ABCF->ab-angle a

Average Accuracy: 18.6% → 46.7%

Time: 1.0min

Precision: binary64

Cost: 34384

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := 2 \cdot t_2\\ t_4 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_5 := C + \left(A + t_0\right)\\ t_6 := \sqrt{2 \cdot t_5}\\ t_7 := \sqrt{A + \left(C + t_0\right)}\\ t_8 := \frac{t_7 \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(B \cdot B + t_1\right)\right)}\right)}{t_4}\\ t_9 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{+64}:\\ \;\;\;\;t_6 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -25000000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_9 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)\right)}}{t_9}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{t_7 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{t_3}\right)\right)}{t_4}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-249}:\\ \;\;\;\;\frac{-\sqrt{t_5}}{\frac{t_2}{\sqrt{F \cdot t_3}}}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-263}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{t_4}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_9 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_9}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;t_8\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

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 (hypot B (- A C)))
        (t_1 (* A (* C -4.0)))
        (t_2 (fma B B t_1))
        (t_3 (* 2.0 t_2))
        (t_4 (+ (* B B) (* -4.0 (* C A))))
        (t_5 (+ C (+ A t_0)))
        (t_6 (sqrt (* 2.0 t_5)))
        (t_7 (sqrt (+ A (+ C t_0))))
        (t_8 (/ (* t_7 (- (sqrt (* F (* 2.0 (+ (* B B) t_1)))))) t_4))
        (t_9 (fma B B (* C (* A -4.0)))))
   (if (<= B -3.7e+64)
     (* t_6 (* (sqrt F) (/ 1.0 B)))
     (if (<= B -25000000000000.0)
       (/
        (- (sqrt (* 2.0 (* t_9 (* F (fma 2.0 C (/ (* -0.5 (* B B)) A)))))))
        t_9)
       (if (<= B -3.6e-34)
         (/ (* t_7 (* (sqrt F) (- (sqrt t_3)))) t_4)
         (if (<= B -1.25e-249)
           (/ (- (sqrt t_5)) (/ t_2 (sqrt (* F t_3))))
           (if (<= B 1.8e-263)
             (/ (- (pow (* (* A (* F (* C A))) -16.0) 0.5)) t_4)
             (if (<= B 1.55e-52)
               t_8
               (if (<= B 8.8e-11)
                 (/
                  (-
                   (sqrt
                    (* 2.0 (* t_9 (* F (fma 2.0 A (* -0.5 (/ (* B B) C))))))))
                  t_9)
                 (if (<= B 4.6e+66)
                   t_8
                   (* t_6 (* (sqrt F) (/ -1.0 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 = hypot(B, (A - C));
	double t_1 = A * (C * -4.0);
	double t_2 = fma(B, B, t_1);
	double t_3 = 2.0 * t_2;
	double t_4 = (B * B) + (-4.0 * (C * A));
	double t_5 = C + (A + t_0);
	double t_6 = sqrt((2.0 * t_5));
	double t_7 = sqrt((A + (C + t_0)));
	double t_8 = (t_7 * -sqrt((F * (2.0 * ((B * B) + t_1))))) / t_4;
	double t_9 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (B <= -3.7e+64) {
		tmp = t_6 * (sqrt(F) * (1.0 / B));
	} else if (B <= -25000000000000.0) {
		tmp = -sqrt((2.0 * (t_9 * (F * fma(2.0, C, ((-0.5 * (B * B)) / A)))))) / t_9;
	} else if (B <= -3.6e-34) {
		tmp = (t_7 * (sqrt(F) * -sqrt(t_3))) / t_4;
	} else if (B <= -1.25e-249) {
		tmp = -sqrt(t_5) / (t_2 / sqrt((F * t_3)));
	} else if (B <= 1.8e-263) {
		tmp = -pow(((A * (F * (C * A))) * -16.0), 0.5) / t_4;
	} else if (B <= 1.55e-52) {
		tmp = t_8;
	} else if (B <= 8.8e-11) {
		tmp = -sqrt((2.0 * (t_9 * (F * fma(2.0, A, (-0.5 * ((B * B) / C))))))) / t_9;
	} else if (B <= 4.6e+66) {
		tmp = t_8;
	} else {
		tmp = t_6 * (sqrt(F) * (-1.0 / 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 = hypot(B, Float64(A - C))
	t_1 = Float64(A * Float64(C * -4.0))
	t_2 = fma(B, B, t_1)
	t_3 = Float64(2.0 * t_2)
	t_4 = Float64(Float64(B * B) + Float64(-4.0 * Float64(C * A)))
	t_5 = Float64(C + Float64(A + t_0))
	t_6 = sqrt(Float64(2.0 * t_5))
	t_7 = sqrt(Float64(A + Float64(C + t_0)))
	t_8 = Float64(Float64(t_7 * Float64(-sqrt(Float64(F * Float64(2.0 * Float64(Float64(B * B) + t_1)))))) / t_4)
	t_9 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B <= -3.7e+64)
		tmp = Float64(t_6 * Float64(sqrt(F) * Float64(1.0 / B)));
	elseif (B <= -25000000000000.0)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_9 * Float64(F * fma(2.0, C, Float64(Float64(-0.5 * Float64(B * B)) / A))))))) / t_9);
	elseif (B <= -3.6e-34)
		tmp = Float64(Float64(t_7 * Float64(sqrt(F) * Float64(-sqrt(t_3)))) / t_4);
	elseif (B <= -1.25e-249)
		tmp = Float64(Float64(-sqrt(t_5)) / Float64(t_2 / sqrt(Float64(F * t_3))));
	elseif (B <= 1.8e-263)
		tmp = Float64(Float64(-(Float64(Float64(A * Float64(F * Float64(C * A))) * -16.0) ^ 0.5)) / t_4);
	elseif (B <= 1.55e-52)
		tmp = t_8;
	elseif (B <= 8.8e-11)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_9 * Float64(F * fma(2.0, A, Float64(-0.5 * Float64(Float64(B * B) / C)))))))) / t_9);
	elseif (B <= 4.6e+66)
		tmp = t_8;
	else
		tmp = Float64(t_6 * Float64(sqrt(F) * Float64(-1.0 / 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[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(2.0 * t$95$5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(N[(t$95$7 * (-N[Sqrt[N[(F * N[(2.0 * N[(N[(B * B), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$9 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.7e+64], N[(t$95$6 * N[(N[Sqrt[F], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -25000000000000.0], N[((-N[Sqrt[N[(2.0 * N[(t$95$9 * N[(F * N[(2.0 * C + N[(N[(-0.5 * N[(B * B), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$9), $MachinePrecision], If[LessEqual[B, -3.6e-34], N[(N[(t$95$7 * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[t$95$3], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, -1.25e-249], N[((-N[Sqrt[t$95$5], $MachinePrecision]) / N[(t$95$2 / N[Sqrt[N[(F * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.8e-263], N[((-N[Power[N[(N[(A * N[(F * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -16.0), $MachinePrecision], 0.5], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[B, 1.55e-52], t$95$8, If[LessEqual[B, 8.8e-11], N[((-N[Sqrt[N[(2.0 * N[(t$95$9 * N[(F * N[(2.0 * A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$9), $MachinePrecision], If[LessEqual[B, 4.6e+66], t$95$8, N[(t$95$6 * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if B < -3.69999999999999983e64

   1. Initial program 8.7%

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

   2. Simplified 10.1%

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

      [Start] 8.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 0.6%

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

      [Start] 10.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      add-sqr-sqrt [=>] 0.4	\[ \color{blue}{\sqrt{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \cdot \sqrt{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      sqrt-unprod [=>] 0.6	\[ \color{blue}{\sqrt{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      frac-times [=>] 0.6	\[ \sqrt{\color{blue}{\frac{\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right) \cdot \left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]

   4. Simplified 0.8%

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

      [Start] 0.6	\[ \sqrt{\frac{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right)}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-*r* [=>] 0.6	\[ \sqrt{\frac{\color{blue}{\left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 0.8	\[ \sqrt{\color{blue}{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}}}} \]
      *-commutative [=>] 0.8	\[ \sqrt{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{\color{blue}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}} \]

   5. Applied egg-rr 1.1%

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

      [Start] 0.8	\[ \sqrt{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      div-inv [=>] 0.8	\[ \sqrt{\color{blue}{\left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}} \]
      sqrt-prod [=>] 0.7	\[ \color{blue}{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}} \]
      associate-+r+ [=>] 0.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{\frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      +-commutative [=>] 0.8	\[ \sqrt{2 \cdot \color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}} \cdot \sqrt{\frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      clear-num [<=] 0.8	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\color{blue}{\frac{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}}} \]
      *-commutative [=>] 0.8	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\color{blue}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 1.1	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F}}}} \]
      associate-*r* [=>] 1.1	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right) \cdot -4}\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F}}} \]
      *-commutative [=>] 1.1	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(C \cdot A\right)}\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F}}} \]

   6. Simplified 4.2%

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

      [Start] 1.1	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      +-commutative [<=] 1.1	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \color{blue}{\left(A + C\right)}\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      +-commutative [=>] 1.1	\[ \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      +-commutative [=>] 1.1	\[ \sqrt{2 \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      associate-+l+ [=>] 1.0	\[ \sqrt{2 \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      associate-/r/ [=>] 1.0	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}} \cdot F}} \]
      unpow2 [=>] 1.0	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}} \cdot F} \]
      associate-/r* [=>] 1.2	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\color{blue}{\frac{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}} \cdot F} \]
      *-inverses [=>] 4.2	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{\color{blue}{1}}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)} \cdot F} \]
      *-commutative [<=] 4.2	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)} \cdot F} \]

   7. Taylor expanded in B around inf 68.4%

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

   if -3.69999999999999983e64 < B < -2.5e13

   1. Initial program 36.8%

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

   2. Simplified 43.1%

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

      [Start] 36.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 13.2%

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

   4. Simplified 13.2%

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

      [Start] 13.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 13.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r/ [=>] 13.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{-0.5 \cdot {B}^{2}}{A}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 13.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if -2.5e13 < B < -3.60000000000000008e-34

   1. Initial program 36.1%

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

   2. Simplified 36.1%

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

      [Start] 36.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 50.4%

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

      [Start] 36.1	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 36.1	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 37.3	\[ \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-neg [=>] 37.3	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-lft-neg-in [=>] 37.3	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 37.3	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      metadata-eval [=>] 37.3	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot \color{blue}{-4}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 37.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 37.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 50.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 50.4%

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

      [Start] 50.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 50.4	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 50.4	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right) \cdot F}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 50.4	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 46.8%

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

      [Start] 50.4	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 46.8	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -3.60000000000000008e-34 < B < -1.24999999999999997e-249

   1. Initial program 20.9%

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

   2. Simplified 20.9%

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

      [Start] 20.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 36.4%

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

      [Start] 20.9	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 20.9	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 23.8	\[ \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-neg [=>] 23.8	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-lft-neg-in [=>] 23.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 23.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      metadata-eval [=>] 23.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot \color{blue}{-4}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 24.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 24.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 36.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 36.4%

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

      [Start] 36.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 36.4	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 36.4	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right) \cdot F}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 36.4	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 36.0%

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

      [Start] 36.4	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      div-inv [=>] 36.0	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      distribute-rgt-neg-in [=>] 36.0	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F}\right)\right)} \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 36.0	\[ \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right)} \]
      *-commutative [=>] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\color{blue}{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot 2\right)} \cdot F}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      associate-*l* [=>] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\color{blue}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      fma-neg [=>] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      distribute-lft-neg-in [=>] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right)}\right) \]
      metadata-eval [=>] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right) \]
      *-commutative [<=] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot -4}\right)}\right) \]
      associate-*r* [<=] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)}\right) \]

   6. Simplified 36.3%

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

      [Start] 36.0	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 36.0	\[ \sqrt{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+l+ [=>] 36.0	\[ \sqrt{\color{blue}{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}} \cdot \left(\left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      *-commutative [=>] 36.0	\[ \sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)\right)} \]
      associate-*l/ [=>] 36.4	\[ \sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \color{blue}{\frac{1 \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

   7. Applied egg-rr 35.3%

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

      [Start] 36.3	\[ \sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \frac{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      distribute-frac-neg [=>] 36.3	\[ \sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      distribute-rgt-neg-out [=>] 36.3	\[ \color{blue}{-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      add-sqr-sqrt [=>] 36.3	\[ -\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \frac{\sqrt{\color{blue}{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqr-neg [<=] 36.3	\[ -\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \frac{\sqrt{\color{blue}{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      sqrt-unprod [<=] 0.3	\[ -\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \frac{\color{blue}{\sqrt{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}} \cdot \sqrt{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Simplified 36.4%

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

      [Start] 35.3	\[ -\frac{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}} \]
      distribute-neg-frac [=>] 35.3	\[ \color{blue}{\frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}}} \]
      associate-+r+ [=>] 36.3	\[ \frac{-\sqrt{\color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}} \]
      +-commutative [=>] 36.3	\[ \frac{-\sqrt{\color{blue}{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}} \]
      +-commutative [=>] 36.3	\[ \frac{-\sqrt{C + \color{blue}{\left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}} \]
      *-commutative [=>] 36.3	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot -4}\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}} \]
      associate-*l* [=>] 36.4	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)}{\sqrt{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)}}} \]
      *-commutative [=>] 36.4	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}}} \]
      *-commutative [=>] 36.4	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\color{blue}{\left(F \cdot 2\right)} \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}} \]
      associate-*l* [=>] 36.4	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\color{blue}{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}}}} \]
      *-commutative [=>] 36.4	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot -4}\right)\right)}}} \]
      associate-*l* [=>] 36.4	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right)}}} \]

   if -1.24999999999999997e-249 < B < 1.8e-263

   1. Initial program 16.8%

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

   2. Simplified 16.8%

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

      [Start] 16.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 15.6%

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

   4. Simplified 17.6%

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

      [Start] 15.6	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 17.6	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \color{blue}{\left(\left({A}^{2} \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 17.6	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(\color{blue}{\left(A \cdot A\right)} \cdot C\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 22.1%

      \[\leadsto \frac{-\color{blue}{{\left(\left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right) \cdot -16\right)}^{0.5}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 17.6	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      pow1/2 [=>] 17.6	\[ \frac{-\color{blue}{{\left(2 \cdot \left(-8 \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)}^{0.5}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 17.6	\[ \frac{-{\color{blue}{\left(\left(2 \cdot -8\right) \cdot \left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right)\right)}}^{0.5}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 17.6	\[ \frac{-{\color{blue}{\left(\left(\left(\left(A \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot -8\right)\right)}}^{0.5}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 19.7	\[ \frac{-{\left(\left(\color{blue}{\left(A \cdot \left(A \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot -8\right)\right)}^{0.5}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 22.1	\[ \frac{-{\left(\color{blue}{\left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)} \cdot \left(2 \cdot -8\right)\right)}^{0.5}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      metadata-eval [=>] 22.1	\[ \frac{-{\left(\left(A \cdot \left(\left(A \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{-16}\right)}^{0.5}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.8e-263 < B < 1.5499999999999999e-52 or 8.8000000000000006e-11 < B < 4.6e66

   1. Initial program 25.2%

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

   2. Simplified 25.2%

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

      [Start] 25.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 40.7%

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

      [Start] 25.2	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 25.2	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 27.0	\[ \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-neg [=>] 27.0	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-lft-neg-in [=>] 27.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 27.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right) \cdot \left(-4\right)}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      metadata-eval [=>] 27.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot \color{blue}{-4}\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 27.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 27.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 40.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 40.7%

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

      [Start] 40.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 40.7	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 40.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right) \cdot F}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 40.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 40.7%

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

      [Start] 40.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-udef [=>] 40.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)}\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 1.5499999999999999e-52 < B < 8.8000000000000006e-11

   1. Initial program 35.3%

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

   2. Simplified 45.0%

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

      [Start] 35.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around -inf 16.2%

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

   4. Simplified 16.2%

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

      [Start] 16.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 16.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 16.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 4.6e66 < B

   1. Initial program 8.6%

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

   2. Simplified 10.0%

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

      [Start] 8.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 0.6%

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

      [Start] 10.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      add-sqr-sqrt [=>] 0.4	\[ \color{blue}{\sqrt{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \cdot \sqrt{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      sqrt-unprod [=>] 0.6	\[ \color{blue}{\sqrt{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      frac-times [=>] 0.7	\[ \sqrt{\color{blue}{\frac{\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right) \cdot \left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]

   4. Simplified 0.8%

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

      [Start] 0.6	\[ \sqrt{\frac{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right)}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-*r* [=>] 0.6	\[ \sqrt{\frac{\color{blue}{\left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 0.8	\[ \sqrt{\color{blue}{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}}}} \]
      *-commutative [=>] 0.8	\[ \sqrt{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{\color{blue}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}} \]

   5. Applied egg-rr 1.2%

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

      [Start] 0.8	\[ \sqrt{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      div-inv [=>] 0.8	\[ \sqrt{\color{blue}{\left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}} \]
      sqrt-prod [=>] 0.8	\[ \color{blue}{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}} \]
      associate-+r+ [=>] 0.8	\[ \sqrt{2 \cdot \color{blue}{\left(\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{\frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      +-commutative [=>] 0.8	\[ \sqrt{2 \cdot \color{blue}{\left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)}} \cdot \sqrt{\frac{1}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}} \]
      clear-num [<=] 0.8	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\color{blue}{\frac{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}}} \]
      *-commutative [=>] 0.8	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\color{blue}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 1.2	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F}}}} \]
      associate-*r* [=>] 1.2	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right) \cdot -4}\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F}}} \]
      *-commutative [=>] 1.2	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(C \cdot A\right)}\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{F}}} \]

   6. Simplified 4.2%

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

      [Start] 1.2	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      +-commutative [<=] 1.2	\[ \sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \color{blue}{\left(A + C\right)}\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      +-commutative [=>] 1.2	\[ \sqrt{2 \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      +-commutative [=>] 1.2	\[ \sqrt{2 \cdot \left(\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      associate-+l+ [=>] 1.1	\[ \sqrt{2 \cdot \color{blue}{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}}{F}}} \]
      associate-/r/ [=>] 1.1	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\color{blue}{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{2}} \cdot F}} \]
      unpow2 [=>] 1.1	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right) \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}} \cdot F} \]
      associate-/r* [=>] 1.2	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\color{blue}{\frac{\frac{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}} \cdot F} \]
      *-inverses [=>] 4.2	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{\color{blue}{1}}{\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)} \cdot F} \]
      *-commutative [<=] 4.2	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)} \cdot F} \]

   7. Taylor expanded in B around -inf 66.0%

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

   8. Simplified 66.0%

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

      [Start] 66.0	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-1 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\right) \]
      mul-1-neg [=>] 66.0	\[ \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \color{blue}{\left(-\sqrt{F} \cdot \frac{1}{B}\right)} \]

3. Recombined 8 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.7 \cdot 10^{+64}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -25000000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-249}:\\ \;\;\;\;\frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-263}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	48.3%
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := C + \left(A + t_0\right)\\ t_3 := \mathsf{fma}\left(B, B, t_1\right)\\ t_4 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_5 := \frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(B \cdot B + t_1\right)\right)}\right)}{t_4}\\ t_6 := \sqrt{2 \cdot t_2}\\ t_7 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.36 \cdot 10^{+92}:\\ \;\;\;\;t_6 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{t_2} \cdot \frac{-\sqrt{2 \cdot \left(F \cdot t_3\right)}}{t_3}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-264}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{t_4}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-53}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_7 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_7}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+65}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 2
Accuracy	48.3%
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_2 := C + \left(A + t_0\right)\\ t_3 := A \cdot \left(C \cdot -4\right)\\ t_4 := \frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(B \cdot B + t_3\right)\right)}\right)}{t_1}\\ t_5 := \mathsf{fma}\left(B, B, t_3\right)\\ t_6 := \sqrt{2 \cdot t_2}\\ t_7 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -7.9 \cdot 10^{+93}:\\ \;\;\;\;t_6 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-252}:\\ \;\;\;\;\frac{-\sqrt{t_2}}{\frac{t_5}{\sqrt{F \cdot \left(2 \cdot t_5\right)}}}\\ \mathbf{elif}\;B \leq 7.7 \cdot 10^{-264}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{t_1}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-55}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_7 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_7}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+65}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 3
Accuracy	49.2%
Cost	27916

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)}\\ t_3 := \frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{if}\;B \leq -4.2 \cdot 10^{+64}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{+67}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 4
Accuracy	50.3%
Cost	21576

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)}\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{+72}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 1.32 \cdot 10^{+67}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)\right)}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 5
Accuracy	42.1%
Cost	21264

\[\begin{array}{l} t_0 := \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{-90}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 6
Accuracy	42.1%
Cost	21264

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5.8 \cdot 10^{-91}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(B \cdot \left(-\sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(2 \cdot A\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 7
Accuracy	45.8%
Cost	21192

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)}\\ \mathbf{if}\;B \leq -5.7 \cdot 10^{-34}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-\sqrt{\left(F \cdot \left(C \cdot A\right)\right) \cdot -8}\right)}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 8
Accuracy	31.5%
Cost	20753

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{-{t_1}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{-73} \lor \neg \left(B \leq 2.25 \cdot 10^{+15}\right):\\ \;\;\;\;\sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-{\left({t_1}^{0.25}\right)}^{2}}{t_0}\\ \end{array} \]

Alternative 9
Accuracy	41.7%
Cost	20752

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_2 := \left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\\ \mathbf{if}\;B \leq -2.05 \cdot 10^{-97}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-144}:\\ \;\;\;\;\frac{-{t_2}^{0.5}}{t_0}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{elif}\;B \leq 10^{+17}:\\ \;\;\;\;\frac{-{\left({t_2}^{0.25}\right)}^{2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{F} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 10
Accuracy	26.0%
Cost	20488

\[\begin{array}{l} \mathbf{if}\;A \leq -1.02 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq 3.6 \cdot 10^{-32}:\\ \;\;\;\;\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\sqrt{F} \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;A \leq 3.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 11
Accuracy	23.3%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;A \leq -2.55 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq 2.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{B}\\ \mathbf{elif}\;A \leq 1.58 \cdot 10^{+171}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 12
Accuracy	21.2%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;A \leq -2.55 \cdot 10^{+109}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq 5.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \mathbf{elif}\;A \leq 2.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 13
Accuracy	19.1%
Cost	13188

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_2 := F \cdot t_1\\ \mathbf{if}\;A \leq -3.6 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq -4.8 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.95 \cdot 10^{-229}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(B + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(C + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 5.6 \cdot 10^{+169}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Accuracy	17.7%
Cost	8848

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ t_2 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -1 \cdot 10^{+53}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{-74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-223}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.9 \cdot 10^{+172}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Accuracy	17.4%
Cost	8716

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -5.8 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.35 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.15 \cdot 10^{+170}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	17.3%
Cost	8272

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 2.2 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 3 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 9.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot -16\right)}^{0.5}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Accuracy	17.0%
Cost	8208

\[\begin{array}{l} t_0 := \sqrt{-\frac{F}{A}}\\ t_1 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -1.26 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 2.15 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 3.4 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 2.8 \cdot 10^{+175}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(-16 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Accuracy	15.2%
Cost	7053

\[\begin{array}{l} \mathbf{if}\;A \leq -1.15 \cdot 10^{+55} \lor \neg \left(A \leq 2.45 \cdot 10^{-238}\right) \land A \leq 2.6 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{-\frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \end{array} \]

Alternative 19
Accuracy	11.5%
Cost	6656

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

Alternative 20
Accuracy	1.2%
Cost	6592

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

Reproduce

herbie shell --seed 2023133 
(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))))