ABCF->ab-angle b

Average Error: 52.3 → 45.4

Time: 1.4min

Precision: binary64

Cost: 41172

Math

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

↓

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 - {B}^{2}\\ t_2 := A \cdot \left(C \cdot 4\right)\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := {B}^{2} - t_0\\ t_5 := A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\\ \mathbf{if}\;C \leq -4.2 \cdot 10^{+230}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -3.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{t_1} \cdot \sqrt{t_4 \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C + C\right)\right)\right)\right)}\\ \mathbf{elif}\;C \leq -4.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{t_2 - {B}^{2}}{\sqrt{2 \cdot \left(t_5 \cdot \left(\left({B}^{2} - t_2\right) \cdot F\right)\right)}}}\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(t_4 \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_3\right) \cdot \left(F \cdot t_5\right)\right)}}{t_3 - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

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 (* A (* C 4.0)))
        (t_3 (* (* 4.0 A) C))
        (t_4 (- (pow B 2.0) t_0))
        (t_5 (+ A (- C (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
   (if (<= C -4.2e+230)
     (* 0.25 (sqrt (* (/ F A) -16.0)))
     (if (<= C -3.4e+155)
       (* (/ 1.0 t_1) (sqrt (* t_4 (* 2.0 (* F (+ A (+ C C)))))))
       (if (<= C -4.5e-294)
         (/
          1.0
          (/
           (- t_2 (pow B 2.0))
           (sqrt (* 2.0 (* t_5 (* (- (pow B 2.0) t_2) F))))))
         (if (<= C 1.45e-248)
           (/ (sqrt (* (* 2.0 F) (* t_4 (+ A (- C (- A)))))) t_1)
           (if (<= C 8.5e-9)
             (/
              (sqrt (* 2.0 (* (- (pow B 2.0) t_3) (* F t_5))))
              (- t_3 (pow B 2.0)))
             (- (sqrt (- (/ F 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 = A * (C * 4.0);
	double t_3 = (4.0 * A) * C;
	double t_4 = pow(B, 2.0) - t_0;
	double t_5 = A + (C - sqrt((pow(B, 2.0) + pow((A - C), 2.0))));
	double tmp;
	if (C <= -4.2e+230) {
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	} else if (C <= -3.4e+155) {
		tmp = (1.0 / t_1) * sqrt((t_4 * (2.0 * (F * (A + (C + C))))));
	} else if (C <= -4.5e-294) {
		tmp = 1.0 / ((t_2 - pow(B, 2.0)) / sqrt((2.0 * (t_5 * ((pow(B, 2.0) - t_2) * F)))));
	} else if (C <= 1.45e-248) {
		tmp = sqrt(((2.0 * F) * (t_4 * (A + (C - -A))))) / t_1;
	} else if (C <= 8.5e-9) {
		tmp = sqrt((2.0 * ((pow(B, 2.0) - t_3) * (F * t_5)))) / (t_3 - pow(B, 2.0));
	} else {
		tmp = -sqrt(-(F / 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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = 4.0d0 * (a * c)
    t_1 = t_0 - (b ** 2.0d0)
    t_2 = a * (c * 4.0d0)
    t_3 = (4.0d0 * a) * c
    t_4 = (b ** 2.0d0) - t_0
    t_5 = a + (c - sqrt(((b ** 2.0d0) + ((a - c) ** 2.0d0))))
    if (c <= (-4.2d+230)) then
        tmp = 0.25d0 * sqrt(((f / a) * (-16.0d0)))
    else if (c <= (-3.4d+155)) then
        tmp = (1.0d0 / t_1) * sqrt((t_4 * (2.0d0 * (f * (a + (c + c))))))
    else if (c <= (-4.5d-294)) then
        tmp = 1.0d0 / ((t_2 - (b ** 2.0d0)) / sqrt((2.0d0 * (t_5 * (((b ** 2.0d0) - t_2) * f)))))
    else if (c <= 1.45d-248) then
        tmp = sqrt(((2.0d0 * f) * (t_4 * (a + (c - -a))))) / t_1
    else if (c <= 8.5d-9) then
        tmp = sqrt((2.0d0 * (((b ** 2.0d0) - t_3) * (f * t_5)))) / (t_3 - (b ** 2.0d0))
    else
        tmp = -sqrt(-(f / 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 = A * (C * 4.0);
	double t_3 = (4.0 * A) * C;
	double t_4 = Math.pow(B, 2.0) - t_0;
	double t_5 = A + (C - Math.sqrt((Math.pow(B, 2.0) + Math.pow((A - C), 2.0))));
	double tmp;
	if (C <= -4.2e+230) {
		tmp = 0.25 * Math.sqrt(((F / A) * -16.0));
	} else if (C <= -3.4e+155) {
		tmp = (1.0 / t_1) * Math.sqrt((t_4 * (2.0 * (F * (A + (C + C))))));
	} else if (C <= -4.5e-294) {
		tmp = 1.0 / ((t_2 - Math.pow(B, 2.0)) / Math.sqrt((2.0 * (t_5 * ((Math.pow(B, 2.0) - t_2) * F)))));
	} else if (C <= 1.45e-248) {
		tmp = Math.sqrt(((2.0 * F) * (t_4 * (A + (C - -A))))) / t_1;
	} else if (C <= 8.5e-9) {
		tmp = Math.sqrt((2.0 * ((Math.pow(B, 2.0) - t_3) * (F * t_5)))) / (t_3 - Math.pow(B, 2.0));
	} else {
		tmp = -Math.sqrt(-(F / 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 = A * (C * 4.0)
	t_3 = (4.0 * A) * C
	t_4 = math.pow(B, 2.0) - t_0
	t_5 = A + (C - math.sqrt((math.pow(B, 2.0) + math.pow((A - C), 2.0))))
	tmp = 0
	if C <= -4.2e+230:
		tmp = 0.25 * math.sqrt(((F / A) * -16.0))
	elif C <= -3.4e+155:
		tmp = (1.0 / t_1) * math.sqrt((t_4 * (2.0 * (F * (A + (C + C))))))
	elif C <= -4.5e-294:
		tmp = 1.0 / ((t_2 - math.pow(B, 2.0)) / math.sqrt((2.0 * (t_5 * ((math.pow(B, 2.0) - t_2) * F)))))
	elif C <= 1.45e-248:
		tmp = math.sqrt(((2.0 * F) * (t_4 * (A + (C - -A))))) / t_1
	elif C <= 8.5e-9:
		tmp = math.sqrt((2.0 * ((math.pow(B, 2.0) - t_3) * (F * t_5)))) / (t_3 - math.pow(B, 2.0))
	else:
		tmp = -math.sqrt(-(F / 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(4.0 * Float64(A * C))
	t_1 = Float64(t_0 - (B ^ 2.0))
	t_2 = Float64(A * Float64(C * 4.0))
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64((B ^ 2.0) - t_0)
	t_5 = Float64(A + Float64(C - sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0)))))
	tmp = 0.0
	if (C <= -4.2e+230)
		tmp = Float64(0.25 * sqrt(Float64(Float64(F / A) * -16.0)));
	elseif (C <= -3.4e+155)
		tmp = Float64(Float64(1.0 / t_1) * sqrt(Float64(t_4 * Float64(2.0 * Float64(F * Float64(A + Float64(C + C)))))));
	elseif (C <= -4.5e-294)
		tmp = Float64(1.0 / Float64(Float64(t_2 - (B ^ 2.0)) / sqrt(Float64(2.0 * Float64(t_5 * Float64(Float64((B ^ 2.0) - t_2) * F))))));
	elseif (C <= 1.45e-248)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(t_4 * Float64(A + Float64(C - Float64(-A)))))) / t_1);
	elseif (C <= 8.5e-9)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_3) * Float64(F * t_5)))) / Float64(t_3 - (B ^ 2.0)));
	else
		tmp = Float64(-sqrt(Float64(-Float64(F / 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 = A * (C * 4.0);
	t_3 = (4.0 * A) * C;
	t_4 = (B ^ 2.0) - t_0;
	t_5 = A + (C - sqrt(((B ^ 2.0) + ((A - C) ^ 2.0))));
	tmp = 0.0;
	if (C <= -4.2e+230)
		tmp = 0.25 * sqrt(((F / A) * -16.0));
	elseif (C <= -3.4e+155)
		tmp = (1.0 / t_1) * sqrt((t_4 * (2.0 * (F * (A + (C + C))))));
	elseif (C <= -4.5e-294)
		tmp = 1.0 / ((t_2 - (B ^ 2.0)) / sqrt((2.0 * (t_5 * (((B ^ 2.0) - t_2) * F)))));
	elseif (C <= 1.45e-248)
		tmp = sqrt(((2.0 * F) * (t_4 * (A + (C - -A))))) / t_1;
	elseif (C <= 8.5e-9)
		tmp = sqrt((2.0 * (((B ^ 2.0) - t_3) * (F * t_5)))) / (t_3 - (B ^ 2.0));
	else
		tmp = -sqrt(-(F / 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[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$5 = 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]}, If[LessEqual[C, -4.2e+230], N[(0.25 * N[Sqrt[N[(N[(F / A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[C, -3.4e+155], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[Sqrt[N[(t$95$4 * N[(2.0 * N[(F * N[(A + N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[C, -4.5e-294], N[(1.0 / N[(N[(t$95$2 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * N[(t$95$5 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 1.45e-248], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(t$95$4 * N[(A + N[(C - (-A)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[C, 8.5e-9], N[(N[Sqrt[N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * N[(F * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision])]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if C < -4.19999999999999986e230

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in B around 0 64.0

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

   4. Simplified 47.4

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

      [Start] 64.0	\[ 0.25 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-8}\right) \cdot \sqrt{\frac{F}{A}}\right) \]
      exponential.json-simplify-20 [=>] 64.0	\[ 0.25 \cdot \left(\color{blue}{\sqrt{-8 \cdot 2}} \cdot \sqrt{\frac{F}{A}}\right) \]
      metadata-eval [=>] 64.0	\[ 0.25 \cdot \left(\sqrt{\color{blue}{-16}} \cdot \sqrt{\frac{F}{A}}\right) \]
      exponential.json-simplify-20 [=>] 47.4	\[ 0.25 \cdot \color{blue}{\sqrt{\frac{F}{A} \cdot -16}} \]

   if -4.19999999999999986e230 < C < -3.4000000000000001e155

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in C around -inf 50.7

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

   4. Simplified 50.7

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

      [Start] 50.7	\[ \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - -1 \cdot C\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-2 [=>] 50.7	\[ \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - \color{blue}{C \cdot -1}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]
      rational.json-simplify-9 [=>] 50.7	\[ \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - \color{blue}{\left(-C\right)}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \]

   5. Applied egg-rr 50.7

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

   if -3.4000000000000001e155 < C < -4.49999999999999981e-294

   1. Initial program 44.4

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

   2. Simplified 44.6

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

      [Start] 44.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [=>] 44.4	\[ \color{blue}{\frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]
      rational.json-simplify-5 [<=] 44.4	\[ \frac{-\left(-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}{\color{blue}{\left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right) - 0}} \]
      rational.json-simplify-50 [<=] 44.4	\[ \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{0 - \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)}} \]

   3. Applied egg-rr 44.5

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

   4. Applied egg-rr 44.5

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

   if -4.49999999999999981e-294 < C < 1.4500000000000001e-248

   1. Initial program 47.0

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

   2. Simplified 48.1

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

      [Start] 47.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 47.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   3. Taylor expanded in A around -inf 57.0

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

   4. Simplified 57.0

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

      [Start] 57.0	\[ \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C - -1 \cdot A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 57.0	\[ \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C - \color{blue}{A \cdot -1}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-9 [=>] 57.0	\[ \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C - \color{blue}{\left(-A\right)}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   if 1.4500000000000001e-248 < C < 8.5e-9

   1. Initial program 48.8

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

   2. Simplified 48.3

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

      [Start] 48.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-50 [<=] 48.8	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   if 8.5e-9 < C

   1. Initial program 61.0

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

   2. Taylor expanded in A around -inf 64.0

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

   3. Simplified 40.3

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{C}}} \]
      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 [=>] 40.3	\[ -\color{blue}{\sqrt{\frac{F}{C} \cdot -1}} \]
      rational.json-simplify-9 [=>] 40.3	\[ -\sqrt{\color{blue}{-\frac{F}{C}}} \]

3. Recombined 6 regimes into one program.

4. Final simplification 45.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -4.2 \cdot 10^{+230}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -3.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \cdot \sqrt{\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C + C\right)\right)\right)\right)}\\ \mathbf{elif}\;C \leq -4.5 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{A \cdot \left(C \cdot 4\right) - {B}^{2}}{\sqrt{2 \cdot \left(\left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right)\right)}}}\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;C \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternatives

Alternative 1
Error	45.4
Cost	41172

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := A \cdot \left(C \cdot 4\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_2\right) \cdot \left(F \cdot \left(A + \left(C - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{t_2 - {B}^{2}}\\ \mathbf{if}\;C \leq -4.5 \cdot 10^{+230}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -5 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{t_1 - {B}^{2}} \cdot \sqrt{\left(\left(A + C\right) - \left(-C\right)\right) \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F + F\right)\right)}\\ \mathbf{elif}\;C \leq -4.5 \cdot 10^{-294}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 1.45 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{elif}\;C \leq 2.15 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 2
Error	47.8
Cost	40532

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t_1 - {B}^{2}\\ t_3 := \frac{\sqrt{2 \cdot \left(\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot \left(F \cdot {B}^{2}\right)\right)}}{t_2}\\ \mathbf{if}\;C \leq -1.15 \cdot 10^{+231}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -3 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F \cdot \left(A + \left(C - \left(-C\right)\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq -4.5 \cdot 10^{-294}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 2.9 \cdot 10^{-239}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(A + \left(C - \left(-A\right)\right)\right)\right)}}{t_0 - {B}^{2}}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{-106}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 3
Error	48.6
Cost	22252

\[\begin{array}{l} t_0 := \frac{0.25}{A \cdot C} \cdot \left(-A \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)\\ t_1 := A \cdot \left(C \cdot 4\right)\\ t_2 := F \cdot \left(A \cdot -16\right)\\ t_3 := -\sqrt{-\frac{F}{A}}\\ t_4 := \left(4 \cdot A\right) \cdot C\\ t_5 := t_4 - {B}^{2}\\ t_6 := \sqrt{t_2}\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_4\right) \cdot \left(F \cdot \left(A + \left(C - \left(-B\right)\right)\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-170}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\frac{1}{-\frac{\frac{t_1 - {B}^{2}}{t_6}}{C}}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-266}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{\frac{C}{\sqrt{\left(A \cdot C\right) \cdot t_2}}}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-229}:\\ \;\;\;\;\frac{C}{{B}^{2} - t_1} \cdot t_6\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-177}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-143}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-130}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A \cdot \left(-8 \cdot \left(A \cdot C\right) + {B}^{2} \cdot 4\right)\right)\right)}}{t_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \end{array} \]

Alternative 4
Error	47.8
Cost	21648

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;C \leq -9 \cdot 10^{+230}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -6 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_1\right) \cdot \left(F \cdot \left(A + \left(C - \left(-C\right)\right)\right)\right)\right)}}{t_1 - {B}^{2}}\\ \mathbf{elif}\;C \leq -4 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-B\right)}\right)\\ \mathbf{elif}\;C \leq 7 \cdot 10^{-211}:\\ \;\;\;\;\frac{1}{t_0 - {B}^{2}} \cdot \sqrt{\left(\left(A + C\right) - \left(-A\right)\right) \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F + F\right)\right)}\\ \mathbf{elif}\;C \leq 2.05 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{-B}{\sqrt{F \cdot \left(A - B\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 5
Error	49.5
Cost	14540

\[\begin{array}{l} t_0 := 0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;A \leq -3.1 \cdot 10^{+190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -1.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(A \cdot C\right) \cdot \left(F \cdot \left(A \cdot -16\right)\right)}}{A}}{C \cdot 4}\\ \mathbf{elif}\;A \leq 2.8 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(C - \left(-C\right)\right) \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;A \leq 8.8 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 6.92 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 1.42 \cdot 10^{+208}:\\ \;\;\;\;\frac{1}{\frac{C \cdot \left(A \cdot 4\right)}{-C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{elif}\;A \leq 8.8 \cdot 10^{+257}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	47.4
Cost	14100

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{if}\;A \leq -3.5 \cdot 10^{+190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -4.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \left(-A \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)\\ \mathbf{elif}\;A \leq 9.5 \cdot 10^{-283}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;A \leq 4.1 \cdot 10^{-204}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 4.6 \cdot 10^{-145}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \mathbf{elif}\;A \leq 3.05 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 1.4 \cdot 10^{+209}:\\ \;\;\;\;\frac{1}{\frac{C \cdot \left(A \cdot 4\right)}{-C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{elif}\;A \leq 9.4 \cdot 10^{+257}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	48.7
Cost	8812

\[\begin{array}{l} t_0 := \frac{0.25}{\frac{A \cdot C}{\sqrt{\left(A \cdot C\right) \cdot \left(F \cdot \left(A \cdot -16\right)\right)}}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ t_3 := \frac{-1}{B} \cdot \sqrt{\left(A - B\right) \cdot \left(F + F\right)}\\ \mathbf{if}\;C \leq -1.65 \cdot 10^{+186}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -4.3 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -2.25 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -1.05 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -7.8 \cdot 10^{-64}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq -1.05 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -3.1 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 2.4 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 3.7 \cdot 10^{-106}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 2.4 \cdot 10^{+160}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq 1.1 \cdot 10^{+242}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \left(-A \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	48.7
Cost	8812

\[\begin{array}{l} t_0 := \sqrt{\left(A \cdot C\right) \cdot \left(F \cdot \left(A \cdot -16\right)\right)}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ t_3 := \frac{-1}{B} \cdot \sqrt{\left(A - B\right) \cdot \left(F + F\right)}\\ \mathbf{if}\;C \leq -1.65 \cdot 10^{+186}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -2.7 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -9.2 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -7.8 \cdot 10^{-64}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq -1.35 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -3.1 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 9.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{\frac{C}{t_0}}\\ \mathbf{elif}\;C \leq 4.9 \cdot 10^{-98}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 4.6 \cdot 10^{+163}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq 3.9 \cdot 10^{+240}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \left(-A \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{A \cdot C}{t_0}}\\ \end{array} \]

Alternative 9
Error	48.6
Cost	8812

\[\begin{array}{l} t_0 := \sqrt{\left(A \cdot C\right) \cdot \left(F \cdot \left(A \cdot -16\right)\right)}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ t_3 := \frac{-1}{B} \cdot \sqrt{\left(A - B\right) \cdot \left(F + F\right)}\\ \mathbf{if}\;C \leq -6.2 \cdot 10^{+185}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -1.15 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -1.2 \cdot 10^{-23}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -6.5 \cdot 10^{-64}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq -2.2 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -3.1 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 4.9 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{t_0}{A}}{C \cdot 4}\\ \mathbf{elif}\;C \leq 1.42 \cdot 10^{-108}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 2.1 \cdot 10^{+161}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq 5.3 \cdot 10^{+240}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \left(-A \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\frac{A \cdot C}{t_0}}\\ \end{array} \]

Alternative 10
Error	47.9
Cost	8556

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ t_3 := \frac{-1}{B} \cdot \sqrt{\left(A - B\right) \cdot \left(F + F\right)}\\ \mathbf{if}\;C \leq -1.65 \cdot 10^{+186}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -1.2 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -3.7 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -3 \cdot 10^{-60}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq -1.2 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -3.5 \cdot 10^{-296}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 9.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \sqrt{A \cdot \left(A \cdot \left(-16 \cdot \left(C \cdot F\right)\right)\right)}\\ \mathbf{elif}\;C \leq 6.8 \cdot 10^{-181}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;C \leq 8.2 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 5.3 \cdot 10^{-107}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	47.4
Cost	8348

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{if}\;A \leq -3.3 \cdot 10^{+190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -3.4 \cdot 10^{-205}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \left(-A \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)\\ \mathbf{elif}\;A \leq 1.28 \cdot 10^{-281}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;A \leq 3.3 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 5.7 \cdot 10^{-145}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(A - B\right) \cdot \left(F + F\right)}\\ \mathbf{elif}\;A \leq 1.8 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 7.2 \cdot 10^{+211}:\\ \;\;\;\;\frac{1}{\frac{C \cdot \left(A \cdot 4\right)}{-C \cdot \sqrt{\left(F \cdot A\right) \cdot -16}}}\\ \mathbf{elif}\;A \leq 1.5 \cdot 10^{+258}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	48.0
Cost	8292

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{C}}\\ t_1 := 0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ t_2 := -\sqrt{-\frac{F}{A}}\\ t_3 := \frac{-1}{B} \cdot \sqrt{\left(A - B\right) \cdot \left(F + F\right)}\\ \mathbf{if}\;C \leq -1.26 \cdot 10^{+186}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -1.12 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -3 \cdot 10^{-61}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq -8.5 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -5.8 \cdot 10^{-305}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 2.9 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.2 \cdot 10^{-100}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	48.1
Cost	8160

\[\begin{array}{l} t_0 := -\sqrt{-2 \cdot \frac{F}{B}}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -9.8 \cdot 10^{+184}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -1.45 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq -1.55 \cdot 10^{-37}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;C \leq -3.8 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq -5.5 \cdot 10^{-293}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.9 \cdot 10^{-238}:\\ \;\;\;\;\frac{0.25}{A \cdot C} \cdot \left(-A \cdot \sqrt{-16 \cdot \left(C \cdot F\right)}\right)\\ \mathbf{elif}\;C \leq 1.65 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.42 \cdot 10^{-98}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{\left(A - B\right) \cdot \left(F + F\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 14
Error	47.4
Cost	7312

\[\begin{array}{l} \mathbf{if}\;C \leq -6.2 \cdot 10^{+185}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -1.26 \cdot 10^{+31}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;C \leq -2.9 \cdot 10^{-39}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{C} \cdot -16}\\ \mathbf{elif}\;C \leq 8.5 \cdot 10^{-177}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 15
Error	47.7
Cost	7248

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -5.8 \cdot 10^{+185}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F}{A} \cdot -16}\\ \mathbf{elif}\;C \leq -5.4 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq -1.3 \cdot 10^{-182}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;C \leq -5.8 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 16
Error	47.5
Cost	7116

\[\begin{array}{l} t_0 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -8.5 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq -3.8 \cdot 10^{-182}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \mathbf{elif}\;C \leq -5.8 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 17
Error	47.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;C \leq -5.8 \cdot 10^{-305}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \end{array} \]

Alternative 18
Error	54.6
Cost	6720

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

Reproduce

herbie shell --seed 2023073 
(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))))