ABCF->ab-angle b

Average Error: 52.2 → 42.6

Time: 1.0min

Precision: binary64

Cost: 42160

Math

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

↓

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ t_1 := \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ t_3 := {B}^{2} - A \cdot \left(C \cdot 4\right)\\ t_4 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ t_5 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_6 := \frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{t_3}\\ \mathbf{if}\;B \leq -3.3 \cdot 10^{+93}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-98}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(t_4 \cdot \left(\left(A + C\right) - t_1\right)\right)}}{t_4}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-159}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -4.6 \cdot 10^{-169}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-271}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(\left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right) \cdot -8\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-218}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(A - \left(t_1 - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

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 (- (/ F C)))))
        (t_1 (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))
        (t_2 (- (sqrt (- (/ F A)))))
        (t_3 (- (pow B 2.0) (* A (* C 4.0))))
        (t_4 (- (pow B 2.0) (* 4.0 (* A C))))
        (t_5 (* 0.25 (sqrt (* (/ F C) -16.0))))
        (t_6 (/ (- (sqrt (* -8.0 (* (- C (- C)) (* F (* A C)))))) t_3)))
   (if (<= B -3.3e+93)
     (- (sqrt (* 2.0 (/ F B))))
     (if (<= B -3e-98)
       (/ (- (sqrt (* (* 2.0 F) (* t_4 (- (+ A C) t_1))))) t_4)
       (if (<= B -6.5e-159)
         t_5
         (if (<= B -4.6e-169)
           t_2
           (if (<= B -1.1e-271)
             t_6
             (if (<= B 1.18e-301)
               (/ (- (sqrt (* A (* (* C (* F (- A (- A)))) -8.0)))) t_3)
               (if (<= B 1.65e-218)
                 t_6
                 (if (<= B 1.7e-167)
                   t_0
                   (if (<= B 6.5e-151)
                     t_2
                     (if (<= B 2.2e-115)
                       t_5
                       (if (<= B 2.5e-39)
                         t_0
                         (if (<= B 1.6e+20)
                           (/
                            (-
                             (sqrt
                              (*
                               (* 2.0 (* F (- (pow B 2.0) (* C (* 4.0 A)))))
                               (- A (- t_1 C)))))
                            (- (pow B 2.0) (* (* 4.0 A) C)))
                           (- (sqrt (* (/ F B) -2.0)))))))))))))))))

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(-(F / C));
	double t_1 = sqrt((pow(B, 2.0) + pow((A - C), 2.0)));
	double t_2 = -sqrt(-(F / A));
	double t_3 = pow(B, 2.0) - (A * (C * 4.0));
	double t_4 = pow(B, 2.0) - (4.0 * (A * C));
	double t_5 = 0.25 * sqrt(((F / C) * -16.0));
	double t_6 = -sqrt((-8.0 * ((C - -C) * (F * (A * C))))) / t_3;
	double tmp;
	if (B <= -3.3e+93) {
		tmp = -sqrt((2.0 * (F / B)));
	} else if (B <= -3e-98) {
		tmp = -sqrt(((2.0 * F) * (t_4 * ((A + C) - t_1)))) / t_4;
	} else if (B <= -6.5e-159) {
		tmp = t_5;
	} else if (B <= -4.6e-169) {
		tmp = t_2;
	} else if (B <= -1.1e-271) {
		tmp = t_6;
	} else if (B <= 1.18e-301) {
		tmp = -sqrt((A * ((C * (F * (A - -A))) * -8.0))) / t_3;
	} else if (B <= 1.65e-218) {
		tmp = t_6;
	} else if (B <= 1.7e-167) {
		tmp = t_0;
	} else if (B <= 6.5e-151) {
		tmp = t_2;
	} else if (B <= 2.2e-115) {
		tmp = t_5;
	} else if (B <= 2.5e-39) {
		tmp = t_0;
	} else if (B <= 1.6e+20) {
		tmp = -sqrt(((2.0 * (F * (pow(B, 2.0) - (C * (4.0 * A))))) * (A - (t_1 - C)))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -sqrt(((F / B) * -2.0));
	}
	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) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = -sqrt(-(f / c))
    t_1 = sqrt(((b ** 2.0d0) + ((a - c) ** 2.0d0)))
    t_2 = -sqrt(-(f / a))
    t_3 = (b ** 2.0d0) - (a * (c * 4.0d0))
    t_4 = (b ** 2.0d0) - (4.0d0 * (a * c))
    t_5 = 0.25d0 * sqrt(((f / c) * (-16.0d0)))
    t_6 = -sqrt(((-8.0d0) * ((c - -c) * (f * (a * c))))) / t_3
    if (b <= (-3.3d+93)) then
        tmp = -sqrt((2.0d0 * (f / b)))
    else if (b <= (-3d-98)) then
        tmp = -sqrt(((2.0d0 * f) * (t_4 * ((a + c) - t_1)))) / t_4
    else if (b <= (-6.5d-159)) then
        tmp = t_5
    else if (b <= (-4.6d-169)) then
        tmp = t_2
    else if (b <= (-1.1d-271)) then
        tmp = t_6
    else if (b <= 1.18d-301) then
        tmp = -sqrt((a * ((c * (f * (a - -a))) * (-8.0d0)))) / t_3
    else if (b <= 1.65d-218) then
        tmp = t_6
    else if (b <= 1.7d-167) then
        tmp = t_0
    else if (b <= 6.5d-151) then
        tmp = t_2
    else if (b <= 2.2d-115) then
        tmp = t_5
    else if (b <= 2.5d-39) then
        tmp = t_0
    else if (b <= 1.6d+20) then
        tmp = -sqrt(((2.0d0 * (f * ((b ** 2.0d0) - (c * (4.0d0 * a))))) * (a - (t_1 - c)))) / ((b ** 2.0d0) - ((4.0d0 * a) * c))
    else
        tmp = -sqrt(((f / b) * (-2.0d0)))
    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(-(F / C));
	double t_1 = Math.sqrt((Math.pow(B, 2.0) + Math.pow((A - C), 2.0)));
	double t_2 = -Math.sqrt(-(F / A));
	double t_3 = Math.pow(B, 2.0) - (A * (C * 4.0));
	double t_4 = Math.pow(B, 2.0) - (4.0 * (A * C));
	double t_5 = 0.25 * Math.sqrt(((F / C) * -16.0));
	double t_6 = -Math.sqrt((-8.0 * ((C - -C) * (F * (A * C))))) / t_3;
	double tmp;
	if (B <= -3.3e+93) {
		tmp = -Math.sqrt((2.0 * (F / B)));
	} else if (B <= -3e-98) {
		tmp = -Math.sqrt(((2.0 * F) * (t_4 * ((A + C) - t_1)))) / t_4;
	} else if (B <= -6.5e-159) {
		tmp = t_5;
	} else if (B <= -4.6e-169) {
		tmp = t_2;
	} else if (B <= -1.1e-271) {
		tmp = t_6;
	} else if (B <= 1.18e-301) {
		tmp = -Math.sqrt((A * ((C * (F * (A - -A))) * -8.0))) / t_3;
	} else if (B <= 1.65e-218) {
		tmp = t_6;
	} else if (B <= 1.7e-167) {
		tmp = t_0;
	} else if (B <= 6.5e-151) {
		tmp = t_2;
	} else if (B <= 2.2e-115) {
		tmp = t_5;
	} else if (B <= 2.5e-39) {
		tmp = t_0;
	} else if (B <= 1.6e+20) {
		tmp = -Math.sqrt(((2.0 * (F * (Math.pow(B, 2.0) - (C * (4.0 * A))))) * (A - (t_1 - C)))) / (Math.pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -Math.sqrt(((F / B) * -2.0));
	}
	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(-(F / C))
	t_1 = math.sqrt((math.pow(B, 2.0) + math.pow((A - C), 2.0)))
	t_2 = -math.sqrt(-(F / A))
	t_3 = math.pow(B, 2.0) - (A * (C * 4.0))
	t_4 = math.pow(B, 2.0) - (4.0 * (A * C))
	t_5 = 0.25 * math.sqrt(((F / C) * -16.0))
	t_6 = -math.sqrt((-8.0 * ((C - -C) * (F * (A * C))))) / t_3
	tmp = 0
	if B <= -3.3e+93:
		tmp = -math.sqrt((2.0 * (F / B)))
	elif B <= -3e-98:
		tmp = -math.sqrt(((2.0 * F) * (t_4 * ((A + C) - t_1)))) / t_4
	elif B <= -6.5e-159:
		tmp = t_5
	elif B <= -4.6e-169:
		tmp = t_2
	elif B <= -1.1e-271:
		tmp = t_6
	elif B <= 1.18e-301:
		tmp = -math.sqrt((A * ((C * (F * (A - -A))) * -8.0))) / t_3
	elif B <= 1.65e-218:
		tmp = t_6
	elif B <= 1.7e-167:
		tmp = t_0
	elif B <= 6.5e-151:
		tmp = t_2
	elif B <= 2.2e-115:
		tmp = t_5
	elif B <= 2.5e-39:
		tmp = t_0
	elif B <= 1.6e+20:
		tmp = -math.sqrt(((2.0 * (F * (math.pow(B, 2.0) - (C * (4.0 * A))))) * (A - (t_1 - C)))) / (math.pow(B, 2.0) - ((4.0 * A) * C))
	else:
		tmp = -math.sqrt(((F / B) * -2.0))
	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 = Float64(-sqrt(Float64(-Float64(F / C))))
	t_1 = sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))
	t_2 = Float64(-sqrt(Float64(-Float64(F / A))))
	t_3 = Float64((B ^ 2.0) - Float64(A * Float64(C * 4.0)))
	t_4 = Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C)))
	t_5 = Float64(0.25 * sqrt(Float64(Float64(F / C) * -16.0)))
	t_6 = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(C - Float64(-C)) * Float64(F * Float64(A * C)))))) / t_3)
	tmp = 0.0
	if (B <= -3.3e+93)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F / B))));
	elseif (B <= -3e-98)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(t_4 * Float64(Float64(A + C) - t_1))))) / t_4);
	elseif (B <= -6.5e-159)
		tmp = t_5;
	elseif (B <= -4.6e-169)
		tmp = t_2;
	elseif (B <= -1.1e-271)
		tmp = t_6;
	elseif (B <= 1.18e-301)
		tmp = Float64(Float64(-sqrt(Float64(A * Float64(Float64(C * Float64(F * Float64(A - Float64(-A)))) * -8.0)))) / t_3);
	elseif (B <= 1.65e-218)
		tmp = t_6;
	elseif (B <= 1.7e-167)
		tmp = t_0;
	elseif (B <= 6.5e-151)
		tmp = t_2;
	elseif (B <= 2.2e-115)
		tmp = t_5;
	elseif (B <= 2.5e-39)
		tmp = t_0;
	elseif (B <= 1.6e+20)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * Float64((B ^ 2.0) - Float64(C * Float64(4.0 * A))))) * Float64(A - Float64(t_1 - C))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B) * -2.0)));
	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(-(F / C));
	t_1 = sqrt(((B ^ 2.0) + ((A - C) ^ 2.0)));
	t_2 = -sqrt(-(F / A));
	t_3 = (B ^ 2.0) - (A * (C * 4.0));
	t_4 = (B ^ 2.0) - (4.0 * (A * C));
	t_5 = 0.25 * sqrt(((F / C) * -16.0));
	t_6 = -sqrt((-8.0 * ((C - -C) * (F * (A * C))))) / t_3;
	tmp = 0.0;
	if (B <= -3.3e+93)
		tmp = -sqrt((2.0 * (F / B)));
	elseif (B <= -3e-98)
		tmp = -sqrt(((2.0 * F) * (t_4 * ((A + C) - t_1)))) / t_4;
	elseif (B <= -6.5e-159)
		tmp = t_5;
	elseif (B <= -4.6e-169)
		tmp = t_2;
	elseif (B <= -1.1e-271)
		tmp = t_6;
	elseif (B <= 1.18e-301)
		tmp = -sqrt((A * ((C * (F * (A - -A))) * -8.0))) / t_3;
	elseif (B <= 1.65e-218)
		tmp = t_6;
	elseif (B <= 1.7e-167)
		tmp = t_0;
	elseif (B <= 6.5e-151)
		tmp = t_2;
	elseif (B <= 2.2e-115)
		tmp = t_5;
	elseif (B <= 2.5e-39)
		tmp = t_0;
	elseif (B <= 1.6e+20)
		tmp = -sqrt(((2.0 * (F * ((B ^ 2.0) - (C * (4.0 * A))))) * (A - (t_1 - C)))) / ((B ^ 2.0) - ((4.0 * A) * C));
	else
		tmp = -sqrt(((F / B) * -2.0));
	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[(F / C), $MachinePrecision])], $MachinePrecision])}, Block[{t$95$1 = N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision])}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(A * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.25 * N[Sqrt[N[(N[(F / C), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[((-N[Sqrt[N[(-8.0 * N[(N[(C - (-C)), $MachinePrecision] * N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[B, -3.3e+93], (-N[Sqrt[N[(2.0 * N[(F / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B, -3e-98], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(t$95$4 * N[(N[(A + C), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, -6.5e-159], t$95$5, If[LessEqual[B, -4.6e-169], t$95$2, If[LessEqual[B, -1.1e-271], t$95$6, If[LessEqual[B, 1.18e-301], N[((-N[Sqrt[N[(A * N[(N[(C * N[(F * N[(A - (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, 1.65e-218], t$95$6, If[LessEqual[B, 1.7e-167], t$95$0, If[LessEqual[B, 6.5e-151], t$95$2, If[LessEqual[B, 2.2e-115], t$95$5, If[LessEqual[B, 2.5e-39], t$95$0, If[LessEqual[B, 1.6e+20], N[((-N[Sqrt[N[(N[(2.0 * N[(F * N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A - N[(t$95$1 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(F / B), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision])]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if B < -3.30000000000000009e93

   1. Initial program 60.5

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

   2. Taylor expanded in A around 0 63.8

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

   3. Simplified 63.8

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

      [Start] 63.8	\[ \frac{-\sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \cdot \left(\sqrt{2} \cdot B\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-43 [=>] 63.8	\[ \frac{-\color{blue}{\sqrt{2} \cdot \left(B \cdot \sqrt{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-2 [=>] 63.8	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Taylor expanded in B around -inf 63.0

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

   5. Simplified 63.0

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

      [Start] 63.0	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(C - \left(-0.5 \cdot \frac{{C}^{2}}{B} + -1 \cdot B\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-1 [=>] 63.0	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(C - \color{blue}{\left(-1 \cdot B + -0.5 \cdot \frac{{C}^{2}}{B}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(C - \left(\color{blue}{B \cdot -1} + -0.5 \cdot \frac{{C}^{2}}{B}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-8 [<=] 63.0	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(C - \left(\color{blue}{\left(-B\right)} + -0.5 \cdot \frac{{C}^{2}}{B}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-2 [=>] 63.0	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(C - \left(\left(-B\right) + \color{blue}{\frac{{C}^{2}}{B} \cdot -0.5}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in C around 0 33.2

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

   7. Simplified 33.0

      \[\leadsto \color{blue}{-\sqrt{2 \cdot \frac{F}{B}}} \]
      Proof

      [Start] 33.2	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      rational.json-simplify-2 [=>] 33.2	\[ \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 33.2	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      exponential.json-simplify-20 [=>] 33.0	\[ -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
      rational.json-simplify-2 [=>] 33.0	\[ -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

   if -3.30000000000000009e93 < B < -3e-98

   1. Initial program 42.8

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

   2. Simplified 43.2

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

      [Start] 42.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -3e-98 < B < -6.5000000000000001e-159 or 6.4999999999999994e-151 < B < 2.1999999999999999e-115

   1. Initial program 51.0

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

   2. Simplified 53.3

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

      [Start] 51.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 53.8

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

   4. Simplified 53.8

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

      [Start] 53.8	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(2 \cdot \left(A \cdot {B}^{2}\right) + -8 \cdot \left({A}^{2} \cdot C\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-1 [=>] 53.8	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-8 \cdot \left({A}^{2} \cdot C\right) + 2 \cdot \left(A \cdot {B}^{2}\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 53.8	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\color{blue}{{A}^{2} \cdot \left(C \cdot -8\right)} + 2 \cdot \left(A \cdot {B}^{2}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 53.8	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({A}^{2} \cdot \left(C \cdot -8\right) + \color{blue}{A \cdot \left({B}^{2} \cdot 2\right)}\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in A around inf 64.0

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

   6. Simplified 54.2

      \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}} \]
      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 [=>] 54.2	\[ 0.25 \cdot \color{blue}{\sqrt{\frac{F}{C} \cdot -16}} \]

   if -6.5000000000000001e-159 < B < -4.6000000000000002e-169 or 1.6999999999999999e-167 < B < 6.4999999999999994e-151

   1. Initial program 52.2

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

   2. Simplified 52.4

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

      [Start] 52.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 64.0

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

   4. Simplified 52.4

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{A}}} \]
      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 [=>] 52.4	\[ -\color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational.json-simplify-9 [=>] 52.4	\[ -\sqrt{\color{blue}{-\frac{F}{A}}} \]

   if -4.6000000000000002e-169 < B < -1.1e-271 or 1.18e-301 < B < 1.65000000000000012e-218

   1. Initial program 52.9

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

   2. Simplified 52.9

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

      [Start] 52.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 52.5

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

   4. Simplified 52.2

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

      [Start] 52.5	\[ \frac{-\sqrt{\left(F + F\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(A - \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - C\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} + 0 \]
      rational.json-simplify-4 [=>] 52.5	\[ \color{blue}{\frac{-\sqrt{\left(F + F\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(A - \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - C\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}} \]

   5. Taylor expanded in A around inf 52.4

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)}}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]

   6. Simplified 49.6

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

      [Start] 52.4	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-43 [=>] 50.3	\[ \frac{-\sqrt{-8 \cdot \color{blue}{\left(\left(C - -1 \cdot C\right) \cdot \left(\left(C \cdot F\right) \cdot A\right)\right)}}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-2 [=>] 50.3	\[ \frac{-\sqrt{-8 \cdot \left(\left(C - \color{blue}{C \cdot -1}\right) \cdot \left(\left(C \cdot F\right) \cdot A\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-9 [=>] 50.3	\[ \frac{-\sqrt{-8 \cdot \left(\left(C - \color{blue}{\left(-C\right)}\right) \cdot \left(\left(C \cdot F\right) \cdot A\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-2 [<=] 50.3	\[ \frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-43 [=>] 52.3	\[ \frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \color{blue}{\left(C \cdot \left(F \cdot A\right)\right)}\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-43 [=>] 49.6	\[ \frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \color{blue}{\left(F \cdot \left(A \cdot C\right)\right)}\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]

   if -1.1e-271 < B < 1.18e-301

   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 54.5

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

      [Start] 54.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 54.2

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

   4. Simplified 53.1

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

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

   5. Taylor expanded in C around inf 52.6

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]

   6. Simplified 51.0

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

      [Start] 52.6	\[ \frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-43 [=>] 52.6	\[ \frac{-\sqrt{\color{blue}{A \cdot \left(\left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot -8\right)}}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-43 [=>] 51.0	\[ \frac{-\sqrt{A \cdot \left(\color{blue}{\left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)} \cdot -8\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-2 [=>] 51.0	\[ \frac{-\sqrt{A \cdot \left(\left(C \cdot \left(F \cdot \left(A - \color{blue}{A \cdot -1}\right)\right)\right) \cdot -8\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]
      rational.json-simplify-9 [=>] 51.0	\[ \frac{-\sqrt{A \cdot \left(\left(C \cdot \left(F \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)\right) \cdot -8\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} \]

   if 1.65000000000000012e-218 < B < 1.6999999999999999e-167 or 2.1999999999999999e-115 < B < 2.4999999999999999e-39

   1. Initial program 49.4

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

   2. Simplified 49.5

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

      [Start] 49.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 64.0

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

   4. Simplified 51.3

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

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{C}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{C}} \]
      exponential.json-simplify-20 [=>] 51.3	\[ -\color{blue}{\sqrt{\frac{F}{C} \cdot -1}} \]
      rational.json-simplify-9 [=>] 51.3	\[ -\sqrt{\color{blue}{-\frac{F}{C}}} \]

   if 2.4999999999999999e-39 < B < 1.6e20

   1. Initial program 39.9

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

   2. Applied egg-rr 39.6

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

   3. Simplified 39.6

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

      [Start] 39.6	\[ \frac{-\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right)\right) \cdot \left(A - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - C\right)\right)} + 0\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-4 [=>] 39.6	\[ \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right)\right) \cdot \left(A - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      exponential.json-simplify-11 [<=] 39.6	\[ \frac{-\sqrt{\color{blue}{{\left(\left(2 \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right)\right) \cdot \left(A - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - C\right)\right)\right)}^{1}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      exponential.json-simplify-11 [=>] 39.6	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right)\right) \cdot \left(A - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-2 [=>] 39.6	\[ \frac{-\sqrt{\left(2 \cdot \color{blue}{\left(F \cdot \left({B}^{2} - A \cdot \left(C \cdot 4\right)\right)\right)}\right) \cdot \left(A - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-43 [=>] 39.6	\[ \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - \color{blue}{C \cdot \left(4 \cdot A\right)}\right)\right)\right) \cdot \left(A - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-1 [=>] 39.6	\[ \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(A - \left(\sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}} - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 1.6e20 < B

   1. Initial program 55.6

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

   2. Simplified 58.2

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

      [Start] 55.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around inf 59.0

      \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-1 \cdot {B}^{3}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 59.0

      \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-{B}^{3}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 59.0	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(-1 \cdot {B}^{3}\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-2 [=>] 59.0	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left({B}^{3} \cdot -1\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-9 [=>] 59.0	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-{B}^{3}\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in A around 0 64.0

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

   6. Simplified 35.3

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot -2}} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{-2}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\sqrt{\frac{F}{B}} \cdot \sqrt{-2}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{-2}} \]
      exponential.json-simplify-20 [=>] 35.3	\[ -\color{blue}{\sqrt{-2 \cdot \frac{F}{B}}} \]
      rational.json-simplify-2 [=>] 35.3	\[ -\sqrt{\color{blue}{\frac{F}{B} \cdot -2}} \]

3. Recombined 9 regimes into one program.

4. Final simplification 42.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.3 \cdot 10^{+93}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-98}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-159}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq -4.6 \cdot 10^{-169}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-271}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.18 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(\left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right) \cdot -8\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-167}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-151}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-39}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(A - \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

Alternatives

Alternative 1
Error	42.9
Cost	40840

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ t_2 := {B}^{2} - A \cdot \left(C \cdot 4\right)\\ t_3 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ t_4 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_5 := \frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+93}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-99}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(t_3 \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq -6.3 \cdot 10^{-161}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-273}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(\left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right) \cdot -8\right)}}{t_2}\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-220}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-115}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

Alternative 2
Error	43.4
Cost	14744

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_1 := \frac{-\left(-A \cdot \sqrt{\left(C \cdot F\right) \cdot -16}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-93}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-228}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-221}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-23}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

Alternative 3
Error	43.1
Cost	14736

\[\begin{array}{l} t_0 := \frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ t_1 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;B \leq -3.2 \cdot 10^{-91}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-288}:\\ \;\;\;\;\frac{-\left(-A \cdot \sqrt{\left(C \cdot F\right) \cdot -16}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-136}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-116}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

Alternative 4
Error	43.2
Cost	14736

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ t_1 := {B}^{2} - A \cdot \left(C \cdot 4\right)\\ t_2 := \frac{-\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-94}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-272}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{-299}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(\left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right) \cdot -8\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-216}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{-151}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

Alternative 5
Error	44.0
Cost	7640

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-91}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -8.8 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-295}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

Alternative 6
Error	43.5
Cost	7312

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-87}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-281}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot -2}\\ \end{array} \]

Alternative 7
Error	47.3
Cost	7048

\[\begin{array}{l} \mathbf{if}\;C \leq -0.265:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{-229}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 8
Error	47.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;C \leq 9 \cdot 10^{-251}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 9
Error	55.2
Cost	6720

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

Reproduce

herbie shell --seed 2023077 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))