ABCF->ab-angle a

Average Error: 52.5 → 46.3

Time: 1.2min

Precision: binary64

Cost: 41368

Math

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

↓

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ t_1 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.6 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -3.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right) + \left(C \cdot \left({B}^{2} \cdot 4\right) + \left(-\frac{0.5 \cdot {B}^{4} + -2 \cdot {\left(C \cdot B\right)}^{2}}{A}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -5.1 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 5.7 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(B + A\right) \cdot F}\right)\\ \mathbf{elif}\;A \leq 1200000:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;A \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(t_1 \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

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 (- (pow B 2.0) (* 4.0 (* A C)))))
   (if (<= A -2.6e+149)
     t_0
     (if (<= A -3.8e+112)
       (/
        (-
         (sqrt
          (*
           (* 2.0 F)
           (+
            (* (pow C 2.0) (* -8.0 A))
            (+
             (* C (* (pow B 2.0) 4.0))
             (- (/ (+ (* 0.5 (pow B 4.0)) (* -2.0 (pow (* C B) 2.0))) A)))))))
        t_1)
       (if (<= A -5.1e-121)
         t_0
         (if (<= A 5.7e-307)
           (* (/ (sqrt 2.0) B) (- (sqrt (* (+ B A) F))))
           (if (<= A 1200000.0)
             (- (sqrt (- (/ F C))))
             (if (<= A 2.25e+79)
               (/
                (-
                 (sqrt
                  (*
                   (* 2.0 F)
                   (*
                    t_1
                    (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0))))))))
                t_1)
               (/ (- (* A (sqrt (* (* F C) -16.0)))) (* A (* C -4.0)))))))))))

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 / A));
	double t_1 = pow(B, 2.0) - (4.0 * (A * C));
	double tmp;
	if (A <= -2.6e+149) {
		tmp = t_0;
	} else if (A <= -3.8e+112) {
		tmp = -sqrt(((2.0 * F) * ((pow(C, 2.0) * (-8.0 * A)) + ((C * (pow(B, 2.0) * 4.0)) + -(((0.5 * pow(B, 4.0)) + (-2.0 * pow((C * B), 2.0))) / A))))) / t_1;
	} else if (A <= -5.1e-121) {
		tmp = t_0;
	} else if (A <= 5.7e-307) {
		tmp = (sqrt(2.0) / B) * -sqrt(((B + A) * F));
	} else if (A <= 1200000.0) {
		tmp = -sqrt(-(F / C));
	} else if (A <= 2.25e+79) {
		tmp = -sqrt(((2.0 * F) * (t_1 * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0))))))) / t_1;
	} else {
		tmp = -(A * sqrt(((F * C) * -16.0))) / (A * (C * -4.0));
	}
	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) :: tmp
    t_0 = -sqrt(-(f / a))
    t_1 = (b ** 2.0d0) - (4.0d0 * (a * c))
    if (a <= (-2.6d+149)) then
        tmp = t_0
    else if (a <= (-3.8d+112)) then
        tmp = -sqrt(((2.0d0 * f) * (((c ** 2.0d0) * ((-8.0d0) * a)) + ((c * ((b ** 2.0d0) * 4.0d0)) + -(((0.5d0 * (b ** 4.0d0)) + ((-2.0d0) * ((c * b) ** 2.0d0))) / a))))) / t_1
    else if (a <= (-5.1d-121)) then
        tmp = t_0
    else if (a <= 5.7d-307) then
        tmp = (sqrt(2.0d0) / b) * -sqrt(((b + a) * f))
    else if (a <= 1200000.0d0) then
        tmp = -sqrt(-(f / c))
    else if (a <= 2.25d+79) then
        tmp = -sqrt(((2.0d0 * f) * (t_1 * ((a + c) + sqrt(((b ** 2.0d0) + ((a - c) ** 2.0d0))))))) / t_1
    else
        tmp = -(a * sqrt(((f * c) * (-16.0d0)))) / (a * (c * (-4.0d0)))
    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 / A));
	double t_1 = Math.pow(B, 2.0) - (4.0 * (A * C));
	double tmp;
	if (A <= -2.6e+149) {
		tmp = t_0;
	} else if (A <= -3.8e+112) {
		tmp = -Math.sqrt(((2.0 * F) * ((Math.pow(C, 2.0) * (-8.0 * A)) + ((C * (Math.pow(B, 2.0) * 4.0)) + -(((0.5 * Math.pow(B, 4.0)) + (-2.0 * Math.pow((C * B), 2.0))) / A))))) / t_1;
	} else if (A <= -5.1e-121) {
		tmp = t_0;
	} else if (A <= 5.7e-307) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt(((B + A) * F));
	} else if (A <= 1200000.0) {
		tmp = -Math.sqrt(-(F / C));
	} else if (A <= 2.25e+79) {
		tmp = -Math.sqrt(((2.0 * F) * (t_1 * ((A + C) + Math.sqrt((Math.pow(B, 2.0) + Math.pow((A - C), 2.0))))))) / t_1;
	} else {
		tmp = -(A * Math.sqrt(((F * C) * -16.0))) / (A * (C * -4.0));
	}
	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 / A))
	t_1 = math.pow(B, 2.0) - (4.0 * (A * C))
	tmp = 0
	if A <= -2.6e+149:
		tmp = t_0
	elif A <= -3.8e+112:
		tmp = -math.sqrt(((2.0 * F) * ((math.pow(C, 2.0) * (-8.0 * A)) + ((C * (math.pow(B, 2.0) * 4.0)) + -(((0.5 * math.pow(B, 4.0)) + (-2.0 * math.pow((C * B), 2.0))) / A))))) / t_1
	elif A <= -5.1e-121:
		tmp = t_0
	elif A <= 5.7e-307:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt(((B + A) * F))
	elif A <= 1200000.0:
		tmp = -math.sqrt(-(F / C))
	elif A <= 2.25e+79:
		tmp = -math.sqrt(((2.0 * F) * (t_1 * ((A + C) + math.sqrt((math.pow(B, 2.0) + math.pow((A - C), 2.0))))))) / t_1
	else:
		tmp = -(A * math.sqrt(((F * C) * -16.0))) / (A * (C * -4.0))
	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 / A))))
	t_1 = Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C)))
	tmp = 0.0
	if (A <= -2.6e+149)
		tmp = t_0;
	elseif (A <= -3.8e+112)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64((C ^ 2.0) * Float64(-8.0 * A)) + Float64(Float64(C * Float64((B ^ 2.0) * 4.0)) + Float64(-Float64(Float64(Float64(0.5 * (B ^ 4.0)) + Float64(-2.0 * (Float64(C * B) ^ 2.0))) / A))))))) / t_1);
	elseif (A <= -5.1e-121)
		tmp = t_0;
	elseif (A <= 5.7e-307)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(Float64(B + A) * F))));
	elseif (A <= 1200000.0)
		tmp = Float64(-sqrt(Float64(-Float64(F / C))));
	elseif (A <= 2.25e+79)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(t_1 * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))))) / t_1);
	else
		tmp = Float64(Float64(-Float64(A * sqrt(Float64(Float64(F * C) * -16.0)))) / Float64(A * Float64(C * -4.0)));
	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 / A));
	t_1 = (B ^ 2.0) - (4.0 * (A * C));
	tmp = 0.0;
	if (A <= -2.6e+149)
		tmp = t_0;
	elseif (A <= -3.8e+112)
		tmp = -sqrt(((2.0 * F) * (((C ^ 2.0) * (-8.0 * A)) + ((C * ((B ^ 2.0) * 4.0)) + -(((0.5 * (B ^ 4.0)) + (-2.0 * ((C * B) ^ 2.0))) / A))))) / t_1;
	elseif (A <= -5.1e-121)
		tmp = t_0;
	elseif (A <= 5.7e-307)
		tmp = (sqrt(2.0) / B) * -sqrt(((B + A) * F));
	elseif (A <= 1200000.0)
		tmp = -sqrt(-(F / C));
	elseif (A <= 2.25e+79)
		tmp = -sqrt(((2.0 * F) * (t_1 * ((A + C) + sqrt(((B ^ 2.0) + ((A - C) ^ 2.0))))))) / t_1;
	else
		tmp = -(A * sqrt(((F * C) * -16.0))) / (A * (C * -4.0));
	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 / A), $MachinePrecision])], $MachinePrecision])}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -2.6e+149], t$95$0, If[LessEqual[A, -3.8e+112], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(N[Power[C, 2.0], $MachinePrecision] * N[(-8.0 * A), $MachinePrecision]), $MachinePrecision] + N[(N[(C * N[(N[Power[B, 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] + (-N[(N[(N[(0.5 * N[Power[B, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[Power[N[(C * B), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[A, -5.1e-121], t$95$0, If[LessEqual[A, 5.7e-307], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(N[(B + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[A, 1200000.0], (-N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision]), If[LessEqual[A, 2.25e+79], N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(t$95$1 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[((-N[(A * N[Sqrt[N[(N[(F * C), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if A < -2.59999999999999979e149 or -3.80000000000000008e112 < A < -5.0999999999999998e-121

   1. Initial program 57.5

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

   2. Simplified 58.5

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

      [Start] 57.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 64.0

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

   4. Simplified 43.1

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

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \cdot -1} \]
      rational.json-simplify-9 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}} \]
      exponential.json-simplify-20 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{A}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{A}} \]
      exponential.json-simplify-20 [=>] 43.1	\[ -\color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational.json-simplify-9 [=>] 43.1	\[ -\sqrt{\color{blue}{-\frac{F}{A}}} \]

   if -2.59999999999999979e149 < A < -3.80000000000000008e112

   1. Initial program 61.7

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

   2. Simplified 62.9

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

      [Start] 61.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 47.7

      \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(-8 \cdot \left(A \cdot {C}^{2}\right) + \left(-1 \cdot \frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A} + 4 \cdot \left(C \cdot {B}^{2}\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 47.7

      \[\leadsto \frac{-\sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left({C}^{2} \cdot \left(-8 \cdot A\right) + \left(C \cdot \left({B}^{2} \cdot 4\right) + \left(-\frac{0.5 \cdot {B}^{4} + -2 \cdot {\left(C \cdot B\right)}^{2}}{A}\right)\right)\right)}}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(-8 \cdot \left(A \cdot {C}^{2}\right) + \left(-1 \cdot \frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A} + 4 \cdot \left(C \cdot {B}^{2}\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\color{blue}{A \cdot \left({C}^{2} \cdot -8\right)} + \left(-1 \cdot \frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A} + 4 \cdot \left(C \cdot {B}^{2}\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\color{blue}{{C}^{2} \cdot \left(-8 \cdot A\right)} + \left(-1 \cdot \frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A} + 4 \cdot \left(C \cdot {B}^{2}\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-1 [=>] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right) + \color{blue}{\left(4 \cdot \left(C \cdot {B}^{2}\right) + -1 \cdot \frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A}\right)}\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right) + \left(\color{blue}{C \cdot \left({B}^{2} \cdot 4\right)} + -1 \cdot \frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-2 [=>] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right) + \left(C \cdot \left({B}^{2} \cdot 4\right) + \color{blue}{\frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A} \cdot -1}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-9 [=>] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right) + \left(C \cdot \left({B}^{2} \cdot 4\right) + \color{blue}{\left(-\frac{0.5 \cdot {B}^{4} + -2 \cdot \left({C}^{2} \cdot {B}^{2}\right)}{A}\right)}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]
      exponential.json-simplify-27 [=>] 47.7	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right) + \left(C \cdot \left({B}^{2} \cdot 4\right) + \left(-\frac{0.5 \cdot {B}^{4} + -2 \cdot \color{blue}{{\left(C \cdot B\right)}^{2}}}{A}\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} \]

   if -5.0999999999999998e-121 < A < 5.70000000000000017e-307

   1. Initial program 49.5

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

   2. Simplified 52.0

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

      [Start] 49.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 51.9

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

   4. Simplified 49.5

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

      [Start] 51.9	\[ \frac{-\sqrt{\left(F + F\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} + 0 \]
      rational.json-simplify-4 [=>] 51.9	\[ \color{blue}{\frac{-\sqrt{\left(F + F\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}} \]

   5. Taylor expanded in B around inf 56.9

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

   6. Taylor expanded in C around 0 50.1

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

   7. Simplified 50.1

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

      [Start] 50.1	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + B\right) \cdot F}\right) \]
      rational.json-simplify-43 [=>] 50.1	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\left(A + B\right) \cdot F} \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 50.1	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(-\sqrt{\left(A + B\right) \cdot F}\right)} \]
      rational.json-simplify-1 [=>] 50.1	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{\left(B + A\right)} \cdot F}\right) \]

   if 5.70000000000000017e-307 < A < 1.2e6

   1. Initial program 46.8

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

   2. Simplified 49.3

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

      [Start] 46.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 64.0

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

   4. Simplified 50.3

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{C}}} \]
      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 [=>] 50.3	\[ -\color{blue}{\sqrt{\frac{F}{C} \cdot -1}} \]
      rational.json-simplify-9 [=>] 50.3	\[ -\sqrt{\color{blue}{-\frac{F}{C}}} \]

   if 1.2e6 < A < 2.24999999999999997e79

   1. Initial program 39.0

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

   2. Simplified 40.1

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

      [Start] 39.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 2.24999999999999997e79 < A

   1. Initial program 57.0

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

   2. Simplified 57.9

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

      [Start] 57.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 57.9

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

   4. Simplified 57.0

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

      [Start] 57.9	\[ \frac{-\sqrt{\left(F + F\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} + 0 \]
      rational.json-simplify-4 [=>] 57.9	\[ \color{blue}{\frac{-\sqrt{\left(F + F\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}} \]

   5. Taylor expanded in A around inf 64.0

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

   6. Simplified 44.9

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

      [Start] 64.0	\[ \frac{-\left(A \cdot \left(\sqrt{2} \cdot \sqrt{-8}\right)\right) \cdot \sqrt{C \cdot F}}{{B}^{2} - C \cdot \left(4 \cdot A\right)} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{-\color{blue}{\sqrt{C \cdot F} \cdot \left(A \cdot \left(\sqrt{2} \cdot \sqrt{-8}\right)\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \frac{-\color{blue}{A \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{C \cdot F}\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)} \]
      exponential.json-simplify-20 [=>] 64.0	\[ \frac{-A \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{C \cdot F}\right)}{{B}^{2} - C \cdot \left(4 \cdot A\right)} \]
      exponential.json-simplify-20 [=>] 44.9	\[ \frac{-A \cdot \color{blue}{\sqrt{\left(C \cdot F\right) \cdot \left(-8 \cdot 2\right)}}}{{B}^{2} - C \cdot \left(4 \cdot A\right)} \]
      rational.json-simplify-2 [=>] 44.9	\[ \frac{-A \cdot \sqrt{\color{blue}{\left(F \cdot C\right)} \cdot \left(-8 \cdot 2\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)} \]
      metadata-eval [=>] 44.9	\[ \frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot \color{blue}{-16}}}{{B}^{2} - C \cdot \left(4 \cdot A\right)} \]

   7. Taylor expanded in B around 0 44.8

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

   8. Simplified 44.8

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

      [Start] 44.8	\[ \frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{-4 \cdot \left(A \cdot C\right)} \]
      rational.json-simplify-43 [=>] 44.8	\[ \frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{\color{blue}{A \cdot \left(C \cdot -4\right)}} \]

3. Recombined 6 regimes into one program.

4. Final simplification 46.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.6 \cdot 10^{+149}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -3.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right) + \left(C \cdot \left({B}^{2} \cdot 4\right) + \left(-\frac{0.5 \cdot {B}^{4} + -2 \cdot {\left(C \cdot B\right)}^{2}}{A}\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;A \leq -5.1 \cdot 10^{-121}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq 5.7 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(B + A\right) \cdot F}\right)\\ \mathbf{elif}\;A \leq 1200000:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;A \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	48.3
Cost	41632

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-195}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-261}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq -7.8 \cdot 10^{-290}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-203}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-C \cdot \sqrt{\left(A \cdot F\right) \cdot -16}}{t_1}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 2
Error	48.3
Cost	41632

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ t_1 := {B}^{2} - C \cdot \left(4 \cdot A\right)\\ \mathbf{if}\;B \leq -1.16 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -2.25 \cdot 10^{-262}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-289}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-204}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-125}:\\ \;\;\;\;\frac{-C \cdot \sqrt{\left(A \cdot F\right) \cdot -16}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \left(A + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(t_1 \cdot \left(F + F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 3
Error	48.8
Cost	40200

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ t_1 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.9 \cdot 10^{+73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({B}^{2} \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -2.5 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-199}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-231}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-263}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B \leq -9.6 \cdot 10^{-290}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left({C}^{2} \cdot \left(-8 \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-204}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{-C \cdot \sqrt{\left(A \cdot F\right) \cdot -16}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \end{array} \]

Alternative 4
Error	47.0
Cost	13704

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;A \leq -5 \cdot 10^{-121}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(B + A\right) \cdot F}\right)\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 3.7 \cdot 10^{+57}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 5
Error	47.2
Cost	7956

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;A \leq -5 \cdot 10^{-121}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-307}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;A \leq 2 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 3.7 \cdot 10^{+57}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-A \cdot \sqrt{\left(F \cdot C\right) \cdot -16}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 6
Error	47.4
Cost	7376

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ \mathbf{if}\;A \leq -6 \cdot 10^{-121}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq 1.08 \cdot 10^{-307}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;A \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 3.3 \cdot 10^{+58}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	46.7
Cost	7048

\[\begin{array}{l} \mathbf{if}\;A \leq -5.6 \cdot 10^{-121}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq 1.05 \cdot 10^{-307}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 8
Error	47.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;A \leq -6.7 \cdot 10^{-304}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 9
Error	55.0
Cost	6720

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

Reproduce

herbie shell --seed 2023077 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))