ABCF->ab-angle a

Average Error: 52.5 → 44.8

Time: 59.4s

Precision: binary64

Cost: 41632

Math

\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

↓

\[\begin{array}{l} t_0 := \sqrt{C \cdot \left(F \cdot -16\right)}\\ t_1 := A \cdot \left(C \cdot 4\right)\\ t_2 := 4 \cdot \left(A \cdot C\right)\\ t_3 := {B}^{2} - t_2\\ \mathbf{if}\;A \leq -7.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{t_2 - {B}^{2}} \cdot \sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right) \cdot \left(F \cdot t_3\right)\right)}\\ \mathbf{elif}\;A \leq -7.5 \cdot 10^{-66}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -2.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -2 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F \cdot 4\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;A \leq -4.4 \cdot 10^{-269}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq -2.1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{elif}\;A \leq 3 \cdot 10^{-118}:\\ \;\;\;\;\frac{0.25}{C} \cdot t_0\\ \mathbf{elif}\;A \leq 3.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{C} \cdot 0.25\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* C (* F -16.0))))
        (t_1 (* A (* C 4.0)))
        (t_2 (* 4.0 (* A C)))
        (t_3 (- (pow B 2.0) t_2)))
   (if (<= A -7.8e+65)
     (*
      (/ 1.0 (- t_2 (pow B 2.0)))
      (sqrt (* 2.0 (* (+ C (+ C (* -0.5 (/ (pow B 2.0) A)))) (* F t_3)))))
     (if (<= A -7.5e-66)
       (- (sqrt (- (/ F A))))
       (if (<= A -2.05e-138)
         (* (/ -1.0 B) (sqrt (* (+ B C) (* 2.0 F))))
         (if (<= A -2e-174)
           (/
            (sqrt (* C (* (- (pow B 2.0) t_1) (* F 4.0))))
            (- t_1 (pow B 2.0)))
           (if (<= A -4.4e-269)
             (* 0.25 (sqrt (* (/ F C) -16.0)))
             (if (<= A -2.1e-302)
               (* (/ (sqrt 2.0) B) (- (sqrt (* F B))))
               (if (<= A 3e-118)
                 (* (/ 0.25 C) t_0)
                 (if (<= A 3.4e+46)
                   (/
                    (-
                     (sqrt
                      (*
                       t_3
                       (*
                        (+ A (+ C (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))
                        (* 2.0 F)))))
                    t_3)
                   (* (/ t_0 C) 0.25)))))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double t_0 = sqrt((C * (F * -16.0)));
	double t_1 = A * (C * 4.0);
	double t_2 = 4.0 * (A * C);
	double t_3 = pow(B, 2.0) - t_2;
	double tmp;
	if (A <= -7.8e+65) {
		tmp = (1.0 / (t_2 - pow(B, 2.0))) * sqrt((2.0 * ((C + (C + (-0.5 * (pow(B, 2.0) / A)))) * (F * t_3))));
	} else if (A <= -7.5e-66) {
		tmp = -sqrt(-(F / A));
	} else if (A <= -2.05e-138) {
		tmp = (-1.0 / B) * sqrt(((B + C) * (2.0 * F)));
	} else if (A <= -2e-174) {
		tmp = sqrt((C * ((pow(B, 2.0) - t_1) * (F * 4.0)))) / (t_1 - pow(B, 2.0));
	} else if (A <= -4.4e-269) {
		tmp = 0.25 * sqrt(((F / C) * -16.0));
	} else if (A <= -2.1e-302) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * B));
	} else if (A <= 3e-118) {
		tmp = (0.25 / C) * t_0;
	} else if (A <= 3.4e+46) {
		tmp = -sqrt((t_3 * ((A + (C + sqrt((pow(B, 2.0) + pow((A - C), 2.0))))) * (2.0 * F)))) / t_3;
	} else {
		tmp = (t_0 / C) * 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((2.0d0 * (((b ** 2.0d0) - ((4.0d0 * a) * c)) * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / ((b ** 2.0d0) - ((4.0d0 * a) * c))
end function

↓

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((c * (f * (-16.0d0))))
    t_1 = a * (c * 4.0d0)
    t_2 = 4.0d0 * (a * c)
    t_3 = (b ** 2.0d0) - t_2
    if (a <= (-7.8d+65)) then
        tmp = (1.0d0 / (t_2 - (b ** 2.0d0))) * sqrt((2.0d0 * ((c + (c + ((-0.5d0) * ((b ** 2.0d0) / a)))) * (f * t_3))))
    else if (a <= (-7.5d-66)) then
        tmp = -sqrt(-(f / a))
    else if (a <= (-2.05d-138)) then
        tmp = ((-1.0d0) / b) * sqrt(((b + c) * (2.0d0 * f)))
    else if (a <= (-2d-174)) then
        tmp = sqrt((c * (((b ** 2.0d0) - t_1) * (f * 4.0d0)))) / (t_1 - (b ** 2.0d0))
    else if (a <= (-4.4d-269)) then
        tmp = 0.25d0 * sqrt(((f / c) * (-16.0d0)))
    else if (a <= (-2.1d-302)) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((f * b))
    else if (a <= 3d-118) then
        tmp = (0.25d0 / c) * t_0
    else if (a <= 3.4d+46) then
        tmp = -sqrt((t_3 * ((a + (c + sqrt(((b ** 2.0d0) + ((a - c) ** 2.0d0))))) * (2.0d0 * f)))) / t_3
    else
        tmp = (t_0 / c) * 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double A, double B, double C, double F) {
	return -Math.sqrt(((2.0 * ((Math.pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / (Math.pow(B, 2.0) - ((4.0 * A) * C));
}

↓

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.sqrt((C * (F * -16.0)));
	double t_1 = A * (C * 4.0);
	double t_2 = 4.0 * (A * C);
	double t_3 = Math.pow(B, 2.0) - t_2;
	double tmp;
	if (A <= -7.8e+65) {
		tmp = (1.0 / (t_2 - Math.pow(B, 2.0))) * Math.sqrt((2.0 * ((C + (C + (-0.5 * (Math.pow(B, 2.0) / A)))) * (F * t_3))));
	} else if (A <= -7.5e-66) {
		tmp = -Math.sqrt(-(F / A));
	} else if (A <= -2.05e-138) {
		tmp = (-1.0 / B) * Math.sqrt(((B + C) * (2.0 * F)));
	} else if (A <= -2e-174) {
		tmp = Math.sqrt((C * ((Math.pow(B, 2.0) - t_1) * (F * 4.0)))) / (t_1 - Math.pow(B, 2.0));
	} else if (A <= -4.4e-269) {
		tmp = 0.25 * Math.sqrt(((F / C) * -16.0));
	} else if (A <= -2.1e-302) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * B));
	} else if (A <= 3e-118) {
		tmp = (0.25 / C) * t_0;
	} else if (A <= 3.4e+46) {
		tmp = -Math.sqrt((t_3 * ((A + (C + Math.sqrt((Math.pow(B, 2.0) + Math.pow((A - C), 2.0))))) * (2.0 * F)))) / t_3;
	} else {
		tmp = (t_0 / C) * 0.25;
	}
	return tmp;
}

Python

def code(A, B, C, F):
	return -math.sqrt(((2.0 * ((math.pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / (math.pow(B, 2.0) - ((4.0 * A) * C))

↓

def code(A, B, C, F):
	t_0 = math.sqrt((C * (F * -16.0)))
	t_1 = A * (C * 4.0)
	t_2 = 4.0 * (A * C)
	t_3 = math.pow(B, 2.0) - t_2
	tmp = 0
	if A <= -7.8e+65:
		tmp = (1.0 / (t_2 - math.pow(B, 2.0))) * math.sqrt((2.0 * ((C + (C + (-0.5 * (math.pow(B, 2.0) / A)))) * (F * t_3))))
	elif A <= -7.5e-66:
		tmp = -math.sqrt(-(F / A))
	elif A <= -2.05e-138:
		tmp = (-1.0 / B) * math.sqrt(((B + C) * (2.0 * F)))
	elif A <= -2e-174:
		tmp = math.sqrt((C * ((math.pow(B, 2.0) - t_1) * (F * 4.0)))) / (t_1 - math.pow(B, 2.0))
	elif A <= -4.4e-269:
		tmp = 0.25 * math.sqrt(((F / C) * -16.0))
	elif A <= -2.1e-302:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * B))
	elif A <= 3e-118:
		tmp = (0.25 / C) * t_0
	elif A <= 3.4e+46:
		tmp = -math.sqrt((t_3 * ((A + (C + math.sqrt((math.pow(B, 2.0) + math.pow((A - C), 2.0))))) * (2.0 * F)))) / t_3
	else:
		tmp = (t_0 / C) * 0.25
	return tmp

Julia

function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
end

↓

function code(A, B, C, F)
	t_0 = sqrt(Float64(C * Float64(F * -16.0)))
	t_1 = Float64(A * Float64(C * 4.0))
	t_2 = Float64(4.0 * Float64(A * C))
	t_3 = Float64((B ^ 2.0) - t_2)
	tmp = 0.0
	if (A <= -7.8e+65)
		tmp = Float64(Float64(1.0 / Float64(t_2 - (B ^ 2.0))) * sqrt(Float64(2.0 * Float64(Float64(C + Float64(C + Float64(-0.5 * Float64((B ^ 2.0) / A)))) * Float64(F * t_3)))));
	elseif (A <= -7.5e-66)
		tmp = Float64(-sqrt(Float64(-Float64(F / A))));
	elseif (A <= -2.05e-138)
		tmp = Float64(Float64(-1.0 / B) * sqrt(Float64(Float64(B + C) * Float64(2.0 * F))));
	elseif (A <= -2e-174)
		tmp = Float64(sqrt(Float64(C * Float64(Float64((B ^ 2.0) - t_1) * Float64(F * 4.0)))) / Float64(t_1 - (B ^ 2.0)));
	elseif (A <= -4.4e-269)
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / C) * -16.0)));
	elseif (A <= -2.1e-302)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * B))));
	elseif (A <= 3e-118)
		tmp = Float64(Float64(0.25 / C) * t_0);
	elseif (A <= 3.4e+46)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * Float64(Float64(A + Float64(C + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))) * Float64(2.0 * F))))) / t_3);
	else
		tmp = Float64(Float64(t_0 / C) * 0.25);
	end
	return tmp
end

MATLAB

function tmp = code(A, B, C, F)
	tmp = -sqrt(((2.0 * (((B ^ 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / ((B ^ 2.0) - ((4.0 * A) * C));
end

↓

function tmp_2 = code(A, B, C, F)
	t_0 = sqrt((C * (F * -16.0)));
	t_1 = A * (C * 4.0);
	t_2 = 4.0 * (A * C);
	t_3 = (B ^ 2.0) - t_2;
	tmp = 0.0;
	if (A <= -7.8e+65)
		tmp = (1.0 / (t_2 - (B ^ 2.0))) * sqrt((2.0 * ((C + (C + (-0.5 * ((B ^ 2.0) / A)))) * (F * t_3))));
	elseif (A <= -7.5e-66)
		tmp = -sqrt(-(F / A));
	elseif (A <= -2.05e-138)
		tmp = (-1.0 / B) * sqrt(((B + C) * (2.0 * F)));
	elseif (A <= -2e-174)
		tmp = sqrt((C * (((B ^ 2.0) - t_1) * (F * 4.0)))) / (t_1 - (B ^ 2.0));
	elseif (A <= -4.4e-269)
		tmp = 0.25 * sqrt(((F / C) * -16.0));
	elseif (A <= -2.1e-302)
		tmp = (sqrt(2.0) / B) * -sqrt((F * B));
	elseif (A <= 3e-118)
		tmp = (0.25 / C) * t_0;
	elseif (A <= 3.4e+46)
		tmp = -sqrt((t_3 * ((A + (C + sqrt(((B ^ 2.0) + ((A - C) ^ 2.0))))) * (2.0 * F)))) / t_3;
	else
		tmp = (t_0 / C) * 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(C * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[A, -7.8e+65], N[(N[(1.0 / N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(C + N[(C + N[(-0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -7.5e-66], (-N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision]), If[LessEqual[A, -2.05e-138], N[(N[(-1.0 / B), $MachinePrecision] * N[Sqrt[N[(N[(B + C), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -2e-174], N[(N[Sqrt[N[(C * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * N[(F * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -4.4e-269], N[(0.25 * N[Sqrt[N[(N[(F / C), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -2.1e-302], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[A, 3e-118], N[(N[(0.25 / C), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[A, 3.4e+46], N[((-N[Sqrt[N[(t$95$3 * N[(N[(A + N[(C + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(N[(t$95$0 / C), $MachinePrecision] * 0.25), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if A < -7.7999999999999996e65

   1. Initial program 62.5

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 63.4

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 62.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 62.5	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around -inf 46.4

      \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   4. Applied egg-rr 44.3

      \[\leadsto \color{blue}{\frac{1}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \cdot \sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right) \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)}} \]

   if -7.7999999999999996e65 < A < -7.49999999999999995e-66

   1. Initial program 54.3

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 53.7

      \[\leadsto \color{blue}{\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
      Proof

      [Start] 54.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 54.3	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 54.3	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 54.3	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in C around inf 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right)} \]

   4. Simplified 43.5

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{A}}} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{A}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{A}} \]
      exponential.json-simplify-20 [=>] 43.5	\[ -\color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational.json-simplify-9 [=>] 43.5	\[ -\sqrt{\color{blue}{-\frac{F}{A}}} \]

   if -7.49999999999999995e-66 < A < -2.05e-138

   1. Initial program 47.3

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 46.6

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 47.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 47.3	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around inf 57.2

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{B}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]

   4. Taylor expanded in A around 0 54.1

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + B\right) \cdot F}\right)} \]

   5. Simplified 54.1

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + B\right)}\right)} \]
      Proof

      [Start] 54.1	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + B\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 54.1	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(C + B\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 54.1	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(C + B\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 54.1	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(C + B\right)}}\right) \]

   6. Applied egg-rr 54.1

      \[\leadsto \color{blue}{\frac{-1}{B} \cdot \sqrt{F \cdot \left(\left(C + B\right) \cdot 2\right)} + 0} \]

   7. Simplified 54.1

      \[\leadsto \color{blue}{\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}} \]
      Proof

      [Start] 54.1	\[ \frac{-1}{B} \cdot \sqrt{F \cdot \left(\left(C + B\right) \cdot 2\right)} + 0 \]
      rational.json-simplify-4 [=>] 54.1	\[ \color{blue}{\frac{-1}{B} \cdot \sqrt{F \cdot \left(\left(C + B\right) \cdot 2\right)}} \]
      rational.json-simplify-43 [=>] 54.1	\[ \frac{-1}{B} \cdot \sqrt{\color{blue}{\left(C + B\right) \cdot \left(2 \cdot F\right)}} \]
      rational.json-simplify-1 [=>] 54.1	\[ \frac{-1}{B} \cdot \sqrt{\color{blue}{\left(B + C\right)} \cdot \left(2 \cdot F\right)} \]

   if -2.05e-138 < A < -2e-174

   1. Initial program 48.7

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 51.3

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 48.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 48.7	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around -inf 56.7

      \[\leadsto \frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \color{blue}{\left(2 \cdot C\right)}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   4. Applied egg-rr 54.0

      \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(\left(F \cdot 4\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} + 0} \]

   5. Simplified 54.2

      \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(F \cdot 4\right)\right)}}{A \cdot \left(C \cdot 4\right) - {B}^{2}}} \]
      Proof

      [Start] 54.0	\[ \frac{\sqrt{C \cdot \left(\left(F \cdot 4\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 54.0	\[ \color{blue}{\frac{\sqrt{C \cdot \left(\left(F \cdot 4\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      rational.json-simplify-2 [=>] 54.0	\[ \frac{\sqrt{C \cdot \color{blue}{\left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot 4\right)\right)}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 54.2	\[ \frac{\sqrt{C \cdot \left(\left({B}^{2} - \color{blue}{A \cdot \left(C \cdot 4\right)}\right) \cdot \left(F \cdot 4\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 54.2	\[ \frac{\sqrt{C \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(F \cdot 4\right)\right)}}{\color{blue}{A \cdot \left(C \cdot 4\right)} - {B}^{2}} \]

   if -2e-174 < A < -4.39999999999999968e-269

   1. Initial program 49.1

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 51.2

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 49.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 49.1	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around 0 64.0

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{C}}\right)} \]

   4. Simplified 56.1

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}} \]
      Proof

      [Start] 64.0	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{C}}\right) \]
      exponential.json-simplify-20 [=>] 64.0	\[ 0.25 \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{\frac{F}{C}}\right) \]
      metadata-eval [=>] 64.0	\[ 0.25 \cdot \left(\sqrt{\color{blue}{-16}} \cdot \sqrt{\frac{F}{C}}\right) \]
      exponential.json-simplify-20 [=>] 56.1	\[ 0.25 \cdot \color{blue}{\sqrt{\frac{F}{C} \cdot -16}} \]

   if -4.39999999999999968e-269 < A < -2.10000000000000013e-302

   1. Initial program 50.2

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 50.1

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 50.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 50.2	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around inf 58.5

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(4 \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C + \color{blue}{B}\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - {B}^{2}} \]

   4. Taylor expanded in A around 0 52.6

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + B\right) \cdot F}\right)} \]

   5. Simplified 52.6

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + B\right)}\right)} \]
      Proof

      [Start] 52.6	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + B\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 52.6	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(C + B\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 52.6	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(C + B\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 52.6	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(C + B\right)}}\right) \]

   6. Taylor expanded in C around 0 52.8

      \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\color{blue}{\sqrt{F \cdot B}}\right) \]

   if -2.10000000000000013e-302 < A < 3.00000000000000018e-118

   1. Initial program 48.9

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 51.0

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 48.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 48.9	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around 0 64.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   4. Simplified 56.7

      \[\leadsto \frac{\color{blue}{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      Proof

      [Start] 64.0	\[ \frac{\left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\color{blue}{\sqrt{C \cdot F} \cdot \left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{\sqrt{C \cdot F} \cdot \color{blue}{\left(A \cdot \left(\sqrt{-8} \cdot \sqrt{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [<=] 64.0	\[ \frac{\sqrt{C \cdot F} \cdot \left(A \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{-8}\right)}\right)}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{\color{blue}{A \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{C \cdot F}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \frac{A \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{C \cdot F}\right)}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      metadata-eval [=>] 64.0	\[ \frac{A \cdot \left(\sqrt{\color{blue}{-16}} \cdot \sqrt{C \cdot F}\right)}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      exponential.json-simplify-20 [=>] 56.7	\[ \frac{A \cdot \color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 56.7	\[ \frac{A \cdot \sqrt{\color{blue}{\left(F \cdot C\right)} \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   5. Taylor expanded in A around inf 53.7

      \[\leadsto \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]

   6. Simplified 53.6

      \[\leadsto \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{A \cdot \left(C \cdot 4\right)}} \]
      Proof

      [Start] 53.7	\[ \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 53.6	\[ \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{A \cdot \left(C \cdot 4\right)}} \]

   7. Applied egg-rr 50.6

      \[\leadsto \color{blue}{\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}} \]

   if 3.00000000000000018e-118 < A < 3.3999999999999998e46

   1. Initial program 43.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 42.4

      \[\leadsto \color{blue}{\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
      Proof

      [Start] 43.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 43.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 43.0	\[ \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [=>] 43.0	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   if 3.3999999999999998e46 < A

   1. Initial program 54.6

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 55.6

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      Proof

      [Start] 54.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 54.6	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around 0 64.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   4. Simplified 44.5

      \[\leadsto \frac{\color{blue}{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      Proof

      [Start] 64.0	\[ \frac{\left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\color{blue}{\sqrt{C \cdot F} \cdot \left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{\sqrt{C \cdot F} \cdot \color{blue}{\left(A \cdot \left(\sqrt{-8} \cdot \sqrt{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [<=] 64.0	\[ \frac{\sqrt{C \cdot F} \cdot \left(A \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{-8}\right)}\right)}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{\color{blue}{A \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{C \cdot F}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \frac{A \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{C \cdot F}\right)}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      metadata-eval [=>] 64.0	\[ \frac{A \cdot \left(\sqrt{\color{blue}{-16}} \cdot \sqrt{C \cdot F}\right)}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      exponential.json-simplify-20 [=>] 44.5	\[ \frac{A \cdot \color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 44.5	\[ \frac{A \cdot \sqrt{\color{blue}{\left(F \cdot C\right)} \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   5. Taylor expanded in A around inf 44.4

      \[\leadsto \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]

   6. Simplified 44.4

      \[\leadsto \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{A \cdot \left(C \cdot 4\right)}} \]
      Proof

      [Start] 44.4	\[ \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 44.4	\[ \frac{A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{A \cdot \left(C \cdot 4\right)}} \]

   7. Applied egg-rr 34.4

      \[\leadsto \color{blue}{\frac{\sqrt{C \cdot \left(F \cdot -16\right)}}{C} \cdot 0.25} \]
3. Recombined 9 regimes into one program.

4. Final simplification 44.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -7.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \cdot \sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right) \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right)}\\ \mathbf{elif}\;A \leq -7.5 \cdot 10^{-66}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -2.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -2 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(F \cdot 4\right)\right)}}{A \cdot \left(C \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;A \leq -4.4 \cdot 10^{-269}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq -2.1 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{elif}\;A \leq 3 \cdot 10^{-118}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \mathbf{elif}\;A \leq 3.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{-\sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(F \cdot -16\right)}}{C} \cdot 0.25\\ \end{array} \]

Alternatives

Alternative 1
Error	45.1
Cost	41568

\[\begin{array}{l} t_0 := \sqrt{C \cdot \left(F \cdot -16\right)}\\ t_1 := A \cdot \left(C \cdot 4\right)\\ t_2 := 4 \cdot \left(A \cdot C\right)\\ t_3 := t_2 - {B}^{2}\\ t_4 := {B}^{2} - t_2\\ \mathbf{if}\;A \leq -3.9 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{t_3} \cdot \sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right) \cdot \left(F \cdot t_4\right)\right)}\\ \mathbf{elif}\;A \leq -3.5 \cdot 10^{-64}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -4.6 \cdot 10^{-175}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F \cdot 4\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;A \leq -3.8 \cdot 10^{-268}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq 7 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{elif}\;A \leq 3.7 \cdot 10^{-118}:\\ \;\;\;\;\frac{0.25}{C} \cdot t_0\\ \mathbf{elif}\;A \leq 5.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_4 \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{C} \cdot 0.25\\ \end{array} \]

Alternative 2
Error	44.7
Cost	41568

\[\begin{array}{l} t_0 := \sqrt{C \cdot \left(F \cdot -16\right)}\\ t_1 := A \cdot \left(C \cdot 4\right)\\ t_2 := 4 \cdot \left(A \cdot C\right)\\ t_3 := A \cdot \left(4 \cdot C\right)\\ \mathbf{if}\;A \leq -2.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{t_2 - {B}^{2}} \cdot \sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right) \cdot \left(F \cdot \left({B}^{2} - t_2\right)\right)\right)}\\ \mathbf{elif}\;A \leq -2.5 \cdot 10^{-66}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -3.3 \cdot 10^{-139}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -1.76 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F \cdot 4\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;A \leq -4.5 \cdot 10^{-266}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq 2.55 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{elif}\;A \leq 6 \cdot 10^{-119}:\\ \;\;\;\;\frac{0.25}{C} \cdot t_0\\ \mathbf{elif}\;A \leq 2.15 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(\left({B}^{2} - t_3\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{t_3 - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{C} \cdot 0.25\\ \end{array} \]

Alternative 3
Error	45.2
Cost	27908

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -6 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{t_1 - {B}^{2}} \cdot \sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right) \cdot \left(F \cdot \left({B}^{2} - t_1\right)\right)\right)}\\ \mathbf{elif}\;A \leq -6.5 \cdot 10^{-66}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -2 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot 4\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{elif}\;A \leq -3.4 \cdot 10^{-271}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq -1.65 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \end{array} \]

Alternative 4
Error	45.2
Cost	27780

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot 4\right)\\ \mathbf{if}\;A \leq -5 \cdot 10^{+65}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{elif}\;A \leq -2.5 \cdot 10^{-64}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -1.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -1.8 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F \cdot 4\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;A \leq -4.3 \cdot 10^{-268}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq 1.9 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \end{array} \]

Alternative 5
Error	46.1
Cost	21332

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := \frac{\sqrt{C \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot 4\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{if}\;A \leq -6.5 \cdot 10^{+225}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -6.8 \cdot 10^{-65}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -1.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -2.6 \cdot 10^{-174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -5.2 \cdot 10^{-266}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq 1.25 \cdot 10^{-306}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \end{array} \]

Alternative 6
Error	46.1
Cost	21332

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot 4\right)\\ \mathbf{if}\;A \leq -1.6 \cdot 10^{+227}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{t_0 - {B}^{2}} \cdot \sqrt{C \cdot \left(\left(F \cdot 4\right) \cdot \left({B}^{2} - t_0\right)\right)}\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-65}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -4.2 \cdot 10^{-138}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -1.65 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F \cdot 4\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;A \leq -3.1 \cdot 10^{-265}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq -3 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \end{array} \]

Alternative 7
Error	45.0
Cost	14236

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;A \leq -6.2 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -7.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{-C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;A \leq -1.4 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.35 \cdot 10^{-135}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -1.25 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -6.5 \cdot 10^{-268}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq 5.4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \end{array} \]

Alternative 8
Error	44.5
Cost	13972

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;A \leq -1.7 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -6.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -7.8 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -4 \cdot 10^{-270}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq -5.2 \cdot 10^{-303}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \end{array} \]

Alternative 9
Error	47.6
Cost	7636

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ t_1 := \frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \mathbf{if}\;B \leq -2 \cdot 10^{-118}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-214}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 0.68:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 10
Error	47.5
Cost	7636

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ t_1 := \sqrt{C \cdot \left(F \cdot -16\right)}\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{-118}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-215}:\\ \;\;\;\;\frac{0.25}{C} \cdot t_1\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-304}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 0.98:\\ \;\;\;\;\frac{t_1}{C} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]

Alternative 11
Error	44.4
Cost	7636

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;A \leq -9 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.15 \cdot 10^{-132}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(B + C\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;A \leq -3.2 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.9 \cdot 10^{-264}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;A \leq -1.15 \cdot 10^{-306}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{C} \cdot \sqrt{C \cdot \left(F \cdot -16\right)}\\ \end{array} \]

Alternative 12
Error	47.6
Cost	7508

\[\begin{array}{l} t_0 := -\sqrt{\frac{F}{B} \cdot 2}\\ t_1 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;A \leq -5.9 \cdot 10^{-223}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq 7.4 \cdot 10^{-202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 4.4 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	47.1
Cost	7048

\[\begin{array}{l} \mathbf{if}\;A \leq -5.6 \cdot 10^{-223}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq 7.4 \cdot 10^{-202}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 14
Error	47.6
Cost	6852

\[\begin{array}{l} \mathbf{if}\;C \leq -3.1 \cdot 10^{-298}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 15
Error	55.1
Cost	6720

\[-\sqrt{-\frac{F}{A}} \]

Reproduce

herbie shell --seed 2023073 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))