ABCF->ab-angle a

Percentage Accurate: 19.2% → 53.2%

Time: 39.5s

Precision: binary64

Cost: 34516

• Unsound rule application detected in e-graph. Results may not be sound.

\[ \begin{array}{c}[A, C] = \mathsf{sort}([A, C])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}\\ t_2 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot F\right) \cdot t_3}}{t_3}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(2, C, B \cdot \frac{-0.5}{\frac{A}{B}}\right)}\right)\right)}{t_2}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{C \cdot 2} \cdot t_4\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot \left(C \cdot 2\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;t_1 \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

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 (/ (sqrt 2.0) B))
        (t_1 (sqrt (+ (hypot B (- A C)) (+ A C))))
        (t_2 (- (* B B) (* 4.0 (* A C))))
        (t_3 (fma A (* C -4.0) (* B B)))
        (t_4 (/ (- (sqrt (* (* 2.0 F) t_3))) t_3)))
   (if (<= B -2.3e-23)
     (* t_1 (* t_0 (sqrt F)))
     (if (<= B -1.55e-128)
       (/
        (*
         (sqrt 2.0)
         (*
          (sqrt F)
          (-
           (sqrt
            (*
             (fma B B (* A (* C -4.0)))
             (fma 2.0 C (* B (/ -0.5 (/ A B)))))))))
        t_2)
       (if (<= B 4.3e-172)
         (* (sqrt (* C 2.0)) t_4)
         (if (<= B 2.3e-123)
           (/ (- (sqrt (* (* 2.0 (* F t_2)) (* C 2.0)))) t_2)
           (if (<= B 1.6e+68) (* t_1 t_4) (* t_1 (* t_0 (- (sqrt F)))))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double t_0 = sqrt(2.0) / B;
	double t_1 = sqrt((hypot(B, (A - C)) + (A + C)));
	double t_2 = (B * B) - (4.0 * (A * C));
	double t_3 = fma(A, (C * -4.0), (B * B));
	double t_4 = -sqrt(((2.0 * F) * t_3)) / t_3;
	double tmp;
	if (B <= -2.3e-23) {
		tmp = t_1 * (t_0 * sqrt(F));
	} else if (B <= -1.55e-128) {
		tmp = (sqrt(2.0) * (sqrt(F) * -sqrt((fma(B, B, (A * (C * -4.0))) * fma(2.0, C, (B * (-0.5 / (A / B)))))))) / t_2;
	} else if (B <= 4.3e-172) {
		tmp = sqrt((C * 2.0)) * t_4;
	} else if (B <= 2.3e-123) {
		tmp = -sqrt(((2.0 * (F * t_2)) * (C * 2.0))) / t_2;
	} else if (B <= 1.6e+68) {
		tmp = t_1 * t_4;
	} else {
		tmp = t_1 * (t_0 * -sqrt(F));
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
end

↓

function code(A, B, C, F)
	t_0 = Float64(sqrt(2.0) / B)
	t_1 = sqrt(Float64(hypot(B, Float64(A - C)) + Float64(A + C)))
	t_2 = Float64(Float64(B * B) - Float64(4.0 * Float64(A * C)))
	t_3 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * F) * t_3))) / t_3)
	tmp = 0.0
	if (B <= -2.3e-23)
		tmp = Float64(t_1 * Float64(t_0 * sqrt(F)));
	elseif (B <= -1.55e-128)
		tmp = Float64(Float64(sqrt(2.0) * Float64(sqrt(F) * Float64(-sqrt(Float64(fma(B, B, Float64(A * Float64(C * -4.0))) * fma(2.0, C, Float64(B * Float64(-0.5 / Float64(A / B))))))))) / t_2);
	elseif (B <= 4.3e-172)
		tmp = Float64(sqrt(Float64(C * 2.0)) * t_4);
	elseif (B <= 2.3e-123)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_2)) * Float64(C * 2.0)))) / t_2);
	elseif (B <= 1.6e+68)
		tmp = Float64(t_1 * t_4);
	else
		tmp = Float64(t_1 * Float64(t_0 * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, If[LessEqual[B, -2.3e-23], N[(t$95$1 * N[(t$95$0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.55e-128], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C + N[(B * N[(-0.5 / N[(A / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B, 4.3e-172], N[(N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[B, 2.3e-123], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.6e+68], N[(t$95$1 * t$95$4), $MachinePrecision], N[(t$95$1 * N[(t$95$0 * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -2.3000000000000001e-23

   1. Initial program 24.5%

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

   2. Simplified 24.5%

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

      [Start] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 36.1%

      \[\leadsto \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 24.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 24.5	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 28.7	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 29.0	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 29.0	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 36.1	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Applied egg-rr 36.1%

      \[\leadsto \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      Step-by-step derivation

      [Start] 36.1	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      div-inv [=>] 36.1	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      distribute-rgt-neg-in [=>] 36.1	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right)\right)} \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 36.1	\[ \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right)} \]
      *-commutative [=>] 36.1	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      associate-*l* [=>] 36.1	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      fma-neg [=>] 36.1	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      distribute-lft-neg-in [=>] 36.1	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right)}\right) \]
      metadata-eval [=>] 36.1	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right) \]

   5. Simplified 35.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      Step-by-step derivation

      [Start] 36.1	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 36.1	\[ \sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 36.1	\[ \sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right)} + A} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+l+ [=>] 35.7	\[ \sqrt{\color{blue}{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 35.8	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\frac{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   6. Taylor expanded in B around -inf 62.8%

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

   if -2.3000000000000001e-23 < B < -1.55000000000000001e-128

   1. Initial program 22.4%

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

   2. Simplified 22.4%

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

      [Start] 22.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 22.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 22.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 22.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 22.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 22.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 22.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 22.8%

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

   4. Simplified 22.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 22.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-def [=>] 22.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 22.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 19.0%

      \[\leadsto \frac{-\color{blue}{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \left(\frac{B}{A} \cdot B\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 22.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 22.8	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 22.9	\[ \frac{-\color{blue}{\sqrt{2} \cdot \sqrt{\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\color{blue}{\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-neg [=>] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, B, -\color{blue}{\left(A \cdot C\right) \cdot 4}\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-rgt-neg-in [=>] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      metadata-eval [=>] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot \color{blue}{-4}\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-/l* [=>] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \color{blue}{\frac{B}{\frac{A}{B}}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-/r/ [=>] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \color{blue}{\left(\frac{B}{A} \cdot B\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 22.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{2} \cdot \sqrt{F \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 19.0	\[ \frac{-\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \left(\frac{B}{A} \cdot B\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 22.9	\[ \frac{-\sqrt{2} \cdot \sqrt{\color{blue}{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \left(\frac{B}{A} \cdot B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 22.9	\[ \frac{-\sqrt{2} \cdot \sqrt{\color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \left(\frac{B}{A} \cdot B\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 22.7	\[ \frac{-\sqrt{2} \cdot \sqrt{\color{blue}{F \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \left(\frac{B}{A} \cdot B\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-/r/ [<=] 22.7	\[ \frac{-\sqrt{2} \cdot \sqrt{F \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \color{blue}{\frac{B}{\frac{A}{B}}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Applied egg-rr 34.6%

      \[\leadsto \frac{-\sqrt{2} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(2, C, \frac{-0.5}{\frac{\frac{A}{B}}{B}}\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 22.7	\[ \frac{-\sqrt{2} \cdot \sqrt{F \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 22.7	\[ \frac{-\sqrt{2} \cdot \sqrt{\color{blue}{\left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)\right) \cdot F}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 34.6	\[ \frac{-\sqrt{2} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)} \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 34.6	\[ \frac{-\sqrt{2} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right) \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}} \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      clear-num [=>] 34.6	\[ \frac{-\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{A}{B}}{B}}}\right) \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)} \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      un-div-inv [=>] 34.6	\[ \frac{-\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(2, C, \color{blue}{\frac{-0.5}{\frac{\frac{A}{B}}{B}}}\right) \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)} \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 34.6	\[ \frac{-\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(2, C, \frac{-0.5}{\frac{\frac{A}{B}}{B}}\right) \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)} \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Simplified 34.6%

      \[\leadsto \frac{-\sqrt{2} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{A}{B}} \cdot B\right)} \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 34.6	\[ \frac{-\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(2, C, \frac{-0.5}{\frac{\frac{A}{B}}{B}}\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 34.6	\[ \frac{-\sqrt{2} \cdot \left(\sqrt{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(2, C, \frac{-0.5}{\frac{\frac{A}{B}}{B}}\right)}} \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-/r/ [=>] 34.6	\[ \frac{-\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{-0.5}{\frac{A}{B}} \cdot B}\right)} \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.55000000000000001e-128 < B < 4.2999999999999997e-172

   1. Initial program 13.7%

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

   2. Simplified 13.7%

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

      [Start] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 30.9%

      \[\leadsto \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 13.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 13.7	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 14.9	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 15.9	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 15.9	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 30.9	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Applied egg-rr 29.5%

      \[\leadsto \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      Step-by-step derivation

      [Start] 30.9	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      div-inv [=>] 29.6	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      distribute-rgt-neg-in [=>] 29.6	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right)\right)} \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 29.5	\[ \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right)} \]
      *-commutative [=>] 29.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      associate-*l* [=>] 29.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      fma-neg [=>] 29.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      distribute-lft-neg-in [=>] 29.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right)}\right) \]
      metadata-eval [=>] 29.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right) \]

   5. Simplified 28.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      Step-by-step derivation

      [Start] 29.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 29.5	\[ \sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 29.5	\[ \sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right)} + A} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+l+ [=>] 26.9	\[ \sqrt{\color{blue}{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 28.3	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\frac{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   6. Taylor expanded in A around -inf 17.0%

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

   if 4.2999999999999997e-172 < B < 2.29999999999999987e-123

   1. Initial program 12.2%

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

   2. Simplified 12.2%

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

      [Start] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 12.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 22.0%

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

   4. Simplified 22.0%

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

      [Start] 22.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 22.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 2.29999999999999987e-123 < B < 1.59999999999999997e68

   1. Initial program 28.0%

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

   2. Simplified 28.0%

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

      [Start] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 55.6%

      \[\leadsto \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 28.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 28.0	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 31.3	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 31.7	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 31.7	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 55.6	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Applied egg-rr 55.6%

      \[\leadsto \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      Step-by-step derivation

      [Start] 55.6	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      div-inv [=>] 55.5	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      distribute-rgt-neg-in [=>] 55.5	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right)\right)} \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 55.5	\[ \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right)} \]
      *-commutative [=>] 55.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      associate-*l* [=>] 55.5	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      fma-neg [=>] 55.6	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      distribute-lft-neg-in [=>] 55.6	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right)}\right) \]
      metadata-eval [=>] 55.6	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right) \]

   5. Simplified 54.9%

      \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      Step-by-step derivation

      [Start] 55.6	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 55.6	\[ \sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 55.6	\[ \sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right)} + A} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+l+ [=>] 54.9	\[ \sqrt{\color{blue}{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 54.9	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\frac{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   if 1.59999999999999997e68 < B

   1. Initial program 10.4%

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

   2. Simplified 10.4%

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

      [Start] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 17.4%

      \[\leadsto \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Step-by-step derivation

      [Start] 10.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 10.4	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 15.0	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 15.0	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 15.0	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 17.4	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Applied egg-rr 17.4%

      \[\leadsto \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      Step-by-step derivation

      [Start] 17.4	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      div-inv [=>] 17.4	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      distribute-rgt-neg-in [=>] 17.4	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right)\right)} \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 17.4	\[ \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right)} \]
      *-commutative [=>] 17.4	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      associate-*l* [=>] 17.4	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      fma-neg [=>] 17.4	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      distribute-lft-neg-in [=>] 17.4	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right)}\right) \]
      metadata-eval [=>] 17.4	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right) \]

   5. Simplified 17.3%

      \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      Step-by-step derivation

      [Start] 17.4	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 17.4	\[ \sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 17.4	\[ \sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right)} + A} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+l+ [=>] 17.4	\[ \sqrt{\color{blue}{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 17.3	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\frac{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   6. Taylor expanded in A around 0 87.1%

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

   7. Simplified 87.1%

      \[\leadsto \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)} \]
      Step-by-step derivation

      [Start] 87.1	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\right) \]
      mul-1-neg [=>] 87.1	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)} \]
      distribute-rgt-neg-in [=>] 87.1	\[ \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)} \]

3. Recombined 6 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \mathsf{fma}\left(2, C, B \cdot \frac{-0.5}{\frac{A}{B}}\right)}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{C \cdot 2} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(C \cdot 2\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	53.8%
Cost	34516

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}\\ t_2 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot \left(C \cdot 2\right)}}{t_2}\\ t_4 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_5 := \frac{-\sqrt{\left(2 \cdot F\right) \cdot t_4}}{t_4}\\ \mathbf{if}\;B \leq -550000:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-249}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{C \cdot 2} \cdot t_5\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+68}:\\ \;\;\;\;t_1 \cdot t_5\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	53.9%
Cost	28244

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{t_1 + \left(A + C\right)}\\ t_3 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(F \cdot t_3\right)\right) \cdot \left(C \cdot 2\right)}}{t_3}\\ t_5 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -14500:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-249}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{C \cdot 2} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot t_5}}{t_5}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-129}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + t_1\right)}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	52.3%
Cost	27532

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}\\ t_2 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -80000:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -3.2 \cdot 10^{-251}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot \left(C \cdot 2\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{C \cdot 2} \cdot \frac{-\sqrt{\left(2 \cdot F\right) \cdot t_3}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	52.2%
Cost	26956

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}\\ t_2 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1450000:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.85 \cdot 10^{-292}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot \left(C \cdot 2\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	48.5%
Cost	26628

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -450000:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-298}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+85}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 6
Accuracy	36.6%
Cost	21260

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.95 \cdot 10^{-294}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7
Accuracy	36.9%
Cost	15240

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+56}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 8
Accuracy	36.8%
Cost	15240

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -2.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.85 \cdot 10^{+56}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9
Accuracy	37.3%
Cost	14788

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -5.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.9 \cdot 10^{+54}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10
Accuracy	34.7%
Cost	13316

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 9.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 11
Accuracy	26.9%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(B + \left(A + C\right)\right)}}{t_0}\\ \end{array} \]

Alternative 12
Accuracy	26.0%
Cost	8192

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(A \cdot C\right)\\ \frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C \cdot 2\right)}}{t_0} \end{array} \]

Alternative 13
Accuracy	8.9%
Cost	6980

\[\begin{array}{l} t_0 := \frac{\sqrt{C \cdot F}}{B}\\ \mathbf{if}\;B \leq -5 \cdot 10^{-311}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot -2\\ \end{array} \]

Alternative 14
Accuracy	0.9%
Cost	6848

\[-2 \cdot \frac{\sqrt{A \cdot F}}{B} \]

Alternative 15
Accuracy	5.4%
Cost	6848

\[\frac{\sqrt{C \cdot F}}{B} \cdot -2 \]

Reproduce

herbie shell --seed 2023162 
(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))))