ABCF->ab-angle b

Average Error: 51.7 → 47.0

Time: 1.4min

Precision: binary64

Cost: 40908

Math

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

↓

\[\begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t_0 - {B}^{2}\\ t_2 := {B}^{2} - t_0\\ \mathbf{if}\;A \leq -1 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + F \cdot \left(\left({B}^{2} \cdot A\right) \cdot 4\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B 2.0)))
        (t_2 (- (pow B 2.0) t_0)))
   (if (<= A -1e+104)
     (/ (sqrt (* 2.0 (* t_2 (* F (+ A (- C (- A))))))) t_1)
     (if (<= A -2.8e-44)
       (/
        (-
         (sqrt
          (+
           (* (* F C) (* -16.0 (pow A 2.0)))
           (* F (* (* (pow B 2.0) A) 4.0)))))
        (- (pow B 2.0) (* 4.0 (* A C))))
       (if (<= A 1.35e+35)
         (/
          (sqrt
           (*
            2.0
            (*
             t_2
             (* F (+ A (- C (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))))
          t_1)
         (*
          (sqrt (* -8.0 (* C (* F (* A (+ C C))))))
          (* (/ 2.0 A) (/ 0.125 C))))))))

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 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B, 2.0);
	double t_2 = pow(B, 2.0) - t_0;
	double tmp;
	if (A <= -1e+104) {
		tmp = sqrt((2.0 * (t_2 * (F * (A + (C - -A)))))) / t_1;
	} else if (A <= -2.8e-44) {
		tmp = -sqrt((((F * C) * (-16.0 * pow(A, 2.0))) + (F * ((pow(B, 2.0) * A) * 4.0)))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else if (A <= 1.35e+35) {
		tmp = sqrt((2.0 * (t_2 * (F * (A + (C - sqrt((pow(B, 2.0) + pow((A - C), 2.0))))))))) / t_1;
	} else {
		tmp = sqrt((-8.0 * (C * (F * (A * (C + C)))))) * ((2.0 / A) * (0.125 / C));
	}
	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) :: tmp
    t_0 = (4.0d0 * a) * c
    t_1 = t_0 - (b ** 2.0d0)
    t_2 = (b ** 2.0d0) - t_0
    if (a <= (-1d+104)) then
        tmp = sqrt((2.0d0 * (t_2 * (f * (a + (c - -a)))))) / t_1
    else if (a <= (-2.8d-44)) then
        tmp = -sqrt((((f * c) * ((-16.0d0) * (a ** 2.0d0))) + (f * (((b ** 2.0d0) * a) * 4.0d0)))) / ((b ** 2.0d0) - (4.0d0 * (a * c)))
    else if (a <= 1.35d+35) then
        tmp = sqrt((2.0d0 * (t_2 * (f * (a + (c - sqrt(((b ** 2.0d0) + ((a - c) ** 2.0d0))))))))) / t_1
    else
        tmp = sqrt(((-8.0d0) * (c * (f * (a * (c + c)))))) * ((2.0d0 / a) * (0.125d0 / c))
    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 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B, 2.0);
	double t_2 = Math.pow(B, 2.0) - t_0;
	double tmp;
	if (A <= -1e+104) {
		tmp = Math.sqrt((2.0 * (t_2 * (F * (A + (C - -A)))))) / t_1;
	} else if (A <= -2.8e-44) {
		tmp = -Math.sqrt((((F * C) * (-16.0 * Math.pow(A, 2.0))) + (F * ((Math.pow(B, 2.0) * A) * 4.0)))) / (Math.pow(B, 2.0) - (4.0 * (A * C)));
	} else if (A <= 1.35e+35) {
		tmp = Math.sqrt((2.0 * (t_2 * (F * (A + (C - Math.sqrt((Math.pow(B, 2.0) + Math.pow((A - C), 2.0))))))))) / t_1;
	} else {
		tmp = Math.sqrt((-8.0 * (C * (F * (A * (C + C)))))) * ((2.0 / A) * (0.125 / C));
	}
	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 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B, 2.0)
	t_2 = math.pow(B, 2.0) - t_0
	tmp = 0
	if A <= -1e+104:
		tmp = math.sqrt((2.0 * (t_2 * (F * (A + (C - -A)))))) / t_1
	elif A <= -2.8e-44:
		tmp = -math.sqrt((((F * C) * (-16.0 * math.pow(A, 2.0))) + (F * ((math.pow(B, 2.0) * A) * 4.0)))) / (math.pow(B, 2.0) - (4.0 * (A * C)))
	elif A <= 1.35e+35:
		tmp = math.sqrt((2.0 * (t_2 * (F * (A + (C - math.sqrt((math.pow(B, 2.0) + math.pow((A - C), 2.0))))))))) / t_1
	else:
		tmp = math.sqrt((-8.0 * (C * (F * (A * (C + C)))))) * ((2.0 / A) * (0.125 / C))
	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(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B ^ 2.0))
	t_2 = Float64((B ^ 2.0) - t_0)
	tmp = 0.0
	if (A <= -1e+104)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(A + Float64(C - Float64(-A))))))) / t_1);
	elseif (A <= -2.8e-44)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(F * C) * Float64(-16.0 * (A ^ 2.0))) + Float64(F * Float64(Float64((B ^ 2.0) * A) * 4.0))))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	elseif (A <= 1.35e+35)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(A + Float64(C - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(-8.0 * Float64(C * Float64(F * Float64(A * Float64(C + C)))))) * Float64(Float64(2.0 / A) * Float64(0.125 / C)));
	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 = (4.0 * A) * C;
	t_1 = t_0 - (B ^ 2.0);
	t_2 = (B ^ 2.0) - t_0;
	tmp = 0.0;
	if (A <= -1e+104)
		tmp = sqrt((2.0 * (t_2 * (F * (A + (C - -A)))))) / t_1;
	elseif (A <= -2.8e-44)
		tmp = -sqrt((((F * C) * (-16.0 * (A ^ 2.0))) + (F * (((B ^ 2.0) * A) * 4.0)))) / ((B ^ 2.0) - (4.0 * (A * C)));
	elseif (A <= 1.35e+35)
		tmp = sqrt((2.0 * (t_2 * (F * (A + (C - sqrt(((B ^ 2.0) + ((A - C) ^ 2.0))))))))) / t_1;
	else
		tmp = sqrt((-8.0 * (C * (F * (A * (C + C)))))) * ((2.0 / A) * (0.125 / C));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[A, -1e+104], N[(N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(A + N[(C - (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[A, -2.8e-44], N[((-N[Sqrt[N[(N[(N[(F * C), $MachinePrecision] * N[(-16.0 * N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F * N[(N[(N[Power[B, 2.0], $MachinePrecision] * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 1.35e+35], N[(N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(A + N[(C - N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(-8.0 * N[(C * N[(F * N[(A * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 / A), $MachinePrecision] * N[(0.125 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -1e104

   1. Initial program 58.0

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

   2. Simplified 58.2

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

      [Start] 58.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 58.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around -inf 50.1

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

   4. Simplified 50.1

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

      [Start] 50.1	\[ \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - -1 \cdot A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [=>] 50.1	\[ \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - \color{blue}{A \cdot -1}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-9 [=>] 50.1	\[ \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - \color{blue}{\left(-A\right)}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]

   if -1e104 < A < -2.8e-44

   1. Initial program 39.6

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

   2. Simplified 39.2

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

      [Start] 39.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [=>] 39.6	\[ \color{blue}{\frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 39.6	\[ \frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [<=] 39.6	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Taylor expanded in A around -inf 43.1

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

   4. Simplified 43.0

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

      [Start] 43.1	\[ \frac{-\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 43.1	\[ \frac{-\sqrt{\color{blue}{{A}^{2} \cdot \left(\left(C \cdot F\right) \cdot -16\right)} + 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 43.1	\[ \frac{-\sqrt{\color{blue}{\left(C \cdot F\right) \cdot \left(-16 \cdot {A}^{2}\right)} + 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-2 [=>] 43.1	\[ \frac{-\sqrt{\color{blue}{\left(F \cdot C\right)} \cdot \left(-16 \cdot {A}^{2}\right) + 4 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 43.0	\[ \frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + 4 \cdot \color{blue}{\left(F \cdot \left({B}^{2} \cdot A\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 43.0	\[ \frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + \color{blue}{F \cdot \left(\left({B}^{2} \cdot A\right) \cdot 4\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   if -2.8e-44 < A < 1.35000000000000001e35

   1. Initial program 47.9

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

   2. Simplified 47.6

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

      [Start] 47.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 47.9	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   if 1.35000000000000001e35 < A

   1. Initial program 61.3

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

   2. Simplified 62.2

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

      [Start] 61.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 61.3	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around inf 51.1

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

   4. Simplified 51.0

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

      [Start] 51.1	\[ \frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 49.1	\[ \frac{\sqrt{-8 \cdot \color{blue}{\left(\left(C - -1 \cdot C\right) \cdot \left(\left(C \cdot F\right) \cdot A\right)\right)}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [<=] 49.1	\[ \frac{\sqrt{-8 \cdot \left(\left(C - -1 \cdot C\right) \cdot \color{blue}{\left(A \cdot \left(C \cdot F\right)\right)}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 51.0	\[ \frac{\sqrt{-8 \cdot \left(\left(C - -1 \cdot C\right) \cdot \color{blue}{\left(C \cdot \left(F \cdot A\right)\right)}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [<=] 51.0	\[ \frac{\sqrt{-8 \cdot \left(\left(C - -1 \cdot C\right) \cdot \left(C \cdot \color{blue}{\left(A \cdot F\right)}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-43 [=>] 51.0	\[ \frac{\sqrt{-8 \cdot \color{blue}{\left(C \cdot \left(\left(A \cdot F\right) \cdot \left(C - -1 \cdot C\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 51.0	\[ \frac{\sqrt{-8 \cdot \left(C \cdot \left(\color{blue}{\left(F \cdot A\right)} \cdot \left(C - -1 \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 51.0	\[ \frac{\sqrt{-8 \cdot \left(C \cdot \left(\left(F \cdot A\right) \cdot \left(C - \color{blue}{C \cdot -1}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-9 [=>] 51.0	\[ \frac{\sqrt{-8 \cdot \left(C \cdot \left(\left(F \cdot A\right) \cdot \left(C - \color{blue}{\left(-C\right)}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   5. Taylor expanded in A around inf 49.4

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

   6. Simplified 49.4

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

      [Start] 49.4	\[ \frac{\sqrt{-8 \cdot \left(C \cdot \left(\left(F \cdot A\right) \cdot \left(C - \left(-C\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 49.4	\[ \frac{\sqrt{-8 \cdot \left(C \cdot \left(\left(F \cdot A\right) \cdot \left(C - \left(-C\right)\right)\right)\right)}}{\color{blue}{A \cdot \left(C \cdot 4\right)}} \]

   7. Applied egg-rr 52.1

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

   8. Simplified 45.6

      \[\leadsto \color{blue}{\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)} \]
      Proof

      [Start] 52.1	\[ \left(\sqrt{-8 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(C + C\right)\right)\right)} \cdot \frac{2}{A}\right) \cdot \frac{1}{C \cdot 8} \]
      rational.json-simplify-2 [=>] 52.1	\[ \color{blue}{\frac{1}{C \cdot 8} \cdot \left(\sqrt{-8 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(C + C\right)\right)\right)} \cdot \frac{2}{A}\right)} \]
      rational.json-simplify-43 [=>] 52.1	\[ \color{blue}{\sqrt{-8 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(C + C\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{1}{C \cdot 8}\right)} \]
      rational.json-simplify-43 [=>] 49.2	\[ \sqrt{-8 \cdot \color{blue}{\left(C \cdot \left(\left(C + C\right) \cdot \left(F \cdot A\right)\right)\right)}} \cdot \left(\frac{2}{A} \cdot \frac{1}{C \cdot 8}\right) \]
      rational.json-simplify-43 [=>] 45.6	\[ \sqrt{-8 \cdot \left(C \cdot \color{blue}{\left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)}\right)} \cdot \left(\frac{2}{A} \cdot \frac{1}{C \cdot 8}\right) \]
      rational.json-simplify-46 [=>] 45.6	\[ \sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \color{blue}{\frac{\frac{1}{C}}{8}}\right) \]
      rational.json-simplify-44 [=>] 45.6	\[ \sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \color{blue}{\frac{\frac{1}{8}}{C}}\right) \]
      metadata-eval [=>] 45.6	\[ \sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{\color{blue}{0.125}}{C}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 47.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + F \cdot \left(\left({B}^{2} \cdot A\right) \cdot 4\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	49.7
Cost	40532

\[\begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := C - \sqrt{{C}^{2} + {B}^{2}}\\ t_2 := t_0 - {B}^{2}\\ t_3 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -1.55 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + F \cdot \left(\left({B}^{2} \cdot A\right) \cdot 4\right)}}{{B}^{2} - t_3}\\ \mathbf{elif}\;A \leq -3.1 \cdot 10^{-242}:\\ \;\;\;\;\frac{\sqrt{\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot \left(\left(A + \left(C + C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq -4.3 \cdot 10^{-280}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot t_1}\right)\\ \mathbf{elif}\;A \leq 5.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{t_1 \cdot \left(\left(F \cdot {B}^{2}\right) \cdot 2\right)}}{t_3 - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 2
Error	51.1
Cost	33820

\[\begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t_0 - {B}^{2}\\ t_2 := {B}^{2} - t_0\\ t_3 := \frac{\sqrt{2}}{B}\\ t_4 := -\frac{\sqrt{2 \cdot \left(A \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot -8\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{if}\;A \leq -2.6 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.3 \cdot 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + F \cdot \left(\left({B}^{2} \cdot A\right) \cdot 4\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;A \leq -3.8 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot \left(\left(A + \left(C + C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 7 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{F \cdot \left(A + C \cdot 2\right)} \cdot \left(-t_3\right)\\ \mathbf{elif}\;A \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(C - \left(-B\right)\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.32 \cdot 10^{-185}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;A \leq 2.5 \cdot 10^{-158}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)\\ \mathbf{elif}\;A \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 3
Error	50.2
Cost	33424

\[\begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t_0 - {B}^{2}\\ \mathbf{if}\;A \leq -5.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + F \cdot \left(\left({B}^{2} \cdot A\right) \cdot 4\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;A \leq -2.1 \cdot 10^{-240}:\\ \;\;\;\;\frac{\sqrt{\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot \left(\left(A + \left(C + C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - \sqrt{{C}^{2} + {B}^{2}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 4
Error	51.2
Cost	27592

\[\begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t_0 - {B}^{2}\\ \mathbf{if}\;A \leq -1.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot C\right) \cdot \left(-16 \cdot {A}^{2}\right) + F \cdot \left(\left({B}^{2} \cdot A\right) \cdot 4\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;A \leq -4.3 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot \left(\left(A + \left(C + C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{F \cdot \left(A + C \cdot 2\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-34}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(A \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot -8\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 5
Error	51.2
Cost	27528

\[\begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t_0 - {B}^{2}\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.75 \cdot 10^{-94}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(\left(F \cdot {A}^{2}\right) \cdot -16\right) + A \cdot \left(\left(F \cdot {B}^{2}\right) \cdot 4\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;A \leq -9.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot \left(\left(A + \left(C + C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{F \cdot \left(A + C \cdot 2\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;A \leq 3.1 \cdot 10^{-34}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(A \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot -8\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 6
Error	51.0
Cost	21192

\[\begin{array}{l} t_0 := -\frac{\sqrt{2 \cdot \left(A \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot -8\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{if}\;A \leq -8 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot \left(\left(A + \left(C + C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;A \leq 1.15 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 7
Error	51.8
Cost	14540

\[\begin{array}{l} t_0 := -\frac{\sqrt{2 \cdot \left(A \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot -8\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}\\ \mathbf{if}\;A \leq -1.05 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 10^{-255}:\\ \;\;\;\;\sqrt{F \cdot \left(A + C \cdot 2\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{elif}\;A \leq 3.05 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 8
Error	52.8
Cost	14084

\[\begin{array}{l} \mathbf{if}\;A \leq -2.35 \cdot 10^{-215}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{-153}:\\ \;\;\;\;\sqrt{F \cdot \left(A + C \cdot 2\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 9
Error	56.5
Cost	13832

\[\begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\frac{2 \cdot \frac{-0.25}{A}}{2 \cdot \frac{-C}{\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}}\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{F \cdot \left(A + C \cdot 2\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{\frac{C}{\sqrt{-16 \cdot \left(C \cdot \left(C \cdot \left(F \cdot A\right)\right)\right)}}}\\ \end{array} \]

Alternative 10
Error	55.8
Cost	13572

\[\begin{array}{l} \mathbf{if}\;C \leq -1.25 \cdot 10^{+171}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{2 \cdot \left(C \cdot F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right)\\ \end{array} \]

Alternative 11
Error	55.4
Cost	7616

\[\sqrt{-8 \cdot \left(C \cdot \left(F \cdot \left(A \cdot \left(C + C\right)\right)\right)\right)} \cdot \left(\frac{2}{A} \cdot \frac{0.125}{C}\right) \]

Alternative 12
Error	55.4
Cost	7492

\[\begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+34}:\\ \;\;\;\;\sqrt{-16 \cdot \left(C \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)} \cdot \frac{\frac{0.25}{C}}{A}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{\frac{C}{\sqrt{-16 \cdot \left(C \cdot \left(C \cdot \left(F \cdot A\right)\right)\right)}}}\\ \end{array} \]

Alternative 13
Error	55.8
Cost	7360

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

Reproduce

herbie shell --seed 2023064 
(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))))