ABCF->ab-angle b

Average Error: 52.7 → 45.8

Time: 38.3s

Precision: binary64

Cost: 136012

Math

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

↓

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 - {B}^{2}\\ t_2 := {\left(A - C\right)}^{2}\\ t_3 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{t_2 + {B}^{2}}\right)}}{t_3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -2\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot \left(C - \left(\sqrt{{B}^{2} + t_2} - A\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(-2 \cdot C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot \left(-B\right)}\right)\\ \end{array} \]

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 (- A C) 2.0))
        (t_3 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt (* (* 2.0 (* t_3 F)) (- (+ A C) (sqrt (+ t_2 (pow B 2.0)))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (sqrt (* A (* 8.0 (* F (* C (* A -2.0)))))) t_1)
     (if (<= t_4 -2e-208)
       (/
        (sqrt
         (*
          2.0
          (*
           (- (pow B 2.0) t_0)
           (* F (- C (- (sqrt (+ (pow B 2.0) t_2)) A))))))
        t_1)
       (if (<= t_4 INFINITY)
         (/ (sqrt (* 8.0 (* C (* A (* F (* -2.0 C)))))) t_1)
         (* (/ (sqrt 2.0) B) (* -1.0 (sqrt (* F (- B))))))))))

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((A - C), 2.0);
	double t_3 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) - sqrt((t_2 + pow(B, 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt((A * (8.0 * (F * (C * (A * -2.0)))))) / t_1;
	} else if (t_4 <= -2e-208) {
		tmp = sqrt((2.0 * ((pow(B, 2.0) - t_0) * (F * (C - (sqrt((pow(B, 2.0) + t_2)) - A)))))) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((8.0 * (C * (A * (F * (-2.0 * C)))))) / t_1;
	} else {
		tmp = (sqrt(2.0) / B) * (-1.0 * sqrt((F * -B)));
	}
	return tmp;
}

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((A - C), 2.0);
	double t_3 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = -Math.sqrt(((2.0 * (t_3 * F)) * ((A + C) - Math.sqrt((t_2 + Math.pow(B, 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((A * (8.0 * (F * (C * (A * -2.0)))))) / t_1;
	} else if (t_4 <= -2e-208) {
		tmp = Math.sqrt((2.0 * ((Math.pow(B, 2.0) - t_0) * (F * (C - (Math.sqrt((Math.pow(B, 2.0) + t_2)) - A)))))) / t_1;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt((8.0 * (C * (A * (F * (-2.0 * C)))))) / t_1;
	} else {
		tmp = (Math.sqrt(2.0) / B) * (-1.0 * Math.sqrt((F * -B)));
	}
	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((A - C), 2.0)
	t_3 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_4 = -math.sqrt(((2.0 * (t_3 * F)) * ((A + C) - math.sqrt((t_2 + math.pow(B, 2.0)))))) / t_3
	tmp = 0
	if t_4 <= -math.inf:
		tmp = math.sqrt((A * (8.0 * (F * (C * (A * -2.0)))))) / t_1
	elif t_4 <= -2e-208:
		tmp = math.sqrt((2.0 * ((math.pow(B, 2.0) - t_0) * (F * (C - (math.sqrt((math.pow(B, 2.0) + t_2)) - A)))))) / t_1
	elif t_4 <= math.inf:
		tmp = math.sqrt((8.0 * (C * (A * (F * (-2.0 * C)))))) / t_1
	else:
		tmp = (math.sqrt(2.0) / B) * (-1.0 * math.sqrt((F * -B)))
	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 - C) ^ 2.0
	t_3 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) - sqrt(Float64(t_2 + (B ^ 2.0))))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(A * Float64(8.0 * Float64(F * Float64(C * Float64(A * -2.0)))))) / t_1);
	elseif (t_4 <= -2e-208)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64((B ^ 2.0) - t_0) * Float64(F * Float64(C - Float64(sqrt(Float64((B ^ 2.0) + t_2)) - A)))))) / t_1);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(8.0 * Float64(C * Float64(A * Float64(F * Float64(-2.0 * C)))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-1.0 * sqrt(Float64(F * Float64(-B)))));
	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) ^ 2.0;
	t_3 = (B ^ 2.0) - ((4.0 * A) * C);
	t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) - sqrt((t_2 + (B ^ 2.0)))))) / t_3;
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = sqrt((A * (8.0 * (F * (C * (A * -2.0)))))) / t_1;
	elseif (t_4 <= -2e-208)
		tmp = sqrt((2.0 * (((B ^ 2.0) - t_0) * (F * (C - (sqrt(((B ^ 2.0) + t_2)) - A)))))) / t_1;
	elseif (t_4 <= Inf)
		tmp = sqrt((8.0 * (C * (A * (F * (-2.0 * C)))))) / t_1;
	else
		tmp = (sqrt(2.0) / B) * (-1.0 * sqrt((F * -B)));
	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[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(t$95$2 + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[N[(A * N[(8.0 * N[(F * N[(C * N[(A * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -2e-208], N[(N[Sqrt[N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * N[(F * N[(C - N[(N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(8.0 * N[(C * N[(A * N[(F * N[(-2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(-1.0 * N[Sqrt[N[(F * (-B)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -inf.0

   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

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right) \cdot \left(\left({B}^{2} + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot -2\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      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-28 [=>] 64.0	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

   3. Taylor expanded in C around inf 56.4

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

   4. Simplified 56.2

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

      [Start] 56.4	\[ \frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-31 [=>] 56.5	\[ \frac{\sqrt{\color{blue}{A \cdot \left(8 \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-31 [=>] 56.2	\[ \frac{\sqrt{A \cdot \left(8 \cdot \color{blue}{\left(F \cdot \left(C \cdot \left(-1 \cdot A - A\right)\right)\right)}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 56.2	\[ \frac{\sqrt{A \cdot \left(8 \cdot \left(F \cdot \left(C \cdot \left(\color{blue}{A \cdot -1} - A\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-18 [=>] 56.2	\[ \frac{\sqrt{A \cdot \left(8 \cdot \left(F \cdot \left(C \cdot \color{blue}{\left(A \cdot \left(-1 + -1\right)\right)}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      metadata-eval [=>] 56.2	\[ \frac{\sqrt{A \cdot \left(8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot \color{blue}{-2}\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2.0000000000000002e-208

   1. Initial program 1.3

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

   2. Simplified 2.4

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

      [Start] 1.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-28 [=>] 1.3	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 1.3	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

   if -2.0000000000000002e-208 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 52.1

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

   2. Simplified 52.1

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

      [Start] 52.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational.json-simplify-28 [=>] 52.1	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 52.1	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

   3. Taylor expanded in A around inf 48.9

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

   4. Simplified 46.9

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

      [Start] 48.9	\[ \frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot C - C\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-31 [=>] 46.9	\[ \frac{\sqrt{8 \cdot \color{blue}{\left(C \cdot \left(A \cdot \left(F \cdot \left(-1 \cdot C - C\right)\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(\color{blue}{C \cdot -1} - C\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-18 [=>] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \color{blue}{\left(C \cdot \left(-1 + -1\right)\right)}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      metadata-eval [=>] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(C \cdot \color{blue}{-2}\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-31 [<=] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \color{blue}{\left(C \cdot \left(F \cdot -2\right)\right)}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [<=] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(C \cdot \color{blue}{\left(-2 \cdot F\right)}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-31 [<=] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \color{blue}{\left(-2 \cdot \left(C \cdot F\right)\right)}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(-2 \cdot \color{blue}{\left(F \cdot C\right)}\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-31 [=>] 46.9	\[ \frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \color{blue}{\left(F \cdot \left(-2 \cdot C\right)\right)}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   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

      \[\leadsto \color{blue}{\frac{\sqrt{2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - A\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
      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-28 [=>] 64.0	\[ \color{blue}{-\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}} \]
      rational.json-simplify-43 [<=] 64.0	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}} \]

   3. Taylor expanded in B around inf 63.8

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

   4. Taylor expanded in C around 0 63.4

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

   5. Simplified 63.4

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

      [Start] 63.4	\[ \frac{\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A - B\right) \cdot F}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 63.4	\[ \frac{\color{blue}{\sqrt{\left(A - B\right) \cdot F} \cdot \left(\sqrt{2} \cdot B\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-31 [=>] 63.4	\[ \frac{\color{blue}{\sqrt{2} \cdot \left(\sqrt{\left(A - B\right) \cdot F} \cdot B\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]
      rational.json-simplify-2 [=>] 63.4	\[ \frac{\sqrt{2} \cdot \left(\sqrt{\color{blue}{F \cdot \left(A - B\right)}} \cdot B\right)}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   6. Taylor expanded in C around 0 54.7

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

   7. Simplified 54.7

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

      [Start] 54.7	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\right) \]
      rational.json-simplify-31 [=>] 54.7	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot \left(A - B\right)}\right)} \]

   8. Taylor expanded in A around 0 54.3

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

   9. Simplified 54.3

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

      [Start] 54.3	\[ \frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{-1 \cdot \left(F \cdot B\right)}\right) \]
      rational.json-simplify-31 [=>] 54.3	\[ \frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{\color{blue}{F \cdot \left(-1 \cdot B\right)}}\right) \]
      rational.json-simplify-2 [=>] 54.3	\[ \frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot \color{blue}{\left(B \cdot -1\right)}}\right) \]
      rational.json-simplify-9 [<=] 54.3	\[ \frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot \color{blue}{\left(-B\right)}}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 45.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -2\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2 \cdot 10^{-208}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - A\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(-2 \cdot C\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot \left(-B\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	47.8
Cost	53704

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;{B}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - C\right)\right) \cdot \left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 2
Error	50.2
Cost	40532

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ t_1 := \frac{\sqrt{\left(\sqrt{{C}^{2} + {B}^{2}} - C\right) \cdot \left(F \cdot \left(-2 \cdot {B}^{2}\right)\right)}}{t_0}\\ t_2 := \frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot \left(-B\right)}\right)\\ \mathbf{if}\;A \leq -5.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C - A\right) + -0.5 \cdot \frac{{B}^{2}}{A}\right) - \left(A + C\right)\right) \cdot \left(\left({B}^{2} + \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot -2\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C \cdot \left(-8 \cdot \left({A}^{2} \cdot F\right)\right) + 2 \cdot \left({B}^{2} \cdot \left(A \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -4.2 \cdot 10^{-224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 5.6 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(C \cdot \left(-16 \cdot \left(F \cdot C\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(-2 \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 3
Error	48.7
Cost	40004

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;B \leq -1.85 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot \left(F \cdot {B}^{2}\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 4
Error	48.7
Cost	40004

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.12 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 5
Error	48.7
Cost	20804

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := t_0 - {B}^{2}\\ \mathbf{if}\;B \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left({B}^{2} - t_0\right) \cdot \left(F \cdot B\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 6
Error	48.8
Cost	20676

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;B \leq -9.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot -1 - C\right) \cdot \left(F \cdot \left(-2 \cdot {B}^{2}\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 7
Error	49.0
Cost	20612

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;B \leq -4 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(A - \left(-B\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 8
Error	49.4
Cost	20292

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot {B}^{3}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 9
Error	50.1
Cost	14472

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;B \leq -1.42 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(-2 \cdot C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(-1 \cdot A - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(C \cdot \left(F \cdot \left(A \cdot -16\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 10
Error	51.0
Cost	14348

\[\begin{array}{l} t_0 := \frac{\sqrt{A \cdot \left(C \cdot \left(-16 \cdot \left(F \cdot C\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - {B}^{2}}\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{-275}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-229}:\\ \;\;\;\;\frac{\sqrt{{A}^{2} \cdot \left(F \cdot \left(-16 \cdot C\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.86 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 11
Error	51.0
Cost	14348

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot 4\right) - {B}^{2}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-272}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(C \cdot \left(-16 \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.9 \cdot 10^{-228}:\\ \;\;\;\;\frac{\sqrt{{A}^{2} \cdot \left(F \cdot \left(-16 \cdot C\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 10^{-109}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(C \cdot \left(F \cdot \left(A \cdot -16\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 12
Error	50.0
Cost	14348

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - {B}^{2}\\ \mathbf{if}\;B \leq -9 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(-2 \cdot C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(8 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -2\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(C \cdot \left(F \cdot \left(A \cdot -16\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 13
Error	49.8
Cost	14212

\[\begin{array}{l} \mathbf{if}\;B \leq 5.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{8 \cdot \left(C \cdot \left(A \cdot \left(F \cdot \left(-2 \cdot C\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 14
Error	52.1
Cost	13828

\[\begin{array}{l} \mathbf{if}\;B \leq 4.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sqrt{{A}^{2} \cdot \left(F \cdot \left(-16 \cdot C\right)\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-1 \cdot \sqrt{F \cdot \left(-B\right)}\right)\\ \end{array} \]

Alternative 15
Error	55.0
Cost	13440

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

Alternative 16
Error	62.8
Cost	13312

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

Alternative 17
Error	62.0
Cost	6976

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

Alternative 18
Error	62.0
Cost	6976

\[2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right) \]

Reproduce

herbie shell --seed 2023053 
(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))))