a) * c)))) / Float64(2.0 * a))
end

↓

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(t_0)) / Float64(2.0 * a)) <= -5.4)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(1.0 / t_0) * Float64(t_0 * t_0)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))) + Float64(Float64(-2.0 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + Float64(-0.25 * Float64(Float64((Float64(c * a) ^ 4.0) * 20.0) / Float64(a * (b ^ 7.0))))));
	end
	return tmp
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

↓

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - ((4.0 * a) * c);
	tmp = 0.0;
	if (((-b + sqrt(t_0)) / (2.0 * a)) <= -5.4)
		tmp = (-b + sqrt(((1.0 / t_0) * (t_0 * t_0)))) / (a * 2.0);
	else
		tmp = -((c / b) + ((a * (c ^ 2.0)) / (b ^ 3.0))) + ((-2.0 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))) + (-0.25 * ((((c * a) ^ 4.0) * 20.0) / (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[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -5.4], N[(N[((-b) + N[Sqrt[N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(-2.0 * N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $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 4 a) c)))) (*.f64 2 a)) < -5.4000000000000004

   1. Initial program 10.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. Simplified 10.0

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
      Proof

      [Start] 10.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational_best-simplify-2 [=>] 10.0	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Applied egg-rr 10.2

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}}}{a \cdot 2} \]

   if -5.4000000000000004 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 30.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

   2. Simplified 30.4

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
      Proof

      [Start] 30.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational_best-simplify-2 [=>] 30.4	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Taylor expanded in b around inf 4.6

      \[\leadsto \color{blue}{-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]

   4. Simplified 4.6

      \[\leadsto \color{blue}{\left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot {\left(c \cdot a\right)}^{2}\right)}^{2} + 16 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}\right)} \]
      Proof

      [Start] 4.6	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
      rational_best-simplify-1 [=>] 4.6	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -1 \cdot \frac{c}{b}\right)}\right) \]
      rational_best-simplify-43 [=>] 4.6	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]
      rational_best-simplify-43 [=>] 4.6	\[ \color{blue}{\left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + \left(-1 \cdot \frac{c}{b} + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
      rational_best-simplify-1 [<=] 4.6	\[ \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -1 \cdot \frac{c}{b}\right)} \]

   5. Taylor expanded in c around 0 4.6

      \[\leadsto \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \color{blue}{\frac{{c}^{4} \cdot \left(16 \cdot {a}^{4} + 4 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right) \]

   6. Simplified 4.6

      \[\leadsto \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \color{blue}{\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}}\right) \]
      Proof

      [Start] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{c}^{4} \cdot \left(16 \cdot {a}^{4} + 4 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-47 [<=] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(4 \cdot {a}^{4}\right) + {c}^{4} \cdot \left(16 \cdot {a}^{4}\right)}}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-44 [<=] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{4 \cdot \left({c}^{4} \cdot {a}^{4}\right)} + {c}^{4} \cdot \left(16 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-44 [<=] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + \color{blue}{16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-2 [=>] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 4} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
      exponential-simplify-27 [=>] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{4}} \cdot 4 + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-2 [<=] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\color{blue}{\left(c \cdot a\right)}}^{4} \cdot 4 + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-2 [=>] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 4 + \color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot 16}}{a \cdot {b}^{7}}\right) \]
      exponential-simplify-27 [=>] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 4 + \color{blue}{{\left(a \cdot c\right)}^{4}} \cdot 16}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-2 [<=] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 4 + {\color{blue}{\left(c \cdot a\right)}}^{4} \cdot 16}{a \cdot {b}^{7}}\right) \]
      rational_best-simplify-47 [=>] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{\color{blue}{{\left(c \cdot a\right)}^{4} \cdot \left(16 + 4\right)}}{a \cdot {b}^{7}}\right) \]
      metadata-eval [=>] 4.6	\[ \left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot \color{blue}{20}}{a \cdot {b}^{7}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5.4:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\frac{1}{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{\left(c \cdot a\right)}^{4} \cdot 20}{a \cdot {b}^{7}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	6.8
Cost	40964

\[\begin{array}{l} t_0 := b \cdot b - \left(4 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{t_0}}{2 \cdot a} \leq -5.4:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\\ \end{array} \]

Alternative 2
Error	6.9
Cost	35076

\[\begin{array}{l} t_0 := b \cdot b - \left(4 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{t_0}}{2 \cdot a} \leq -5.4:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c \cdot a}{b} + \left(-4 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}} + -2 \cdot \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}\right)}{a \cdot 2}\\ \end{array} \]

Alternative 3
Error	6.9
Cost	34948

\[\begin{array}{l} t_0 := b \cdot b - \left(4 \cdot a\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{t_0}}{2 \cdot a} \leq -5.4:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \left(\frac{c \cdot a}{b} + \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}\right) + -4 \cdot \frac{{\left(c \cdot a\right)}^{3}}{{b}^{5}}}{a \cdot 2}\\ \end{array} \]

Alternative 4
Error	9.5
Cost	29508

\[\begin{array}{l} t_0 := \left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;\frac{t_0}{2 \cdot a} \leq -0.052:\\ \;\;\;\;\frac{\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5
Error	9.5
Cost	21060

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{if}\;t_0 \leq -0.052:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 6
Error	15.3
Cost	14916

\[\begin{array}{l} t_0 := \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{if}\;t_0 \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 7
Error	23.1
Cost	256

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

Reproduce

herbie shell --seed 2023092 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))