ABCF->ab-angle a

Average Error: 52.2 → 40.7

Time: 59.6s

Precision: binary64

Cost: 47180

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_3 := \sqrt{t_0}\\ t_4 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_5 := \frac{\sqrt{2 \cdot t_4}}{t_0}\\ t_6 := t_5 \cdot \left(\sqrt{F} \cdot \left(-t_3\right)\right)\\ t_7 := -\frac{\sqrt{-2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq -3.3 \cdot 10^{-90}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;t_5 \cdot \left(\sqrt{F} \cdot \left(-{\left({t_0}^{0.25}\right)}^{2}\right)\right)\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-269}:\\ \;\;\;\;t_5 \cdot \left(-\sqrt{F \cdot t_1}\right)\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\left(\sqrt{F} \cdot t_3\right) \cdot \left(0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{C}\right)\right)\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_4 \cdot t_2\right)\right)}}{-t_2}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-76}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{B \cdot B + t_1}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+136}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* A C) (* B B)))
        (t_1 (* -4.0 (* A C)))
        (t_2 (fma B B (* C (* -4.0 A))))
        (t_3 (sqrt t_0))
        (t_4 (+ C (+ A (hypot B (- A C)))))
        (t_5 (/ (sqrt (* 2.0 t_4)) t_0))
        (t_6 (* t_5 (* (sqrt F) (- t_3))))
        (t_7 (- (/ (sqrt (* -2.0 (* t_2 (* F (* C -2.0))))) t_2))))
   (if (<= B -1.9e-36)
     t_6
     (if (<= B -3.3e-90)
       t_7
       (if (<= B -8.2e-110)
         (* t_5 (* (sqrt F) (- (pow (pow t_0 0.25) 2.0))))
         (if (<= B -2.6e-136)
           (/ (- (sqrt (* 2.0 (* t_2 (* F (* A 2.0)))))) t_2)
           (if (<= B 1.7e-269)
             (* t_5 (- (sqrt (* F t_1))))
             (if (<= B 2.5e-228)
               (*
                (* (sqrt F) t_3)
                (* 0.25 (* (sqrt (/ 1.0 A)) (/ (pow (sqrt 2.0) 2.0) C))))
               (if (<= B 3.25e-174)
                 (/ (sqrt (* 2.0 (* F (* t_4 t_2)))) (- t_2))
                 (if (<= B 5.4e-76)
                   t_7
                   (if (<= B 4.4e-73)
                     (/
                      (- (sqrt (* -2.0 (* (* (* A A) (* C F)) 8.0))))
                      (+ (* B B) t_1))
                     (if (<= B 7.5e+136)
                       t_6
                       (*
                        (sqrt (* F (+ A (hypot B A))))
                        (/ (- (sqrt 2.0)) B))))))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (A * C), (B * B));
	double t_1 = -4.0 * (A * C);
	double t_2 = fma(B, B, (C * (-4.0 * A)));
	double t_3 = sqrt(t_0);
	double t_4 = C + (A + hypot(B, (A - C)));
	double t_5 = sqrt((2.0 * t_4)) / t_0;
	double t_6 = t_5 * (sqrt(F) * -t_3);
	double t_7 = -(sqrt((-2.0 * (t_2 * (F * (C * -2.0))))) / t_2);
	double tmp;
	if (B <= -1.9e-36) {
		tmp = t_6;
	} else if (B <= -3.3e-90) {
		tmp = t_7;
	} else if (B <= -8.2e-110) {
		tmp = t_5 * (sqrt(F) * -pow(pow(t_0, 0.25), 2.0));
	} else if (B <= -2.6e-136) {
		tmp = -sqrt((2.0 * (t_2 * (F * (A * 2.0))))) / t_2;
	} else if (B <= 1.7e-269) {
		tmp = t_5 * -sqrt((F * t_1));
	} else if (B <= 2.5e-228) {
		tmp = (sqrt(F) * t_3) * (0.25 * (sqrt((1.0 / A)) * (pow(sqrt(2.0), 2.0) / C)));
	} else if (B <= 3.25e-174) {
		tmp = sqrt((2.0 * (F * (t_4 * t_2)))) / -t_2;
	} else if (B <= 5.4e-76) {
		tmp = t_7;
	} else if (B <= 4.4e-73) {
		tmp = -sqrt((-2.0 * (((A * A) * (C * F)) * 8.0))) / ((B * B) + t_1);
	} else if (B <= 7.5e+136) {
		tmp = t_6;
	} else {
		tmp = sqrt((F * (A + hypot(B, A)))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_1 = Float64(-4.0 * Float64(A * C))
	t_2 = fma(B, B, Float64(C * Float64(-4.0 * A)))
	t_3 = sqrt(t_0)
	t_4 = Float64(C + Float64(A + hypot(B, Float64(A - C))))
	t_5 = Float64(sqrt(Float64(2.0 * t_4)) / t_0)
	t_6 = Float64(t_5 * Float64(sqrt(F) * Float64(-t_3)))
	t_7 = Float64(-Float64(sqrt(Float64(-2.0 * Float64(t_2 * Float64(F * Float64(C * -2.0))))) / t_2))
	tmp = 0.0
	if (B <= -1.9e-36)
		tmp = t_6;
	elseif (B <= -3.3e-90)
		tmp = t_7;
	elseif (B <= -8.2e-110)
		tmp = Float64(t_5 * Float64(sqrt(F) * Float64(-((t_0 ^ 0.25) ^ 2.0))));
	elseif (B <= -2.6e-136)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(A * 2.0)))))) / t_2);
	elseif (B <= 1.7e-269)
		tmp = Float64(t_5 * Float64(-sqrt(Float64(F * t_1))));
	elseif (B <= 2.5e-228)
		tmp = Float64(Float64(sqrt(F) * t_3) * Float64(0.25 * Float64(sqrt(Float64(1.0 / A)) * Float64((sqrt(2.0) ^ 2.0) / C))));
	elseif (B <= 3.25e-174)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(t_4 * t_2)))) / Float64(-t_2));
	elseif (B <= 5.4e-76)
		tmp = t_7;
	elseif (B <= 4.4e-73)
		tmp = Float64(Float64(-sqrt(Float64(-2.0 * Float64(Float64(Float64(A * A) * Float64(C * F)) * 8.0)))) / Float64(Float64(B * B) + t_1));
	elseif (B <= 7.5e+136)
		tmp = t_6;
	else
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B, A)))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$4 = N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[(N[Sqrt[F], $MachinePrecision] * (-t$95$3)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = (-N[(N[Sqrt[N[(-2.0 * N[(t$95$2 * N[(F * N[(C * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision])}, If[LessEqual[B, -1.9e-36], t$95$6, If[LessEqual[B, -3.3e-90], t$95$7, If[LessEqual[B, -8.2e-110], N[(t$95$5 * N[(N[Sqrt[F], $MachinePrecision] * (-N[Power[N[Power[t$95$0, 0.25], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.6e-136], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(A * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.7e-269], N[(t$95$5 * (-N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, 2.5e-228], N[(N[(N[Sqrt[F], $MachinePrecision] * t$95$3), $MachinePrecision] * N[(0.25 * N[(N[Sqrt[N[(1.0 / A), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.25e-174], N[(N[Sqrt[N[(2.0 * N[(F * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[B, 5.4e-76], t$95$7, If[LessEqual[B, 4.4e-73], N[((-N[Sqrt[N[(-2.0 * N[(N[(N[(A * A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 7.5e+136], t$95$6, N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 9 regimes

2. if B < -1.89999999999999985e-36 or 4.4e-73 < B < 7.5000000000000002e136

   1. Initial program 50.1

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

   2. Simplified 46.8

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

      [Start] 50.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 50.1	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 50.1	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 50.1	\[ \frac{-\sqrt{\left(\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 50.1	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 50.1	\[ \frac{-\sqrt{\left(\left(\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 50.1	\[ \frac{-\sqrt{\left(\left(\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 50.1	\[ \frac{-\sqrt{\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 50.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 50.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 50.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 50.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 50.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 50.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      associate-*l* [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}} \]
      fma-def [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \]
      metadata-eval [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right)} \]
      unpow2 [=>] 46.8	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 41.7

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

   4. Applied egg-rr 41.5

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

   5. Simplified 41.5

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

      [Start] 41.5	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 41.5	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 41.5	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   6. Applied egg-rr 38.1

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

   if -1.89999999999999985e-36 < B < -3.3e-90 or 3.25000000000000004e-174 < B < 5.4000000000000001e-76

   1. Initial program 49.2

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

   2. Simplified 42.7

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

      [Start] 49.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 49.3	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)} \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-\color{blue}{A \cdot 4}\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot \left(-4\right)\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 49.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot \color{blue}{-4}\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 48.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 48.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 48.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 48.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)}} \]
      distribute-lft-neg-in [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)} \]
      *-commutative [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \]
      *-commutative [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-\color{blue}{A \cdot 4}\right)\right)} \]
      distribute-rgt-neg-in [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot \left(-4\right)\right)}\right)} \]
      metadata-eval [=>] 42.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot \color{blue}{-4}\right)\right)} \]

   3. Taylor expanded in A around -inf 48.6

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

   if -3.3e-90 < B < -8.19999999999999965e-110

   1. Initial program 49.6

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

   2. Simplified 42.3

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

      [Start] 49.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 49.6	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 49.6	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 49.6	\[ \frac{-\sqrt{\left(\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 49.6	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 49.6	\[ \frac{-\sqrt{\left(\left(\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 49.6	\[ \frac{-\sqrt{\left(\left(\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 49.6	\[ \frac{-\sqrt{\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 49.6	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 49.6	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 48.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 48.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 48.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 48.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      associate-*l* [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}} \]
      fma-def [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \]
      metadata-eval [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right)} \]
      unpow2 [=>] 42.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 37.4

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

   4. Applied egg-rr 37.3

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

   5. Simplified 37.3

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

      [Start] 37.3	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 37.3	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 37.3	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   6. Applied egg-rr 40.0

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

   7. Applied egg-rr 40.0

      \[\leadsto \left(\color{blue}{{\left({\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{0.25}\right)}^{2}} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if -8.19999999999999965e-110 < B < -2.59999999999999997e-136

   1. Initial program 48.7

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

   2. Simplified 43.9

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

      [Start] 48.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 48.7	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)} \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-\color{blue}{A \cdot 4}\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot \left(-4\right)\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 49.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot \color{blue}{-4}\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 48.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 48.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 48.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 48.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)}} \]
      distribute-lft-neg-in [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)} \]
      *-commutative [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \]
      *-commutative [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-\color{blue}{A \cdot 4}\right)\right)} \]
      distribute-rgt-neg-in [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot \left(-4\right)\right)}\right)} \]
      metadata-eval [=>] 43.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot \color{blue}{-4}\right)\right)} \]

   3. Taylor expanded in A around inf 48.1

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

   if -2.59999999999999997e-136 < B < 1.6999999999999999e-269

   1. Initial program 51.7

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

   2. Simplified 46.3

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

      [Start] 51.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 51.7	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 51.7	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 51.7	\[ \frac{-\sqrt{\left(\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 51.7	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 51.7	\[ \frac{-\sqrt{\left(\left(\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 51.7	\[ \frac{-\sqrt{\left(\left(\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 51.7	\[ \frac{-\sqrt{\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 51.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 51.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 51.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 51.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 51.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 51.0	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      associate-*l* [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}} \]
      fma-def [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \]
      metadata-eval [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right)} \]
      unpow2 [=>] 46.3	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 43.5

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

   4. Applied egg-rr 42.8

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

   5. Simplified 42.8

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

      [Start] 42.8	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 42.8	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 42.8	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   6. Taylor expanded in A around inf 45.4

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

   7. Simplified 43.1

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

      [Start] 45.4	\[ \sqrt{-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-*r* [=>] 43.1	\[ \sqrt{-4 \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot F\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-*l* [<=] 43.1	\[ \sqrt{\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [<=] 43.1	\[ \sqrt{\color{blue}{F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 43.1	\[ \sqrt{F \cdot \color{blue}{\left(\left(A \cdot C\right) \cdot -4\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if 1.6999999999999999e-269 < B < 2.49999999999999986e-228

   1. Initial program 54.9

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

   2. Simplified 50.7

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

      [Start] 54.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 54.9	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 54.9	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 54.9	\[ \frac{-\sqrt{\left(\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 54.9	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 54.9	\[ \frac{-\sqrt{\left(\left(\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 54.9	\[ \frac{-\sqrt{\left(\left(\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 54.9	\[ \frac{-\sqrt{\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 54.9	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 54.9	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 54.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 54.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 54.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 54.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      associate-*l* [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}} \]
      fma-def [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \]
      metadata-eval [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right)} \]
      unpow2 [=>] 50.7	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 49.2

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

   4. Applied egg-rr 48.6

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

   5. Simplified 48.6

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

      [Start] 48.6	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 48.6	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 48.6	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   6. Applied egg-rr 52.9

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

   7. Taylor expanded in C around -inf 53.3

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{F}\right) \cdot \color{blue}{\left(0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{C}\right)\right)} \]

   if 2.49999999999999986e-228 < B < 3.25000000000000004e-174

   1. Initial program 51.1

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

   2. Simplified 45.1

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

      [Start] 51.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 51.1	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)} \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-\color{blue}{A \cdot 4}\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot \left(-4\right)\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 50.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot \color{blue}{-4}\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 50.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 50.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 50.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 50.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)}} \]
      distribute-lft-neg-in [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)} \]
      *-commutative [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \]
      *-commutative [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-\color{blue}{A \cdot 4}\right)\right)} \]
      distribute-rgt-neg-in [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot \left(-4\right)\right)}\right)} \]
      metadata-eval [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot \color{blue}{-4}\right)\right)} \]

   3. Applied egg-rr 49.5

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

   4. Simplified 48.4

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

      [Start] 49.5	\[ \sqrt{2 \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r/ [=>] 49.4	\[ \color{blue}{\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot 1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      *-rgt-identity [=>] 49.4	\[ \frac{\color{blue}{\sqrt{2 \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r+ [=>] 48.5	\[ \frac{\sqrt{2 \cdot \left(F \cdot \left(\color{blue}{\left(\left(\mathsf{hypot}\left(B, A - C\right) + C\right) + A\right)} \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [<=] 48.5	\[ \frac{\sqrt{2 \cdot \left(F \cdot \left(\left(\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right)} + A\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+l+ [=>] 48.4	\[ \frac{\sqrt{2 \cdot \left(F \cdot \left(\color{blue}{\left(C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)\right)} \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 5.4000000000000001e-76 < B < 4.4e-73

   1. Initial program 32.0

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

   2. Simplified 32.0

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

      [Start] 32.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 32.0	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 32.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 32.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 32.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 32.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 32.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 32.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]

   3. Taylor expanded in A around inf 47.2

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

   4. Taylor expanded in A around inf 58.6

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

   5. Simplified 58.6

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

      [Start] 58.6	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 58.6	\[ \frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\color{blue}{\left(A \cdot A\right)} \cdot \left(C \cdot F\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 7.5000000000000002e136 < B

   1. Initial program 63.2

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

   2. Simplified 63.1

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

      [Start] 63.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 63.2	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 63.2	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 63.2	\[ \frac{-\sqrt{\left(\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 63.2	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 63.2	\[ \frac{-\sqrt{\left(\left(\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 63.2	\[ \frac{-\sqrt{\left(\left(\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 63.2	\[ \frac{-\sqrt{\left(\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 63.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 63.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right) \cdot F\right) \cdot \left(2 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l+ [=>] 63.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \color{blue}{\left(A + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 63.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 63.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 63.2	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      associate-*l* [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\left(-\color{blue}{4 \cdot \left(A \cdot C\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)} + {B}^{2}} \]
      fma-def [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \]
      metadata-eval [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(\color{blue}{-4}, A \cdot C, {B}^{2}\right)} \]
      unpow2 [=>] 63.1	\[ \frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 62.2

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

   4. Applied egg-rr 62.2

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

   5. Simplified 62.2

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

      [Start] 62.2	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 62.2	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 62.2	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   6. Applied egg-rr 59.8

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

   7. Taylor expanded in C around 0 61.4

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

   8. Simplified 32.8

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

      [Start] 61.4	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      associate-*r* [=>] 61.4	\[ \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      *-commutative [=>] 61.4	\[ \color{blue}{\sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right)} \]
      *-commutative [=>] 61.4	\[ \sqrt{\color{blue}{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)}} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      unpow2 [=>] 61.4	\[ \sqrt{F \cdot \left(A + \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      unpow2 [=>] 61.4	\[ \sqrt{F \cdot \left(A + \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      hypot-def [=>] 32.8	\[ \sqrt{F \cdot \left(A + \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)} \cdot \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \]
      mul-1-neg [=>] 32.8	\[ \sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \]

3. Recombined 9 regimes into one program.

4. Final simplification 40.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right)\right)\\ \mathbf{elif}\;B \leq -3.3 \cdot 10^{-90}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(\sqrt{F} \cdot \left(-{\left({\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)}^{0.25}\right)}^{2}\right)\right)\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-269}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(-\sqrt{F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)}\right)\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right) \cdot \left(0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{C}\right)\right)\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	40.7
Cost	41576

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := \sqrt{t_0}\\ t_3 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_4 := \frac{\sqrt{2 \cdot t_3}}{t_0}\\ t_5 := t_4 \cdot \left(\sqrt{F} \cdot \left(-t_2\right)\right)\\ t_6 := -4 \cdot \left(A \cdot C\right)\\ t_7 := -\frac{\sqrt{-2 \cdot \left(t_1 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{-41}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-88}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-109}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-137}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-279}:\\ \;\;\;\;t_4 \cdot \left(-\sqrt{F \cdot t_6}\right)\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-215}:\\ \;\;\;\;\left(\sqrt{F} \cdot t_2\right) \cdot \left(0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{C}\right)\right)\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_3 \cdot t_1\right)\right)}}{-t_1}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-75}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{B \cdot B + t_6}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2
Error	40.3
Cost	40076

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := C + \left(A + t_0\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_3 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_4 := \frac{\sqrt{2 \cdot t_1}}{t_3}\\ t_5 := \sqrt{t_3}\\ \mathbf{if}\;B \leq -4 \cdot 10^{+18}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot t_4\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-269}:\\ \;\;\;\;t_4 \cdot \left(-\sqrt{t_3 \cdot F}\right)\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-215}:\\ \;\;\;\;\left(\sqrt{F} \cdot t_5\right) \cdot \left(0.25 \cdot \left(\sqrt{\frac{1}{A}} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{C}\right)\right)\\ \mathbf{elif}\;B \leq 3.25 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(t_1 \cdot t_2\right)\right)}}{-t_2}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 0.00025:\\ \;\;\;\;\frac{t_5 \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(A + \left(C + t_0\right)\right)}\right)}{t_3}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-B\right)\right) \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Error	39.8
Cost	34384

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_3 := \frac{\sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -6.8 \cdot 10^{+19}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot t_3\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-175}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{t_1 \cdot F}\right)\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 0.00025:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(A + \left(C + t_0\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-B\right)\right) \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Error	39.4
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_2 := -4 \cdot \left(A \cdot C\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_4 := \frac{\sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{+17}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot t_4\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-174}:\\ \;\;\;\;t_4 \cdot \left(-\sqrt{t_1 \cdot F}\right)\\ \mathbf{elif}\;B \leq 2 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(t_3 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_2\right)\right)} \cdot \left(-\sqrt{A + \left(C + t_0\right)}\right)}{B \cdot B + t_2}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-B\right)\right) \cdot t_4\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Error	39.7
Cost	28112

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \frac{\sqrt{2 \cdot \left(C + \left(A + t_2\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ t_4 := \frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_1\right)\right)} \cdot \left(-\sqrt{A + \left(C + t_2\right)}\right)}{B \cdot B + t_1}\\ \mathbf{if}\;B \leq -9 \cdot 10^{-19}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot t_3\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-176}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-B\right)\right) \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Error	42.3
Cost	27732

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{-19}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot t_2\\ \mathbf{elif}\;B \leq 3.45 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.35 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-B\right)\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	41.0
Cost	27720

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \frac{\sqrt{2 \cdot \left(C + \left(A + t_1\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{-18}:\\ \;\;\;\;\left(B \cdot \sqrt{F}\right) \cdot t_2\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(C + t_1\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-B\right)\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8
Error	44.4
Cost	27600

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ \mathbf{if}\;B \leq 3.25 \cdot 10^{-174}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(t_1 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-21}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+137}:\\ \;\;\;\;\left(\sqrt{F} \cdot \left(-B\right)\right) \cdot \frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Error	44.6
Cost	21000

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-76}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(t_1 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+72}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 10
Error	44.4
Cost	20036

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.25 \cdot 10^{+72}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 11
Error	44.4
Cost	15044

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.5 \cdot 10^{+72}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 12
Error	49.9
Cost	14856

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{-75}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 13
Error	49.9
Cost	14348

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-174}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 14
Error	49.9
Cost	13448

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 15
Error	53.1
Cost	9352

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -0.016:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right) \cdot \left(\left(\left(C + \frac{B \cdot B}{A - C} \cdot -0.5\right) - A\right) - \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \left(C - A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 16
Error	53.1
Cost	8712

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -3:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A - C\right) + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	53.1
Cost	8712

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -0.023:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{elif}\;C \leq 1.8 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A - C\right) + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) + \left(C - A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	53.0
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -1.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{elif}\;C \leq 6.6 \cdot 10^{-95}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	55.9
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -2.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	57.0
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -9 \cdot 10^{-36}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(C + A \cdot 2\right)\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 21
Error	56.7
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq -1.35 \cdot 10^{-33}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot F\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	58.3
Cost	7808

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

Alternative 23
Error	62.6
Cost	7680

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

Alternative 24
Error	62.6
Cost	7680

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

Reproduce

herbie shell --seed 2022354 
(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))))