Quadratic roots, narrow range

Average Error: 28.2 → 5.1

Time: 12.9s

Precision: binary64

Cost: 74628

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -30:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot \frac{1}{t_0 \cdot \left(t_0 \cdot \frac{1}{t_0}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{{c}^{4} \cdot \left(4 \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)\\ \end{array} \]

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 (/ (- (sqrt (+ (* b b) (* c (* a -4.0)))) b) (+ a a))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -30.0)
     (* t_0 (* t_0 (/ 1.0 (* t_0 (* t_0 (/ 1.0 t_0))))))
     (+
      (*
       -0.25
       (/
        (+
         (* (pow c 4.0) (* 4.0 (pow a 4.0)))
         (* 16.0 (* (pow c 4.0) (pow a 4.0))))
        (* a (pow b 7.0))))
      (+
       (* -1.0 (/ c b))
       (+
        (* -1.0 (/ (* a (pow c 2.0)) (pow b 3.0)))
        (* -2.0 (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)))))))))

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 = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a + a);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -30.0) {
		tmp = t_0 * (t_0 * (1.0 / (t_0 * (t_0 * (1.0 / t_0)))));
	} else {
		tmp = (-0.25 * (((pow(c, 4.0) * (4.0 * pow(a, 4.0))) + (16.0 * (pow(c, 4.0) * pow(a, 4.0)))) / (a * pow(b, 7.0)))) + ((-1.0 * (c / b)) + ((-1.0 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-2.0 * ((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0)))));
	}
	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 = (sqrt(((b * b) + (c * (a * (-4.0d0))))) - b) / (a + a)
    if (((-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)) <= (-30.0d0)) then
        tmp = t_0 * (t_0 * (1.0d0 / (t_0 * (t_0 * (1.0d0 / t_0)))))
    else
        tmp = ((-0.25d0) * ((((c ** 4.0d0) * (4.0d0 * (a ** 4.0d0))) + (16.0d0 * ((c ** 4.0d0) * (a ** 4.0d0)))) / (a * (b ** 7.0d0)))) + (((-1.0d0) * (c / b)) + (((-1.0d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-2.0d0) * (((c ** 3.0d0) * (a ** 2.0d0)) / (b ** 5.0d0)))))
    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 = (Math.sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a + a);
	double tmp;
	if (((-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -30.0) {
		tmp = t_0 * (t_0 * (1.0 / (t_0 * (t_0 * (1.0 / t_0)))));
	} else {
		tmp = (-0.25 * (((Math.pow(c, 4.0) * (4.0 * Math.pow(a, 4.0))) + (16.0 * (Math.pow(c, 4.0) * Math.pow(a, 4.0)))) / (a * Math.pow(b, 7.0)))) + ((-1.0 * (c / b)) + ((-1.0 * ((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)))));
	}
	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 = (math.sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a + a)
	tmp = 0
	if ((-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -30.0:
		tmp = t_0 * (t_0 * (1.0 / (t_0 * (t_0 * (1.0 / t_0)))))
	else:
		tmp = (-0.25 * (((math.pow(c, 4.0) * (4.0 * math.pow(a, 4.0))) + (16.0 * (math.pow(c, 4.0) * math.pow(a, 4.0)))) / (a * math.pow(b, 7.0)))) + ((-1.0 * (c / b)) + ((-1.0 * ((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)))))
	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(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -4.0)))) - b) / Float64(a + a))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -30.0)
		tmp = Float64(t_0 * Float64(t_0 * Float64(1.0 / Float64(t_0 * Float64(t_0 * Float64(1.0 / t_0))))));
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(Float64((c ^ 4.0) * Float64(4.0 * (a ^ 4.0))) + Float64(16.0 * Float64((c ^ 4.0) * (a ^ 4.0)))) / Float64(a * (b ^ 7.0)))) + Float64(Float64(-1.0 * Float64(c / b)) + Float64(Float64(-1.0 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-2.0 * Float64(Float64((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0))))));
	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 = (sqrt(((b * b) + (c * (a * -4.0)))) - b) / (a + a);
	tmp = 0.0;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -30.0)
		tmp = t_0 * (t_0 * (1.0 / (t_0 * (t_0 * (1.0 / t_0)))));
	else
		tmp = (-0.25 * ((((c ^ 4.0) * (4.0 * (a ^ 4.0))) + (16.0 * ((c ^ 4.0) * (a ^ 4.0)))) / (a * (b ^ 7.0)))) + ((-1.0 * (c / b)) + ((-1.0 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-2.0 * (((c ^ 3.0) * (a ^ 2.0)) / (b ^ 5.0)))));
	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[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], -30.0], N[(t$95$0 * N[(t$95$0 * N[(1.0 / N[(t$95$0 * N[(t$95$0 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(4.0 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(16.0 * N[(N[Power[c, 4.0], $MachinePrecision] * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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]), $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)) < -30

   1. Initial program 8.7

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

   2. Simplified 8.7

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

      [Start] 8.7	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 8.7	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(0 - b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 8.7	\[ \frac{\color{blue}{\left(0 + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) - b}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 8.7	\[ \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + 0\right)} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 8.7	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      metadata-eval [=>] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \color{blue}{1} + a \cdot \frac{4}{4}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{a} + a \cdot \frac{4}{4}} \]
      metadata-eval [=>] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a + a \cdot \color{blue}{1}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 8.7	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a + \color{blue}{a}} \]

   3. Applied egg-rr 8.7

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

   4. Applied egg-rr 8.7

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

   if -30 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 4 a) c)))) (*.f64 2 a))

   1. Initial program 30.1

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

   2. Simplified 30.1

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

      [Start] 30.1	\[ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 30.1	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-97 [=>] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \color{blue}{\left(0 - b\right)}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-108 [=>] 30.1	\[ \frac{\color{blue}{\left(0 + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) - b}}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 30.1	\[ \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + 0\right)} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 30.1	\[ \frac{\color{blue}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{2 \cdot a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{a \cdot 2}} \]
      metadata-eval [<=] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \color{blue}{\left(1 + 1\right)}} \]
      metadata-eval [<=] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \left(\color{blue}{\frac{4}{4}} + 1\right)} \]
      metadata-eval [<=] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \left(\frac{4}{4} + \color{blue}{\frac{4}{4}}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-23 [<=] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{\frac{4}{4} \cdot a + a \cdot \frac{4}{4}}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [<=] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{a \cdot \frac{4}{4}} + a \cdot \frac{4}{4}} \]
      metadata-eval [=>] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot \color{blue}{1} + a \cdot \frac{4}{4}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{\color{blue}{a} + a \cdot \frac{4}{4}} \]
      metadata-eval [=>] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a + a \cdot \color{blue}{1}} \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 30.1	\[ \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a + \color{blue}{a}} \]

   3. Taylor expanded in b around inf 4.8

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

   4. Simplified 4.7

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

      [Start] 4.8	\[ -1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 4.8	\[ \color{blue}{-0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + \left(-1 \cdot \frac{c}{b} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 4.7	\[ -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \color{blue}{\left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 4.7	\[ -0.25 \cdot \frac{{\left(-2 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

   5. Taylor expanded in c around 0 4.7

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

   6. Simplified 4.7

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

      [Start] 4.7	\[ -0.25 \cdot \frac{4 \cdot \left({c}^{4} \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 4.7	\[ -0.25 \cdot \frac{\color{blue}{{c}^{4} \cdot \left(4 \cdot {a}^{4}\right)} + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -30:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a} \cdot \left(\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a} \cdot \frac{1}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a} \cdot \left(\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a} \cdot \frac{1}{\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{{c}^{4} \cdot \left(4 \cdot {a}^{4}\right) + 16 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	5.1
Cost	74500

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a}\\ t_1 := \frac{{c}^{4}}{{b}^{6}}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -30:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot \frac{1}{t_0 \cdot \left(t_0 \cdot \frac{1}{t_0}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + \left(-2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + -0.25 \cdot \frac{{a}^{3} \cdot \left(16 \cdot t_1 + 4 \cdot t_1\right)}{b}\right)\\ \end{array} \]

Alternative 2
Error	6.7
Cost	44804

\[\begin{array}{l} t_0 := \frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}{a + a}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -30:\\ \;\;\;\;t_0 \cdot \left(t_0 \cdot \frac{1}{t_0 \cdot \left(t_0 \cdot \frac{1}{t_0}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\\ \end{array} \]

Alternative 3
Error	6.7
Cost	41156

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;\frac{\left(-b\right) + t_0}{2 \cdot a} \leq -30:\\ \;\;\;\;\frac{t_0 - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b} + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)\\ \end{array} \]

Alternative 4
Error	6.7
Cost	41028

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;\frac{\left(-b\right) + t_0}{2 \cdot a} \leq -6:\\ \;\;\;\;\frac{t_0 - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right) + -2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\\ \end{array} \]

Alternative 5
Error	15.2
Cost	14852

\[\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;\frac{\left(-b\right) + t_0}{2 \cdot a} \leq -4.47 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Alternative 6
Error	10.0
Cost	13764

\[\begin{array}{l} \mathbf{if}\;b \leq 15:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{c}{b} + \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 7
Error	23.1
Cost	256

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

Reproduce

herbie shell --seed 2023090 
(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)))