ABCF->ab-angle a

Percentage Accurate: 18.6% → 43.0%

Time: 54.3s

Precision: binary64

Cost: 46852

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

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

Math

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

↓

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \sqrt{A + \left(C + t_2\right)}\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_5 := \sqrt{t_4}\\ \mathbf{if}\;B \leq -4 \cdot 10^{+28}:\\ \;\;\;\;\frac{B}{\frac{t_5}{\sqrt{F \cdot 2}}} \cdot \frac{t_3}{t_5}\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-t_3\right)}{t_0}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{t_2 + \left(C + A\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(-\sqrt{F}\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

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 (- (* B B) (* 4.0 (* C A))))
        (t_1 (* 2.0 (* F t_0)))
        (t_2 (hypot B (- A C)))
        (t_3 (sqrt (+ A (+ C t_2))))
        (t_4 (fma B B (* C (* A -4.0))))
        (t_5 (sqrt t_4)))
   (if (<= B -4e+28)
     (* (/ B (/ t_5 (sqrt (* F 2.0)))) (/ t_3 t_5))
     (if (<= B -1.06e-8)
       (/ (- (sqrt (* t_1 (fma 2.0 C (* -0.5 (/ (* B B) A)))))) t_0)
       (if (<= B -5.8e-112)
         (/ (* (sqrt (* 2.0 (* F (fma B B (* -4.0 (* C A)))))) (- t_3)) t_0)
         (if (<= B 6e-118)
           (/ (- (sqrt (* t_1 (* C 2.0)))) t_0)
           (if (<= B 1.25e+143)
             (/
              (*
               (sqrt (+ t_2 (+ C A)))
               (* (sqrt (* 2.0 (fma B B (* A (* C -4.0))))) (- (sqrt F))))
              t_4)
             (* (sqrt (* F (+ C (hypot B C)))) (/ (- (sqrt 2.0)) 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 = (B * B) - (4.0 * (C * A));
	double t_1 = 2.0 * (F * t_0);
	double t_2 = hypot(B, (A - C));
	double t_3 = sqrt((A + (C + t_2)));
	double t_4 = fma(B, B, (C * (A * -4.0)));
	double t_5 = sqrt(t_4);
	double tmp;
	if (B <= -4e+28) {
		tmp = (B / (t_5 / sqrt((F * 2.0)))) * (t_3 / t_5);
	} else if (B <= -1.06e-8) {
		tmp = -sqrt((t_1 * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_0;
	} else if (B <= -5.8e-112) {
		tmp = (sqrt((2.0 * (F * fma(B, B, (-4.0 * (C * A)))))) * -t_3) / t_0;
	} else if (B <= 6e-118) {
		tmp = -sqrt((t_1 * (C * 2.0))) / t_0;
	} else if (B <= 1.25e+143) {
		tmp = (sqrt((t_2 + (C + A))) * (sqrt((2.0 * fma(B, B, (A * (C * -4.0))))) * -sqrt(F))) / t_4;
	} else {
		tmp = sqrt((F * (C + hypot(B, C)))) * (-sqrt(2.0) / 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(Float64(B * B) - Float64(4.0 * Float64(C * A)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	t_2 = hypot(B, Float64(A - C))
	t_3 = sqrt(Float64(A + Float64(C + t_2)))
	t_4 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_5 = sqrt(t_4)
	tmp = 0.0
	if (B <= -4e+28)
		tmp = Float64(Float64(B / Float64(t_5 / sqrt(Float64(F * 2.0)))) * Float64(t_3 / t_5));
	elseif (B <= -1.06e-8)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_0);
	elseif (B <= -5.8e-112)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B, B, Float64(-4.0 * Float64(C * A)))))) * Float64(-t_3)) / t_0);
	elseif (B <= 6e-118)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(C * 2.0)))) / t_0);
	elseif (B <= 1.25e+143)
		tmp = Float64(Float64(sqrt(Float64(t_2 + Float64(C + A))) * Float64(sqrt(Float64(2.0 * fma(B, B, Float64(A * Float64(C * -4.0))))) * Float64(-sqrt(F)))) / t_4);
	else
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(B, C)))) * Float64(Float64(-sqrt(2.0)) / B));
	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[(B * B), $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(A + N[(C + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$4], $MachinePrecision]}, If[LessEqual[B, -4e+28], N[(N[(B / N[(t$95$5 / N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.06e-8], N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, -5.8e-112], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B * B + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-t$95$3)), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 6e-118], N[((-N[Sqrt[N[(t$95$1 * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 1.25e+143], N[(N[(N[Sqrt[N[(t$95$2 + N[(C + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -3.99999999999999983e28

   1. Initial program 8.0%

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

   2. Simplified 8.0%

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

      [Start] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 7.9%

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

      [Start] 8.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      add-sqr-sqrt [=>] 7.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      pow2 [=>] 7.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \color{blue}{{\left(\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}}}\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 7.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}}\right)}^{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 7.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(B, A - C\right)}}\right)}^{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Taylor expanded in B around inf 7.9%

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

   5. Simplified 7.9%

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

      [Start] 7.9	\[ \frac{-\sqrt{\left(2 \cdot \left(F \cdot {B}^{2}\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 7.9	\[ \frac{-\sqrt{\left(2 \cdot \left(F \cdot \color{blue}{\left(B \cdot B\right)}\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Applied egg-rr 23.6%

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

      [Start] 7.9	\[ \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-un-lft-identity [=>] 7.9	\[ \frac{\color{blue}{1 \cdot \left(-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      add-sqr-sqrt [=>] 7.9	\[ \frac{1 \cdot \left(-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}\right)}{\color{blue}{\sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot \sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)}}} \]
      times-frac [=>] 8.0	\[ \color{blue}{\frac{1}{\sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \cdot \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{\sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)}}} \]
      cancel-sign-sub-inv [=>] 8.0	\[ \frac{1}{\sqrt{\color{blue}{B \cdot B + \left(-4\right) \cdot \left(A \cdot C\right)}}} \cdot \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{\sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      fma-def [=>] 8.0	\[ \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(B, B, \left(-4\right) \cdot \left(A \cdot C\right)\right)}}} \cdot \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{\sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      associate-*r* [=>] 8.0	\[ \frac{1}{\sqrt{\mathsf{fma}\left(B, B, \color{blue}{\left(\left(-4\right) \cdot A\right) \cdot C}\right)}} \cdot \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{\sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      metadata-eval [=>] 8.0	\[ \frac{1}{\sqrt{\mathsf{fma}\left(B, B, \left(\color{blue}{-4} \cdot A\right) \cdot C\right)}} \cdot \frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) \cdot \left(\left(A + C\right) + {\left(\sqrt{\mathsf{hypot}\left(B, A - C\right)}\right)}^{2}\right)}}{\sqrt{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   7. Simplified 28.5%

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

      [Start] 23.6	\[ \frac{1}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}} \cdot \frac{\left(\sqrt{2 \cdot F} \cdot B\right) \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}} \]
      associate-*l/ [=>] 23.7	\[ \color{blue}{\frac{1 \cdot \frac{\left(\sqrt{2 \cdot F} \cdot B\right) \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}}} \]
      *-lft-identity [=>] 23.7	\[ \frac{\color{blue}{\frac{\left(\sqrt{2 \cdot F} \cdot B\right) \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}}}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}} \]
      associate-*r/ [<=] 27.9	\[ \frac{\color{blue}{\left(\sqrt{2 \cdot F} \cdot B\right) \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}}}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}} \]
      associate-*l/ [<=] 27.9	\[ \color{blue}{\frac{\sqrt{2 \cdot F} \cdot B}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}} \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}}} \]

   if -3.99999999999999983e28 < B < -1.06000000000000006e-8

   1. Initial program 10.2%

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

   2. Simplified 10.2%

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

      [Start] 10.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 10.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 10.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 10.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 10.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 10.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 10.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 34.5%

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

   4. Simplified 34.5%

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

      [Start] 34.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-def [=>] 34.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 34.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.06000000000000006e-8 < B < -5.79999999999999985e-112

   1. Initial program 28.5%

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

   2. Simplified 28.5%

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

      [Start] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 45.6%

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

      [Start] 28.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 28.5	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 32.8	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 32.7	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 32.7	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 45.6	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -5.79999999999999985e-112 < B < 6.00000000000000035e-118

   1. Initial program 14.6%

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

   2. Simplified 14.6%

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

      [Start] 14.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 14.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 14.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 14.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 14.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 14.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 14.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 29.7%

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

   4. Simplified 29.7%

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

      [Start] 29.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 29.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(C \cdot 2\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 6.00000000000000035e-118 < B < 1.25000000000000003e143

   1. Initial program 41.8%

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

   2. Simplified 47.8%

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

      [Start] 41.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 51.0%

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

      [Start] 47.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 47.8	\[ \frac{-\sqrt{\color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      sqrt-prod [=>] 51.4	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 51.4	\[ \frac{-\sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 51.4	\[ \frac{-\sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + C\right)} + A} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+l+ [=>] 51.0	\[ \frac{-\sqrt{\color{blue}{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 66.1%

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

      [Start] 51.0	\[ \frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r* [=>] 51.4	\[ \frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot F}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      sqrt-prod [=>] 66.1	\[ \frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F}\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 66.1	\[ \frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot -4\right) \cdot C}\right)} \cdot \sqrt{F}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*l* [=>] 66.1	\[ \frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(-4 \cdot C\right)}\right)} \cdot \sqrt{F}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 1.25000000000000003e143 < B

   1. Initial program 0.1%

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

   2. Simplified 0.1%

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

      [Start] 0.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 2.4%

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

   4. Simplified 57.1%

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

      [Start] 2.4	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 2.4	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]
      distribute-rgt-neg-in [=>] 2.4	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right)} \]
      *-commutative [=>] 2.4	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}}\right) \]
      unpow2 [=>] 2.4	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right)}\right) \]
      unpow2 [=>] 2.4	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right)}\right) \]
      hypot-def [=>] 57.1	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \color{blue}{\mathsf{hypot}\left(B, C\right)}\right)}\right) \]

3. Recombined 6 regimes into one program.

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4 \cdot 10^{+28}:\\ \;\;\;\;\frac{B}{\frac{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}{\sqrt{F \cdot 2}}} \cdot \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)\right) \cdot \left(C \cdot 2\right)}}{B \cdot B - 4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	42.9%
Cost	40916

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \sqrt{A + \left(C + t_2\right)}\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5 \cdot 10^{+33}:\\ \;\;\;\;\frac{B \cdot \sqrt{F \cdot 2}}{\frac{t_4}{t_3}}\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq -4.8 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-t_3\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-117}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\sqrt{t_2 + \left(C + A\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(-\sqrt{F}\right)\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Accuracy	43.1%
Cost	27980

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_3 := \sqrt{F \cdot 2}\\ t_4 := \frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{t_2}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{B \cdot t_3}{t_4}\\ \mathbf{elif}\;B \leq -1.06 \cdot 10^{-8}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-t_2\right)}{t_0}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_3 \cdot \left(-B\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Accuracy	41.3%
Cost	27468

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \sqrt{F \cdot 2}\\ t_3 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{+31}:\\ \;\;\;\;\left(B \cdot t_2\right) \cdot \frac{t_3}{t_1}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_2 \cdot \left(-B\right)}{\frac{t_1}{t_3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Accuracy	41.1%
Cost	27468

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{F \cdot 2}\\ t_2 := \frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}\\ \mathbf{if}\;B \leq -4 \cdot 10^{+28}:\\ \;\;\;\;\frac{B \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-115}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1 \cdot \left(-B\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Accuracy	40.9%
Cost	27140

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{F \cdot 2}\\ t_2 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{+29}:\\ \;\;\;\;\left(B \cdot t_1\right) \cdot \frac{t_2}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1 \cdot \left(t_2 \cdot \left(-B\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Accuracy	40.2%
Cost	21196

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{F \cdot 2}\\ t_2 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -3.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{B \cdot \left(t_1 \cdot t_2\right)}{t_0}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{t_1 \cdot \left(t_2 \cdot \left(-B\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Accuracy	36.4%
Cost	21064

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 1.25 \cdot 10^{-109}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2} \cdot \left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-B\right)\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8
Accuracy	36.8%
Cost	20168

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq 6.8 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 40000000000:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Accuracy	35.8%
Cost	15108

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq 8.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10
Accuracy	36.0%
Cost	14920

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq 6.4 \cdot 10^{-111}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 11
Accuracy	35.7%
Cost	13316

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq 1.42 \cdot 10^{+19}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 12
Accuracy	27.3%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B - 4 \cdot \left(C \cdot A\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq 1.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot 2\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(B + \left(C + A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 13
Accuracy	26.2%
Cost	8192

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

Alternative 14
Accuracy	6.6%
Cost	8064

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

Alternative 15
Accuracy	6.7%
Cost	7808

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

Alternative 16
Accuracy	3.1%
Cost	7680

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

Alternative 17
Accuracy	2.3%
Cost	7616

\[\frac{\sqrt{\left(B \cdot -8\right) \cdot \left(A \cdot \left(C \cdot F\right)\right)}}{B \cdot B + 4 \cdot \left(C \cdot A\right)} \]

Alternative 18
Accuracy	2.4%
Cost	7616

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

Alternative 19
Accuracy	2.3%
Cost	7104

\[\left(0.25 \cdot \frac{B \cdot 2}{A}\right) \cdot \sqrt{\frac{F}{C}} \]

Alternative 20
Accuracy	1.1%
Cost	6912

\[-2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B} \]

Alternative 21
Accuracy	0.8%
Cost	6848

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

Alternative 22
Accuracy	0.8%
Cost	6848

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

Reproduce

herbie shell --seed 2023159 
(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))))