ABCF->ab-angle a

Percentage Accurate: 19.6% → 42.2%

Time: 48.2s

Precision: binary64

Cost: 20816

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

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

Math

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

↓

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := F \cdot t_0\\ t_2 := 2 \cdot t_1\\ t_3 := \sqrt{t_2}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{2} \cdot \left(-0.5 \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (* B B) (* (* C A) 4.0)))
        (t_1 (* F t_0))
        (t_2 (* 2.0 t_1))
        (t_3 (sqrt t_2)))
   (if (<= B -1.32e+154)
     (* 2.0 (* (sqrt (* F C)) (/ 1.0 B)))
     (if (<= B -5e+67)
       (/ (* t_3 (- (sqrt (- C B)))) t_0)
       (if (<= B -1.05e-245)
         (/ (- (sqrt (* t_2 (fma 2.0 C (* -0.5 (/ (* B B) A)))))) t_0)
         (if (<= B -1.25e-279)
           (*
            (sqrt 2.0)
            (* -0.5 (sqrt (/ F (/ A (fma 0.5 (/ (* B B) (* A A)) -2.0))))))
           (if (<= B 4.6e-68)
             (/ (* t_3 (- (sqrt (+ C C)))) t_0)
             (if (<= B 7e+106)
               (*
                (sqrt (* 2.0 (* t_1 (+ C (+ A (hypot B (- A C)))))))
                (/ -1.0 t_0))
               (* (/ (sqrt 2.0) B) (* (sqrt F) (- (sqrt 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) - ((C * A) * 4.0);
	double t_1 = F * t_0;
	double t_2 = 2.0 * t_1;
	double t_3 = sqrt(t_2);
	double tmp;
	if (B <= -1.32e+154) {
		tmp = 2.0 * (sqrt((F * C)) * (1.0 / B));
	} else if (B <= -5e+67) {
		tmp = (t_3 * -sqrt((C - B))) / t_0;
	} else if (B <= -1.05e-245) {
		tmp = -sqrt((t_2 * fma(2.0, C, (-0.5 * ((B * B) / A))))) / t_0;
	} else if (B <= -1.25e-279) {
		tmp = sqrt(2.0) * (-0.5 * sqrt((F / (A / fma(0.5, ((B * B) / (A * A)), -2.0)))));
	} else if (B <= 4.6e-68) {
		tmp = (t_3 * -sqrt((C + C))) / t_0;
	} else if (B <= 7e+106) {
		tmp = sqrt((2.0 * (t_1 * (C + (A + hypot(B, (A - C))))))) * (-1.0 / t_0);
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt(F) * -sqrt(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(Float64(C * A) * 4.0))
	t_1 = Float64(F * t_0)
	t_2 = Float64(2.0 * t_1)
	t_3 = sqrt(t_2)
	tmp = 0.0
	if (B <= -1.32e+154)
		tmp = Float64(2.0 * Float64(sqrt(Float64(F * C)) * Float64(1.0 / B)));
	elseif (B <= -5e+67)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(C - B)))) / t_0);
	elseif (B <= -1.05e-245)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * fma(2.0, C, Float64(-0.5 * Float64(Float64(B * B) / A)))))) / t_0);
	elseif (B <= -1.25e-279)
		tmp = Float64(sqrt(2.0) * Float64(-0.5 * sqrt(Float64(F / Float64(A / fma(0.5, Float64(Float64(B * B) / Float64(A * A)), -2.0))))));
	elseif (B <= 4.6e-68)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(Float64(C + C)))) / t_0);
	elseif (B <= 7e+106)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t_1 * Float64(C + Float64(A + hypot(B, Float64(A - C))))))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(F) * Float64(-sqrt(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[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[B, -1.32e+154], N[(2.0 * N[(N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -5e+67], N[(N[(t$95$3 * (-N[Sqrt[N[(C - B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, -1.05e-245], N[((-N[Sqrt[N[(t$95$2 * 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, -1.25e-279], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-0.5 * N[Sqrt[N[(F / N[(A / N[(0.5 * N[(N[(B * B), $MachinePrecision] / N[(A * A), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.6e-68], N[(N[(t$95$3 * (-N[Sqrt[N[(C + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B, 7e+106], N[(N[Sqrt[N[(2.0 * N[(t$95$1 * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -1.31999999999999998e154

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 0.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 0.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 0.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 0.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 0.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 0.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 0.0%

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

   4. Taylor expanded in B around -inf 2.8%

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

   5. Simplified 2.8%

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

      [Start] 2.8%	\[ 2 \cdot \left(\sqrt{C \cdot F} \cdot \frac{1}{B}\right) \]
      *-commutative [=>] 2.8%	\[ 2 \cdot \left(\sqrt{\color{blue}{F \cdot C}} \cdot \frac{1}{B}\right) \]

   if -1.31999999999999998e154 < B < -4.99999999999999976e67

   1. Initial program 44.9%

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

   2. Simplified 44.9%

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

      [Start] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 74.9%

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

      [Start] 44.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 69.0%	\[ \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 69.0%	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 69.0%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 69.0%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 69.0%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-udef [<=] 75.2%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+r+ [=>] 75.2%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      +-commutative [=>] 75.2%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+r+ [<=] 74.9%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Taylor expanded in B around -inf 70.4%

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

   5. Simplified 70.4%

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

      [Start] 70.4%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + -1 \cdot B}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      mul-1-neg [=>] 70.4%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{C + \color{blue}{\left(-B\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -4.99999999999999976e67 < B < -1.05000000000000005e-245

   1. Initial program 21.5%

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

   2. Simplified 21.5%

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

      [Start] 21.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 21.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 21.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 21.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 21.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 21.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 21.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 22.0%

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

   4. Simplified 22.0%

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

      [Start] 22.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-def [=>] 22.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 22.0%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -1.05000000000000005e-245 < B < -1.24999999999999992e-279

   1. Initial program 1.6%

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

   2. Simplified 1.6%

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

      [Start] 1.6%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 1.6%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 1.6%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 1.6%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 1.6%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 1.6%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 1.6%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 1.1%

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

   4. Simplified 0.9%

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

      [Start] 1.1%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + \left(-0.5 \cdot \frac{{B}^{2}}{A} + -1 \cdot A\right)\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+r+ [=>] 1.1%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \color{blue}{\left(\left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + -0.5 \cdot \frac{{B}^{2}}{A}\right) + -1 \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      mul-1-neg [=>] 1.1%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(\left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + -0.5 \cdot \frac{{B}^{2}}{A}\right) + \color{blue}{\left(-A\right)}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unsub-neg [=>] 1.1%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \color{blue}{\left(\left(-0.5 \cdot \frac{C \cdot {B}^{2}}{{A}^{2}} + -0.5 \cdot \frac{{B}^{2}}{A}\right) - A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-lft-out [=>] 1.1%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(\color{blue}{-0.5 \cdot \left(\frac{C \cdot {B}^{2}}{{A}^{2}} + \frac{{B}^{2}}{A}\right)} - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 1.1%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(-0.5 \cdot \left(\frac{\color{blue}{{B}^{2} \cdot C}}{{A}^{2}} + \frac{{B}^{2}}{A}\right) - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 1.1%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(-0.5 \cdot \left(\frac{{B}^{2} \cdot C}{\color{blue}{A \cdot A}} + \frac{{B}^{2}}{A}\right) - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      times-frac [=>] 0.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(-0.5 \cdot \left(\color{blue}{\frac{{B}^{2}}{A} \cdot \frac{C}{A}} + \frac{{B}^{2}}{A}\right) - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 0.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(-0.5 \cdot \left(\frac{\color{blue}{B \cdot B}}{A} \cdot \frac{C}{A} + \frac{{B}^{2}}{A}\right) - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 0.9%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(C + \left(-0.5 \cdot \left(\frac{B \cdot B}{A} \cdot \frac{C}{A} + \frac{\color{blue}{B \cdot B}}{A}\right) - A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in C around -inf 29.2%

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

   6. Simplified 29.2%

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

      [Start] 29.2%	\[ -0.5 \cdot \left(\sqrt{\frac{F \cdot \left(0.5 \cdot \frac{{B}^{2}}{{A}^{2}} - 2\right)}{A}} \cdot \sqrt{2}\right) \]
      associate-*r* [=>] 29.2%	\[ \color{blue}{\left(-0.5 \cdot \sqrt{\frac{F \cdot \left(0.5 \cdot \frac{{B}^{2}}{{A}^{2}} - 2\right)}{A}}\right) \cdot \sqrt{2}} \]
      associate-/l* [=>] 29.2%	\[ \left(-0.5 \cdot \sqrt{\color{blue}{\frac{F}{\frac{A}{0.5 \cdot \frac{{B}^{2}}{{A}^{2}} - 2}}}}\right) \cdot \sqrt{2} \]
      fma-neg [=>] 29.2%	\[ \left(-0.5 \cdot \sqrt{\frac{F}{\frac{A}{\color{blue}{\mathsf{fma}\left(0.5, \frac{{B}^{2}}{{A}^{2}}, -2\right)}}}}\right) \cdot \sqrt{2} \]
      unpow2 [=>] 29.2%	\[ \left(-0.5 \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{\color{blue}{B \cdot B}}{{A}^{2}}, -2\right)}}}\right) \cdot \sqrt{2} \]
      unpow2 [=>] 29.2%	\[ \left(-0.5 \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{\color{blue}{A \cdot A}}, -2\right)}}}\right) \cdot \sqrt{2} \]
      metadata-eval [=>] 29.2%	\[ \left(-0.5 \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, \color{blue}{-2}\right)}}}\right) \cdot \sqrt{2} \]

   if -1.24999999999999992e-279 < B < 4.59999999999999994e-68

   1. Initial program 20.5%

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

   2. Simplified 20.5%

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

      [Start] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 48.1%

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

      [Start] 20.5%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 25.0%	\[ \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 25.0%	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 25.0%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 25.2%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 25.2%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-udef [<=] 46.5%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+r+ [=>] 46.3%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      +-commutative [=>] 46.3%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+r+ [<=] 48.1%	\[ \frac{-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \sqrt{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Taylor expanded in A around -inf 34.0%

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

   if 4.59999999999999994e-68 < B < 6.99999999999999962e106

   1. Initial program 47.7%

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

   2. Simplified 47.7%

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

      [Start] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 64.7%

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

      [Start] 47.7%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      div-inv [=>] 47.5%	\[ \color{blue}{\left(-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]

   if 6.99999999999999962e106 < B

   1. Initial program 2.8%

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

   2. Simplified 2.8%

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

      [Start] 2.8%	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 13.9%

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

   4. Simplified 56.0%

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

      [Start] 13.9%	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 13.9%	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      *-commutative [=>] 13.9%	\[ -\color{blue}{\sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot \frac{\sqrt{2}}{B}} \]
      distribute-rgt-neg-in [=>] 13.9%	\[ \color{blue}{\sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot \left(-\frac{\sqrt{2}}{B}\right)} \]
      unpow2 [=>] 13.9%	\[ \sqrt{\left(A + \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right) \cdot F} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]
      unpow2 [=>] 13.9%	\[ \sqrt{\left(A + \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right) \cdot F} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]
      hypot-def [=>] 56.0%	\[ \sqrt{\left(A + \color{blue}{\mathsf{hypot}\left(B, A\right)}\right) \cdot F} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]

   5. Applied egg-rr 87.9%

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

      [Start] 56.0%	\[ \sqrt{\left(A + \mathsf{hypot}\left(B, A\right)\right) \cdot F} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]
      sqrt-prod [=>] 87.9%	\[ \color{blue}{\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right) \]

   6. Taylor expanded in A around 0 85.9%

      \[\leadsto \left(\color{blue}{\sqrt{B}} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - \left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(-\sqrt{C - B}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(C \cdot A\right) \cdot 4\right)\right)\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{2} \cdot \left(-0.5 \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - \left(C \cdot A\right) \cdot 4\right)\right)} \cdot \left(-\sqrt{C + C}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B - \left(C \cdot A\right) \cdot 4\right)\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	42.2%
Cost	20816

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := F \cdot t_0\\ t_2 := 2 \cdot t_1\\ t_3 := \sqrt{t_2}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-245}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-279}:\\ \;\;\;\;\sqrt{2} \cdot \left(-0.5 \cdot \sqrt{\frac{F}{\frac{A}{\mathsf{fma}\left(0.5, \frac{B \cdot B}{A \cdot A}, -2\right)}}}\right)\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	43.9%
Cost	34252

\[\begin{array}{l} t_0 := \sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)}\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\\ \mathbf{if}\;B \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F}\right) \cdot \left(t_0 \cdot \frac{1}{t_2}\right)\\ \mathbf{elif}\;B \leq 10^{-253}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.75 \cdot 10^{+71}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot t_2\right)}}{\frac{t_2}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	43.9%
Cost	33668

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq -2 \cdot 10^{+67}:\\ \;\;\;\;\left(\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F}\right) \cdot \left(\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\right)\\ \mathbf{elif}\;B \leq 10^{-253}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_1\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	43.4%
Cost	27268

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -2.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{t_2 \cdot \left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_1\right)} \cdot \left(-t_2\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	43.4%
Cost	26832

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -2.15 \cdot 10^{+67}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-254}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-t_1\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{A + \mathsf{hypot}\left(B, A\right)}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	43.2%
Cost	21840

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.62 \cdot 10^{+72}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 10^{-253}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-t_1\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	43.0%
Cost	21192

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_2} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_2 \cdot t_1\right)} \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	42.6%
Cost	20240

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := F \cdot t_0\\ t_2 := \sqrt{2 \cdot t_1}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq 9.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+106}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	38.8%
Cost	15636

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := F \cdot t_0\\ t_2 := 2 \cdot t_1\\ t_3 := \sqrt{t_2}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq -1.66 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+114}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 10
Accuracy	38.9%
Cost	15504

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := F \cdot t_0\\ t_2 := \sqrt{2 \cdot t_1}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -9.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-64}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)} \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 11
Accuracy	38.7%
Cost	15372

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := \sqrt{t_1}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -8.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-306}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 12
Accuracy	39.8%
Cost	15184

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -4.4 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 13
Accuracy	39.1%
Cost	15184

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := \sqrt{t_1}\\ \mathbf{if}\;B \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C - B}\right)}{t_0}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+113}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 14
Accuracy	36.5%
Cost	14988

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 3.35 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+72}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 15
Accuracy	36.1%
Cost	14988

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{+148}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -5.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+71}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 16
Accuracy	36.5%
Cost	13845

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -1.82 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-58} \lor \neg \left(B \leq 5.5 \cdot 10^{-28}\right) \land B \leq 4.2 \cdot 10^{+71}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 17
Accuracy	36.5%
Cost	13844

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ t_2 := -\frac{\sqrt{t_1 \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+149}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + \left(C - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 6.9 \cdot 10^{-28}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 18
Accuracy	26.7%
Cost	8256

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

Alternative 19
Accuracy	26.7%
Cost	8192

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

Alternative 20
Accuracy	16.8%
Cost	7680

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

Alternative 21
Accuracy	9.0%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;B \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(\sqrt{F \cdot C} \cdot \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot {\left(F \cdot C\right)}^{0.5}}{B}\\ \end{array} \]

Alternative 22
Accuracy	5.2%
Cost	6912

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

Alternative 23
Accuracy	5.1%
Cost	6848

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

Reproduce

herbie shell --seed 2023272 
(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))))