ABCF->ab-angle b

Average Error: 52.6 → 45.5

Time: 1.1min

Precision: binary64

Cost: 114208

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := {C}^{2} + {B}^{2}\\ t_4 := C - \sqrt{t_3}\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-272}:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right) + A \cdot \left(\left(F \cdot {B}^{2}\right) \cdot 4\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot t_4\right)} + 0.5 \cdot \left(\left(\sqrt{2} \cdot \left({B}^{2} \cdot \left(1 - \sqrt{\frac{1}{t_3}} \cdot \left(-C\right)\right) + C \cdot \left(-4 \cdot t_4\right)\right)\right) \cdot \left(\frac{A}{B} \cdot \sqrt{\frac{F}{t_4}}\right)\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(\frac{{B}^{2}}{A} \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

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 A))))
        (t_1 (/ (- (* A (sqrt (* (* F C) -16.0)))) (* (* A C) 4.0)))
        (t_2 (/ (sqrt 2.0) B))
        (t_3 (+ (pow C 2.0) (pow B 2.0)))
        (t_4 (- C (sqrt t_3))))
   (if (<= B -3.8e-9)
     (- (sqrt (* 2.0 (/ F B))))
     (if (<= B -1.95e-272)
       (- t_0)
       (if (<= B 1.95e-296)
         (* 0.25 (sqrt (* -16.0 (/ F C))))
         (if (<= B 2.2e-229)
           t_1
           (if (<= B 1.9e-173)
             t_0
             (if (<= B 1.8e-69)
               (/
                (sqrt
                 (+
                  (* C (* (* F (pow A 2.0)) -16.0))
                  (* A (* (* F (pow B 2.0)) 4.0))))
                (- (* 4.0 (* A C)) (pow B 2.0)))
               (if (<= B 1.4e+47)
                 t_1
                 (if (<= B 5.2e+79)
                   (/
                    (+
                     (* B (sqrt (* 2.0 (* F t_4))))
                     (*
                      0.5
                      (*
                       (*
                        (sqrt 2.0)
                        (+
                         (* (pow B 2.0) (- 1.0 (* (sqrt (/ 1.0 t_3)) (- C))))
                         (* C (* -4.0 t_4))))
                       (* (/ A B) (sqrt (/ F t_4))))))
                    (- (* (* 4.0 A) C) (pow B 2.0)))
                   (if (<= B 2.4e+131)
                     (* t_2 (- (sqrt (* F (* (/ (pow B 2.0) A) -0.5)))))
                     (* t_2 (- (sqrt (- (* B F))))))))))))))))

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 / A));
	double t_1 = -(A * sqrt(((F * C) * -16.0))) / ((A * C) * 4.0);
	double t_2 = sqrt(2.0) / B;
	double t_3 = pow(C, 2.0) + pow(B, 2.0);
	double t_4 = C - sqrt(t_3);
	double tmp;
	if (B <= -3.8e-9) {
		tmp = -sqrt((2.0 * (F / B)));
	} else if (B <= -1.95e-272) {
		tmp = -t_0;
	} else if (B <= 1.95e-296) {
		tmp = 0.25 * sqrt((-16.0 * (F / C)));
	} else if (B <= 2.2e-229) {
		tmp = t_1;
	} else if (B <= 1.9e-173) {
		tmp = t_0;
	} else if (B <= 1.8e-69) {
		tmp = sqrt(((C * ((F * pow(A, 2.0)) * -16.0)) + (A * ((F * pow(B, 2.0)) * 4.0)))) / ((4.0 * (A * C)) - pow(B, 2.0));
	} else if (B <= 1.4e+47) {
		tmp = t_1;
	} else if (B <= 5.2e+79) {
		tmp = ((B * sqrt((2.0 * (F * t_4)))) + (0.5 * ((sqrt(2.0) * ((pow(B, 2.0) * (1.0 - (sqrt((1.0 / t_3)) * -C))) + (C * (-4.0 * t_4)))) * ((A / B) * sqrt((F / t_4)))))) / (((4.0 * A) * C) - pow(B, 2.0));
	} else if (B <= 2.4e+131) {
		tmp = t_2 * -sqrt((F * ((pow(B, 2.0) / A) * -0.5)));
	} else {
		tmp = t_2 * -sqrt(-(B * F));
	}
	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) :: tmp
    t_0 = sqrt(-(f / a))
    t_1 = -(a * sqrt(((f * c) * (-16.0d0)))) / ((a * c) * 4.0d0)
    t_2 = sqrt(2.0d0) / b
    t_3 = (c ** 2.0d0) + (b ** 2.0d0)
    t_4 = c - sqrt(t_3)
    if (b <= (-3.8d-9)) then
        tmp = -sqrt((2.0d0 * (f / b)))
    else if (b <= (-1.95d-272)) then
        tmp = -t_0
    else if (b <= 1.95d-296) then
        tmp = 0.25d0 * sqrt(((-16.0d0) * (f / c)))
    else if (b <= 2.2d-229) then
        tmp = t_1
    else if (b <= 1.9d-173) then
        tmp = t_0
    else if (b <= 1.8d-69) then
        tmp = sqrt(((c * ((f * (a ** 2.0d0)) * (-16.0d0))) + (a * ((f * (b ** 2.0d0)) * 4.0d0)))) / ((4.0d0 * (a * c)) - (b ** 2.0d0))
    else if (b <= 1.4d+47) then
        tmp = t_1
    else if (b <= 5.2d+79) then
        tmp = ((b * sqrt((2.0d0 * (f * t_4)))) + (0.5d0 * ((sqrt(2.0d0) * (((b ** 2.0d0) * (1.0d0 - (sqrt((1.0d0 / t_3)) * -c))) + (c * ((-4.0d0) * t_4)))) * ((a / b) * sqrt((f / t_4)))))) / (((4.0d0 * a) * c) - (b ** 2.0d0))
    else if (b <= 2.4d+131) then
        tmp = t_2 * -sqrt((f * (((b ** 2.0d0) / a) * (-0.5d0))))
    else
        tmp = t_2 * -sqrt(-(b * f))
    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 / A));
	double t_1 = -(A * Math.sqrt(((F * C) * -16.0))) / ((A * C) * 4.0);
	double t_2 = Math.sqrt(2.0) / B;
	double t_3 = Math.pow(C, 2.0) + Math.pow(B, 2.0);
	double t_4 = C - Math.sqrt(t_3);
	double tmp;
	if (B <= -3.8e-9) {
		tmp = -Math.sqrt((2.0 * (F / B)));
	} else if (B <= -1.95e-272) {
		tmp = -t_0;
	} else if (B <= 1.95e-296) {
		tmp = 0.25 * Math.sqrt((-16.0 * (F / C)));
	} else if (B <= 2.2e-229) {
		tmp = t_1;
	} else if (B <= 1.9e-173) {
		tmp = t_0;
	} else if (B <= 1.8e-69) {
		tmp = Math.sqrt(((C * ((F * Math.pow(A, 2.0)) * -16.0)) + (A * ((F * Math.pow(B, 2.0)) * 4.0)))) / ((4.0 * (A * C)) - Math.pow(B, 2.0));
	} else if (B <= 1.4e+47) {
		tmp = t_1;
	} else if (B <= 5.2e+79) {
		tmp = ((B * Math.sqrt((2.0 * (F * t_4)))) + (0.5 * ((Math.sqrt(2.0) * ((Math.pow(B, 2.0) * (1.0 - (Math.sqrt((1.0 / t_3)) * -C))) + (C * (-4.0 * t_4)))) * ((A / B) * Math.sqrt((F / t_4)))))) / (((4.0 * A) * C) - Math.pow(B, 2.0));
	} else if (B <= 2.4e+131) {
		tmp = t_2 * -Math.sqrt((F * ((Math.pow(B, 2.0) / A) * -0.5)));
	} else {
		tmp = t_2 * -Math.sqrt(-(B * F));
	}
	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 / A))
	t_1 = -(A * math.sqrt(((F * C) * -16.0))) / ((A * C) * 4.0)
	t_2 = math.sqrt(2.0) / B
	t_3 = math.pow(C, 2.0) + math.pow(B, 2.0)
	t_4 = C - math.sqrt(t_3)
	tmp = 0
	if B <= -3.8e-9:
		tmp = -math.sqrt((2.0 * (F / B)))
	elif B <= -1.95e-272:
		tmp = -t_0
	elif B <= 1.95e-296:
		tmp = 0.25 * math.sqrt((-16.0 * (F / C)))
	elif B <= 2.2e-229:
		tmp = t_1
	elif B <= 1.9e-173:
		tmp = t_0
	elif B <= 1.8e-69:
		tmp = math.sqrt(((C * ((F * math.pow(A, 2.0)) * -16.0)) + (A * ((F * math.pow(B, 2.0)) * 4.0)))) / ((4.0 * (A * C)) - math.pow(B, 2.0))
	elif B <= 1.4e+47:
		tmp = t_1
	elif B <= 5.2e+79:
		tmp = ((B * math.sqrt((2.0 * (F * t_4)))) + (0.5 * ((math.sqrt(2.0) * ((math.pow(B, 2.0) * (1.0 - (math.sqrt((1.0 / t_3)) * -C))) + (C * (-4.0 * t_4)))) * ((A / B) * math.sqrt((F / t_4)))))) / (((4.0 * A) * C) - math.pow(B, 2.0))
	elif B <= 2.4e+131:
		tmp = t_2 * -math.sqrt((F * ((math.pow(B, 2.0) / A) * -0.5)))
	else:
		tmp = t_2 * -math.sqrt(-(B * F))
	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(-Float64(F / A)))
	t_1 = Float64(Float64(-Float64(A * sqrt(Float64(Float64(F * C) * -16.0)))) / Float64(Float64(A * C) * 4.0))
	t_2 = Float64(sqrt(2.0) / B)
	t_3 = Float64((C ^ 2.0) + (B ^ 2.0))
	t_4 = Float64(C - sqrt(t_3))
	tmp = 0.0
	if (B <= -3.8e-9)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F / B))));
	elseif (B <= -1.95e-272)
		tmp = Float64(-t_0);
	elseif (B <= 1.95e-296)
		tmp = Float64(0.25 * sqrt(Float64(-16.0 * Float64(F / C))));
	elseif (B <= 2.2e-229)
		tmp = t_1;
	elseif (B <= 1.9e-173)
		tmp = t_0;
	elseif (B <= 1.8e-69)
		tmp = Float64(sqrt(Float64(Float64(C * Float64(Float64(F * (A ^ 2.0)) * -16.0)) + Float64(A * Float64(Float64(F * (B ^ 2.0)) * 4.0)))) / Float64(Float64(4.0 * Float64(A * C)) - (B ^ 2.0)));
	elseif (B <= 1.4e+47)
		tmp = t_1;
	elseif (B <= 5.2e+79)
		tmp = Float64(Float64(Float64(B * sqrt(Float64(2.0 * Float64(F * t_4)))) + Float64(0.5 * Float64(Float64(sqrt(2.0) * Float64(Float64((B ^ 2.0) * Float64(1.0 - Float64(sqrt(Float64(1.0 / t_3)) * Float64(-C)))) + Float64(C * Float64(-4.0 * t_4)))) * Float64(Float64(A / B) * sqrt(Float64(F / t_4)))))) / Float64(Float64(Float64(4.0 * A) * C) - (B ^ 2.0)));
	elseif (B <= 2.4e+131)
		tmp = Float64(t_2 * Float64(-sqrt(Float64(F * Float64(Float64((B ^ 2.0) / A) * -0.5)))));
	else
		tmp = Float64(t_2 * Float64(-sqrt(Float64(-Float64(B * F)))));
	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 / A));
	t_1 = -(A * sqrt(((F * C) * -16.0))) / ((A * C) * 4.0);
	t_2 = sqrt(2.0) / B;
	t_3 = (C ^ 2.0) + (B ^ 2.0);
	t_4 = C - sqrt(t_3);
	tmp = 0.0;
	if (B <= -3.8e-9)
		tmp = -sqrt((2.0 * (F / B)));
	elseif (B <= -1.95e-272)
		tmp = -t_0;
	elseif (B <= 1.95e-296)
		tmp = 0.25 * sqrt((-16.0 * (F / C)));
	elseif (B <= 2.2e-229)
		tmp = t_1;
	elseif (B <= 1.9e-173)
		tmp = t_0;
	elseif (B <= 1.8e-69)
		tmp = sqrt(((C * ((F * (A ^ 2.0)) * -16.0)) + (A * ((F * (B ^ 2.0)) * 4.0)))) / ((4.0 * (A * C)) - (B ^ 2.0));
	elseif (B <= 1.4e+47)
		tmp = t_1;
	elseif (B <= 5.2e+79)
		tmp = ((B * sqrt((2.0 * (F * t_4)))) + (0.5 * ((sqrt(2.0) * (((B ^ 2.0) * (1.0 - (sqrt((1.0 / t_3)) * -C))) + (C * (-4.0 * t_4)))) * ((A / B) * sqrt((F / t_4)))))) / (((4.0 * A) * C) - (B ^ 2.0));
	elseif (B <= 2.4e+131)
		tmp = t_2 * -sqrt((F * (((B ^ 2.0) / A) * -0.5)));
	else
		tmp = t_2 * -sqrt(-(B * F));
	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 / A), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[((-N[(A * N[Sqrt[N[(N[(F * C), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[(A * C), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[C, 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(C - N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.8e-9], (-N[Sqrt[N[(2.0 * N[(F / B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[B, -1.95e-272], (-t$95$0), If[LessEqual[B, 1.95e-296], N[(0.25 * N[Sqrt[N[(-16.0 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.2e-229], t$95$1, If[LessEqual[B, 1.9e-173], t$95$0, If[LessEqual[B, 1.8e-69], N[(N[Sqrt[N[(N[(C * N[(N[(F * N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision] + N[(A * N[(N[(F * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.4e+47], t$95$1, If[LessEqual[B, 5.2e+79], N[(N[(N[(B * N[Sqrt[N[(2.0 * N[(F * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(N[Power[B, 2.0], $MachinePrecision] * N[(1.0 - N[(N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision] * (-C)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(C * N[(-4.0 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A / B), $MachinePrecision] * N[Sqrt[N[(F / t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.4e+131], N[(t$95$2 * (-N[Sqrt[N[(F * N[(N[(N[Power[B, 2.0], $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$2 * (-N[Sqrt[(-N[(B * F), $MachinePrecision])], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if B < -3.80000000000000011e-9

   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 54.2

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

   3. Taylor expanded in B around -inf 57.8

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

   4. Taylor expanded in A around 0 36.5

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

   5. Simplified 36.3

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

      [Start] 36.5	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      rational.json-simplify-2 [=>] 36.5	\[ \color{blue}{\left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 36.5	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      exponential.json-simplify-20 [=>] 36.3	\[ -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
      rational.json-simplify-2 [=>] 36.3	\[ -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

   if -3.80000000000000011e-9 < B < -1.9499999999999999e-272

   1. Initial program 49.9

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

   2. Simplified 49.3

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

      [Start] 49.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [=>] 49.9	\[ \color{blue}{\frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 49.9	\[ \frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [<=] 49.9	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   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 52.3

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{A}}} \]
      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 [=>] 52.3	\[ -\color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational.json-simplify-9 [=>] 52.3	\[ -\sqrt{\color{blue}{-\frac{F}{A}}} \]

   if -1.9499999999999999e-272 < B < 1.95000000000000005e-296

   1. Initial program 52.4

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

   2. Simplified 52.5

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

      [Start] 52.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 52.4	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around -inf 55.9

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

   4. Simplified 55.9

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

      [Start] 55.9	\[ \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C - -1 \cdot A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 55.9	\[ \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C - \color{blue}{A \cdot -1}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-9 [=>] 55.9	\[ \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C - \color{blue}{\left(-A\right)}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   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 50.8

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

      [Start] 64.0	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{C}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ 0.25 \cdot \color{blue}{\left(\sqrt{\frac{F}{C}} \cdot \left(\sqrt{2} \cdot \sqrt{-8}\right)\right)} \]
      exponential.json-simplify-20 [=>] 64.0	\[ 0.25 \cdot \left(\sqrt{\frac{F}{C}} \cdot \color{blue}{\sqrt{-8 \cdot 2}}\right) \]
      metadata-eval [=>] 64.0	\[ 0.25 \cdot \left(\sqrt{\frac{F}{C}} \cdot \sqrt{\color{blue}{-16}}\right) \]
      exponential.json-simplify-20 [=>] 50.8	\[ 0.25 \cdot \color{blue}{\sqrt{-16 \cdot \frac{F}{C}}} \]

   if 1.95000000000000005e-296 < B < 2.1999999999999999e-229 or 1.80000000000000009e-69 < B < 1.39999999999999994e47

   1. Initial program 46.5

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

   2. Simplified 46.3

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

      [Start] 46.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 46.5	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around -inf 64.0

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

   4. Simplified 53.7

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

      [Start] 64.0	\[ \frac{-1 \cdot \left(\left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\color{blue}{\left(\left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}\right) \cdot -1}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-9 [=>] 64.0	\[ \frac{\color{blue}{-\left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{-\color{blue}{\sqrt{C \cdot F} \cdot \left(\sqrt{2} \cdot \left(A \cdot \sqrt{-8}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{-\sqrt{C \cdot F} \cdot \color{blue}{\left(A \cdot \left(\sqrt{-8} \cdot \sqrt{2}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [<=] 64.0	\[ \frac{-\sqrt{C \cdot F} \cdot \left(A \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{-8}\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{-\color{blue}{A \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{C \cdot F}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \frac{-A \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{C \cdot F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      metadata-eval [=>] 64.0	\[ \frac{-A \cdot \left(\sqrt{\color{blue}{-16}} \cdot \sqrt{C \cdot F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      exponential.json-simplify-20 [=>] 53.7	\[ \frac{-A \cdot \color{blue}{\sqrt{\left(C \cdot F\right) \cdot -16}}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [=>] 53.7	\[ \frac{-A \cdot \sqrt{\color{blue}{\left(F \cdot C\right)} \cdot -16}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]

   5. Taylor expanded in A around inf 51.9

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

   6. Simplified 51.9

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

      [Start] 51.9	\[ \frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-2 [=>] 51.9	\[ \frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{\left(A \cdot C\right) \cdot 4}} \]

   if 2.1999999999999999e-229 < B < 1.90000000000000015e-173

   1. Initial program 53.1

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

   2. Simplified 52.4

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

      [Start] 53.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [=>] 53.1	\[ \color{blue}{\frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 53.1	\[ \frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [<=] 53.1	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in C around 0 62.4

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

   4. Simplified 62.4

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

      [Start] 62.4	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 62.4	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 62.4	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 62.4	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]

   5. Taylor expanded in A around inf 61.9

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

   6. Simplified 61.9

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

      [Start] 61.9	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A}\right)}\right) \]
      rational.json-simplify-2 [=>] 61.9	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \color{blue}{\left(\frac{{B}^{2}}{A} \cdot -0.5\right)}}\right) \]

   7. Taylor expanded in B around -inf 64.0

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

   8. Simplified 53.5

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

      [Start] 64.0	\[ \left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}} \]
      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 [=>] 53.5	\[ \color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational.json-simplify-9 [=>] 53.5	\[ \sqrt{\color{blue}{-\frac{F}{A}}} \]

   if 1.90000000000000015e-173 < B < 1.80000000000000009e-69

   1. Initial program 51.4

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

   2. Simplified 52.7

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

      [Start] 51.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 51.4	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around -inf 55.1

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

   4. Simplified 55.3

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

      [Start] 55.1	\[ \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 55.3	\[ \frac{\sqrt{-16 \cdot \color{blue}{\left(C \cdot \left(F \cdot {A}^{2}\right)\right)} + 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 55.3	\[ \frac{\sqrt{\color{blue}{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right)} + 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 55.3	\[ \frac{\sqrt{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right) + \color{blue}{A \cdot \left(\left(F \cdot {B}^{2}\right) \cdot 4\right)}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   if 1.39999999999999994e47 < B < 5.20000000000000029e79

   1. Initial program 46.2

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

   2. Simplified 46.0

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

      [Start] 46.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 46.2	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around 0 42.6

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

   4. Simplified 42.4

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

      [Start] 42.6

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

      rational.json-simplify-1 [=>] 42.6

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

   if 5.20000000000000029e79 < B < 2.3999999999999999e131

   1. Initial program 49.0

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

   2. Simplified 49.0

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

      [Start] 49.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [=>] 49.0	\[ \color{blue}{\frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 49.0	\[ \frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [<=] 49.0	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in C around 0 38.2

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

   4. Simplified 38.2

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

      [Start] 38.2	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 38.2	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 38.2	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 38.2	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]

   5. Taylor expanded in A around inf 55.2

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

   6. Simplified 55.2

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

      [Start] 55.2	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A}\right)}\right) \]
      rational.json-simplify-2 [=>] 55.2	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \color{blue}{\left(\frac{{B}^{2}}{A} \cdot -0.5\right)}}\right) \]

   if 2.3999999999999999e131 < B

   1. Initial program 62.9

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

   2. Simplified 62.9

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

      [Start] 62.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [=>] 62.9	\[ \color{blue}{\frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 62.9	\[ \frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [<=] 62.9	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in C around 0 60.0

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

   4. Simplified 60.0

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

      [Start] 60.0	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 60.0	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 60.0	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      rational.json-simplify-2 [=>] 60.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]

   5. Taylor expanded in A around 0 64.0

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

   6. Simplified 33.3

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

      [Start] 64.0	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-1} \cdot \sqrt{F \cdot B}\right) \]
      exponential.json-simplify-20 [=>] 33.3	\[ \frac{\sqrt{2}}{B} \cdot \left(-\color{blue}{\sqrt{\left(F \cdot B\right) \cdot -1}}\right) \]
      rational.json-simplify-9 [=>] 33.3	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{-F \cdot B}}\right) \]
      rational.json-simplify-2 [=>] 33.3	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-\color{blue}{B \cdot F}}\right) \]

3. Recombined 9 regimes into one program.

4. Final simplification 45.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.8 \cdot 10^{-9}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-272}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-229}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right) + A \cdot \left(\left(F \cdot {B}^{2}\right) \cdot 4\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{{C}^{2} + {B}^{2}}\right)\right)} + 0.5 \cdot \left(\left(\sqrt{2} \cdot \left({B}^{2} \cdot \left(1 - \sqrt{\frac{1}{{C}^{2} + {B}^{2}}} \cdot \left(-C\right)\right) + C \cdot \left(-4 \cdot \left(C - \sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right) \cdot \left(\frac{A}{B} \cdot \sqrt{\frac{F}{C - \sqrt{{C}^{2} + {B}^{2}}}}\right)\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(\frac{{B}^{2}}{A} \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	45.5
Cost	33952

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -65:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-272}:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.76 \cdot 10^{-177}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 9.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right) + A \cdot \left(\left(F \cdot {B}^{2}\right) \cdot 4\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+79}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(C - \sqrt{{C}^{2} + {B}^{2}}\right)}\right)\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{+134}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(\frac{{B}^{2}}{A} \cdot -0.5\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

Alternative 2
Error	44.4
Cost	28452

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ t_1 := \sqrt{-\frac{F}{A}}\\ t_2 := F \cdot {B}^{2}\\ \mathbf{if}\;B \leq -7 \cdot 10^{-58}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(F \cdot {C}^{2}\right) \cdot -16\right) + 4 \cdot \left(C \cdot t_2\right)}}{t_0}\\ \mathbf{elif}\;B \leq -8.5 \cdot 10^{-186}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq -1.18 \cdot 10^{-209}:\\ \;\;\;\;\frac{A \cdot \sqrt{\left(C \cdot F\right) \cdot -16}}{t_0}\\ \mathbf{elif}\;B \leq -2.55 \cdot 10^{-272}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-297}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-225}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right) + A \cdot \left(t_2 \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

Alternative 3
Error	44.4
Cost	28452

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ t_1 := \sqrt{-\frac{F}{A}}\\ \mathbf{if}\;B \leq -1.85 \cdot 10^{-57}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(F \cdot {C}^{2}\right) \cdot -16\right) + 2 \cdot \left(\left(2 \cdot \left(F \cdot C + \left(F \cdot C\right) \cdot \frac{A}{A - C}\right)\right) \cdot {B}^{2}\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-186}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-209}:\\ \;\;\;\;\frac{A \cdot \sqrt{\left(C \cdot F\right) \cdot -16}}{t_0}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-272}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-228}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right) + A \cdot \left(\left(F \cdot {B}^{2}\right) \cdot 4\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

Alternative 4
Error	44.6
Cost	27528

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ t_1 := \sqrt{-\frac{F}{A}}\\ \mathbf{if}\;B \leq -1.24 \cdot 10^{-57}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(\left(F \cdot {C}^{2}\right) \cdot -16\right) + 4 \cdot \left(C \cdot \left(F \cdot {B}^{2}\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-186}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-209}:\\ \;\;\;\;\frac{A \cdot \sqrt{\left(C \cdot F\right) \cdot -16}}{t_0}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-272}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-231}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 5.3 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(C \cdot \left(\left(F \cdot \left(A - \left(-A\right)\right)\right) \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

Alternative 5
Error	44.5
Cost	14936

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ \mathbf{if}\;B \leq -8.2 \cdot 10^{-10}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-272}:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-296}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-229}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(C \cdot \left(\left(F \cdot \left(A - \left(-A\right)\right)\right) \cdot A\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

Alternative 6
Error	44.7
Cost	14168

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := 0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-272}:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-296}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-230}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 66000000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-B \cdot F}\right)\\ \end{array} \]

Alternative 7
Error	47.2
Cost	7576

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \frac{F}{B}}\\ t_1 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;A \leq -1.1 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -1.75 \cdot 10^{+23}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\left(A \cdot C\right) \cdot 4}\\ \mathbf{elif}\;A \leq -4.4 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 10^{-307}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 2.6 \cdot 10^{-234}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 8
Error	48.5
Cost	7376

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -1.8 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq -1.3 \cdot 10^{-184}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;C \leq 2.3 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 2.9 \cdot 10^{-109}:\\ \;\;\;\;-1 \cdot \sqrt{F \cdot \frac{2}{B}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 9
Error	48.5
Cost	7312

\[\begin{array}{l} t_0 := -\sqrt{2 \cdot \frac{F}{B}}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -2 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -7.5 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.05 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 8.2 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 10
Error	49.4
Cost	6984

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{-125}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;A \leq 3.55 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-t_0\\ \end{array} \]

Alternative 11
Error	52.4
Cost	6852

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ \mathbf{if}\;A \leq 3.9 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-t_0\\ \end{array} \]

Alternative 12
Error	56.8
Cost	6656

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

Reproduce

herbie shell --seed 2023074 
(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))))