Quadratic roots, medium range

Average Error: 43.8 → 2.9

Time: 1.2min

Precision: binary64

Cost: 47168

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

↓

\[\left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(-{\left(c \cdot a\right)}^{4}\right) \cdot \frac{5}{a \cdot {b}^{7}} \]

FPCore

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

↓

(FPCore (a b c)
 :precision binary64
 (+
  (+
   (- 0.0 (+ (/ c b) (/ (* a (pow c 2.0)) (pow b 3.0))))
   (* (* (pow c 3.0) (pow a 2.0)) (/ -2.0 (pow b 5.0))))
  (* (- (pow (* c a) 4.0)) (/ 5.0 (* 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) {
	return ((0.0 - ((c / b) + ((a * pow(c, 2.0)) / pow(b, 3.0)))) + ((pow(c, 3.0) * pow(a, 2.0)) * (-2.0 / pow(b, 5.0)))) + (-pow((c * a), 4.0) * (5.0 / (a * pow(b, 7.0))));
}

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
    code = ((0.0d0 - ((c / b) + ((a * (c ** 2.0d0)) / (b ** 3.0d0)))) + (((c ** 3.0d0) * (a ** 2.0d0)) * ((-2.0d0) / (b ** 5.0d0)))) + (-((c * a) ** 4.0d0) * (5.0d0 / (a * (b ** 7.0d0))))
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) {
	return ((0.0 - ((c / b) + ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)))) + ((Math.pow(c, 3.0) * Math.pow(a, 2.0)) * (-2.0 / Math.pow(b, 5.0)))) + (-Math.pow((c * a), 4.0) * (5.0 / (a * Math.pow(b, 7.0))));
}

Python

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

↓

def code(a, b, c):
	return ((0.0 - ((c / b) + ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))) + ((math.pow(c, 3.0) * math.pow(a, 2.0)) * (-2.0 / math.pow(b, 5.0)))) + (-math.pow((c * a), 4.0) * (5.0 / (a * math.pow(b, 7.0))))

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)
	return Float64(Float64(Float64(0.0 - Float64(Float64(c / b) + Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))) + Float64(Float64((c ^ 3.0) * (a ^ 2.0)) * Float64(-2.0 / (b ^ 5.0)))) + Float64(Float64(-(Float64(c * a) ^ 4.0)) * Float64(5.0 / Float64(a * (b ^ 7.0)))))
end

MATLAB

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

↓

function tmp = code(a, b, c)
	tmp = ((0.0 - ((c / b) + ((a * (c ^ 2.0)) / (b ^ 3.0)))) + (((c ^ 3.0) * (a ^ 2.0)) * (-2.0 / (b ^ 5.0)))) + (-((c * a) ^ 4.0) * (5.0 / (a * (b ^ 7.0))));
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_] := N[(N[(N[(0.0 - N[(N[(c / b), $MachinePrecision] + N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[c, 3.0], $MachinePrecision] * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[(-2.0 / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]) * N[(5.0 / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.8

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

2. Simplified 43.8

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

   [Start] 43.8	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   rational_best-simplify-3 [=>] 43.8	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{2 \cdot a} \]
   rational_best-simplify-9 [<=] 43.8	\[ \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - 0\right)} + \left(-b\right)}{2 \cdot a} \]
   rational_best-simplify-56 [=>] 43.8	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \left(0 + b\right)}}{2 \cdot a} \]
   rational_best-simplify-6 [=>] 43.8	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - \color{blue}{b}}{2 \cdot a} \]
   rational_best-simplify-1 [=>] 43.8	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{a \cdot 2}} \]

3. Taylor expanded in b around inf 2.9

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

4. Simplified 2.9

   \[\leadsto \color{blue}{\left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(16 \cdot {\left(c \cdot a\right)}^{4} - \frac{{\left(c \cdot a\right)}^{4}}{-0.25}\right) \cdot \frac{-0.25}{a \cdot {b}^{7}}} \]
   Proof

   [Start] 2.9	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
   rational_best-simplify-3 [=>] 2.9	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \color{blue}{\left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)} \]
   rational_best-simplify-47 [=>] 2.9	\[ \color{blue}{-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(\left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right) + -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)} \]
   rational_best-simplify-3 [<=] 2.9	\[ -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
   rational_best-simplify-3 [=>] 2.9	\[ \color{blue}{\left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) + -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]

5. Taylor expanded in c around 0 2.9

   \[\leadsto \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{-0.25 \cdot \frac{{c}^{4} \cdot \left(16 \cdot {a}^{4} - -4 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}} \]

6. Simplified 2.9

   \[\leadsto \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{\left(-{\left(c \cdot a\right)}^{4}\right) \cdot \frac{5}{a \cdot {b}^{7}}} \]
   Proof

   [Start] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + -0.25 \cdot \frac{{c}^{4} \cdot \left(16 \cdot {a}^{4} - -4 \cdot {a}^{4}\right)}{a \cdot {b}^{7}} \]
   rational_best-simplify-55 [=>] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{\left({c}^{4} \cdot \left(16 \cdot {a}^{4} - -4 \cdot {a}^{4}\right)\right) \cdot \frac{-0.25}{a \cdot {b}^{7}}} \]
   rational_best-simplify-1 [=>] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left({c}^{4} \cdot \left(16 \cdot {a}^{4} - \color{blue}{{a}^{4} \cdot -4}\right)\right) \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-62 [=>] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left({c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(16 - -4\right)\right)}\right) \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   metadata-eval [=>] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left({c}^{4} \cdot \left({a}^{4} \cdot \color{blue}{20}\right)\right) \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-50 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{\left(20 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)} \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-1 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(20 \cdot \color{blue}{\left({c}^{4} \cdot {a}^{4}\right)}\right) \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   exponential-simplify-27 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(20 \cdot \color{blue}{{\left(c \cdot a\right)}^{4}}\right) \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-1 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{\left({\left(c \cdot a\right)}^{4} \cdot 20\right)} \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-9 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{\left({\left(c \cdot a\right)}^{4} \cdot 20 - 0\right)} \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   metadata-eval [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left({\left(c \cdot a\right)}^{4} \cdot 20 - \color{blue}{\left(0 - 0\right)}\right) \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-51 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{\left(0 - \left(0 - {\left(c \cdot a\right)}^{4} \cdot 20\right)\right)} \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-14 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(0 - \color{blue}{\left(-{\left(c \cdot a\right)}^{4} \cdot 20\right)}\right) \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   rational_best-simplify-14 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \color{blue}{\left(-\left(-{\left(c \cdot a\right)}^{4} \cdot 20\right)\right)} \cdot \frac{-0.25}{a \cdot {b}^{7}} \]
   metadata-eval [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(-\left(-{\left(c \cdot a\right)}^{4} \cdot 20\right)\right) \cdot \frac{\color{blue}{\frac{-1}{4}}}{a \cdot {b}^{7}} \]
   rational_best-simplify-54 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(-\left(-{\left(c \cdot a\right)}^{4} \cdot 20\right)\right) \cdot \color{blue}{\frac{-1}{4 \cdot \left(a \cdot {b}^{7}\right)}} \]
   rational_best-simplify-1 [<=] 2.9	\[ \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(-\left(-{\left(c \cdot a\right)}^{4} \cdot 20\right)\right) \cdot \frac{-1}{\color{blue}{\left(a \cdot {b}^{7}\right) \cdot 4}} \]

7. Final simplification 2.9

   \[\leadsto \left(\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}}\right) + \left(-{\left(c \cdot a\right)}^{4}\right) \cdot \frac{5}{a \cdot {b}^{7}} \]

Alternatives

Alternative 1
Error	3.9
Cost	33536

\[\left(0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\right) + \left({c}^{3} \cdot {a}^{2}\right) \cdot \frac{-2}{{b}^{5}} \]

Alternative 2
Error	6.3
Cost	29700

\[\begin{array}{l} t_0 := \sqrt{b \cdot b + \frac{c \cdot a}{-0.25}}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5:\\ \;\;\;\;\frac{\frac{t_0 + \left(2 \cdot t_0 - b\right)}{a + a} - \frac{b + \frac{b - t_0}{2}}{a}}{4}\\ \mathbf{else}:\\ \;\;\;\;0 - \left(\left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b + b}\right) - \frac{\frac{c}{-2}}{b}\right)\\ \end{array} \]

Alternative 3
Error	4.2
Cost	27456

\[\frac{-2 \cdot \left(\frac{c \cdot a}{b} + \frac{{\left(c \cdot a\right)}^{2}}{{b}^{3}}\right) + {\left(c \cdot a\right)}^{3} \cdot \frac{-4}{{b}^{5}}}{a \cdot 2} \]

Alternative 4
Error	4.2
Cost	27392

\[\left(-0.5 \cdot \left(\frac{c}{a \cdot b} + \frac{{c}^{2}}{{b}^{3}}\right) + \left(-\frac{a \cdot {c}^{3}}{{b}^{5}}\right)\right) \cdot \frac{1}{\frac{0.5}{a}} \]

Alternative 5
Error	6.3
Cost	22724

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5:\\ \;\;\;\;\frac{\frac{0.5}{a} \cdot \left(\left(-b\right) + t_0\right)}{2} - \left(\frac{b}{a \cdot 4} - \frac{t_0}{a \cdot 4}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - \left(\left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b + b}\right) - \frac{\frac{c}{-2}}{b}\right)\\ \end{array} \]

Alternative 6
Error	6.3
Cost	22276

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -5:\\ \;\;\;\;\frac{t_0}{4 \cdot a} - \frac{b + \left(b - t_0\right)}{4 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \left(\left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b + b}\right) - \frac{\frac{c}{-2}}{b}\right)\\ \end{array} \]

Alternative 7
Error	6.1
Cost	21636

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -170:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot 2}{4 \cdot a} - \frac{b}{4 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \left(\left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b + b}\right) - \frac{\frac{c}{-2}}{b}\right)\\ \end{array} \]

Alternative 8
Error	6.1
Cost	21124

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -170:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot 2}{4 \cdot a} - \frac{b}{4 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;0 - \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 9
Error	10.8
Cost	15428

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot 2}{4 \cdot a} - \frac{b}{4 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 10
Error	10.8
Cost	15364

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b} - b\right) \cdot \left(\left(a \cdot \frac{-2}{a}\right) \cdot \frac{1}{a \cdot -4}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 11
Error	10.8
Cost	15364

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(\left(\sqrt{a \cdot \left(c \cdot -4\right) + b \cdot b} - b\right) \cdot \left(\left(a + a\right) \cdot \frac{-0.5}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 12
Error	10.8
Cost	15236

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + \frac{c \cdot a}{-0.25}} - b}{a}}{\frac{\frac{8}{\frac{4}{a}}}{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 13
Error	10.8
Cost	15108

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(a \cdot \frac{-0.5}{a}\right) \cdot \frac{b - \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 14
Error	10.8
Cost	15108

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-0.5}{a}}{a} \cdot \left(a \cdot \left(b - \sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 15
Error	10.8
Cost	14980

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \frac{c \cdot a}{-0.25}} - b}{\frac{8}{\frac{4}{a}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 16
Error	10.8
Cost	14852

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 17
Error	12.0
Cost	256

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

Alternative 18
Error	62.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))