ABCF->ab-angle b

Average Error: 52.6 → 38.2

Time: 53.3s

Precision: binary64

Cost: 34648

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - t_0\right)}\\ t_2 := \frac{t_1}{-B}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.08 \cdot 10^{+278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -3.4 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -8.8 \cdot 10^{-263}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_3 \cdot \left(F \cdot \left(\left(t_0 - A\right) - C\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-97}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(\frac{-0.5}{A - C} \cdot \left(B \cdot B\right) + 2 \cdot C\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_0\right)\right)} \cdot \left(-\sqrt{t_4}\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (hypot B (- A C)))
        (t_1 (- (sqrt (* (* 2.0 F) (- (+ A C) t_0)))))
        (t_2 (/ t_1 (- B)))
        (t_3 (fma B B (* A (* C -4.0))))
        (t_4 (fma A (* C -4.0) (* B B))))
   (if (<= B -1.08e+278)
     t_2
     (if (<= B -3.1e+204)
       (* (sqrt (/ F B)) (- (sqrt 2.0)))
       (if (<= B -3.4e+40)
         t_2
         (if (<= B -8.8e-263)
           (/ (- (sqrt (* -2.0 (* t_3 (* F (- (- t_0 A) C)))))) t_3)
           (if (<= B 7.2e-97)
             (-
              (/
               (sqrt
                (*
                 2.0
                 (* (* F t_3) (+ (* (/ -0.5 (- A C)) (* B B)) (* 2.0 C)))))
               t_3))
             (if (<= B 9.8e+58)
               (/ (* (sqrt (* (* 2.0 F) (+ C (- A t_0)))) (- (sqrt t_4))) t_4)
               (/ t_1 (fma -2.0 (* C (/ A B)) B))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = -sqrt(((2.0 * F) * ((A + C) - t_0)));
	double t_2 = t_1 / -B;
	double t_3 = fma(B, B, (A * (C * -4.0)));
	double t_4 = fma(A, (C * -4.0), (B * B));
	double tmp;
	if (B <= -1.08e+278) {
		tmp = t_2;
	} else if (B <= -3.1e+204) {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	} else if (B <= -3.4e+40) {
		tmp = t_2;
	} else if (B <= -8.8e-263) {
		tmp = -sqrt((-2.0 * (t_3 * (F * ((t_0 - A) - C))))) / t_3;
	} else if (B <= 7.2e-97) {
		tmp = -(sqrt((2.0 * ((F * t_3) * (((-0.5 / (A - C)) * (B * B)) + (2.0 * C))))) / t_3);
	} else if (B <= 9.8e+58) {
		tmp = (sqrt(((2.0 * F) * (C + (A - t_0)))) * -sqrt(t_4)) / t_4;
	} else {
		tmp = t_1 / fma(-2.0, (C * (A / B)), B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(A + C) - t_0))))
	t_2 = Float64(t_1 / Float64(-B))
	t_3 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_4 = fma(A, Float64(C * -4.0), Float64(B * B))
	tmp = 0.0
	if (B <= -1.08e+278)
		tmp = t_2;
	elseif (B <= -3.1e+204)
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	elseif (B <= -3.4e+40)
		tmp = t_2;
	elseif (B <= -8.8e-263)
		tmp = Float64(Float64(-sqrt(Float64(-2.0 * Float64(t_3 * Float64(F * Float64(Float64(t_0 - A) - C)))))) / t_3);
	elseif (B <= 7.2e-97)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_3) * Float64(Float64(Float64(-0.5 / Float64(A - C)) * Float64(B * B)) + Float64(2.0 * C))))) / t_3));
	elseif (B <= 9.8e+58)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + Float64(A - t_0)))) * Float64(-sqrt(t_4))) / t_4);
	else
		tmp = Float64(t_1 / fma(-2.0, Float64(C * Float64(A / B)), B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(t$95$1 / (-B)), $MachinePrecision]}, Block[{t$95$3 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.08e+278], t$95$2, If[LessEqual[B, -3.1e+204], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -3.4e+40], t$95$2, If[LessEqual[B, -8.8e-263], N[((-N[Sqrt[N[(-2.0 * N[(t$95$3 * N[(F * N[(N[(t$95$0 - A), $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], If[LessEqual[B, 7.2e-97], (-N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$3), $MachinePrecision] * N[(N[(N[(-0.5 / N[(A - C), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), If[LessEqual[B, 9.8e+58], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$4], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], N[(t$95$1 / N[(-2.0 * N[(C * N[(A / B), $MachinePrecision]), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -1.07999999999999995e278 or -3.1000000000000002e204 < B < -3.39999999999999989e40

   1. Initial program 54.5

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

   2. Simplified 52.8

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

      [Start] 54.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 54.5	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 54.5	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot 2\right)\right)} \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 54.5	\[ \frac{-\sqrt{\color{blue}{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 54.5	\[ \frac{-\sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \color{blue}{\left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 54.5	\[ \frac{-\sqrt{\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 54.5	\[ \frac{-\sqrt{\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 54.5	\[ \frac{-\sqrt{\left(\left(-\color{blue}{C \cdot \left(4 \cdot A\right)}\right) + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 54.5	\[ \frac{-\sqrt{\left(\color{blue}{\left(-C\right) \cdot \left(4 \cdot A\right)} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*r* [=>] 54.5	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-C\right) \cdot 4\right) \cdot A} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 54.5	\[ \frac{-\sqrt{\left(\color{blue}{A \cdot \left(\left(-C\right) \cdot 4\right)} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 54.5	\[ \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(A, \left(-C\right) \cdot 4, {B}^{2}\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [<=] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, \color{blue}{-C \cdot 4}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, \color{blue}{C \cdot \left(-4\right)}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot \color{blue}{-4}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-rgt-identity [<=] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\left(A + \color{blue}{\left(C + 0\right)}\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+r- [<=] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\color{blue}{\left(A + \left(\left(C + 0\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-rgt-identity [=>] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(\color{blue}{C} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 54.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{\left(2 \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      *-commutative [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(-\color{blue}{C \cdot \left(4 \cdot A\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(-C\right) \cdot \left(4 \cdot A\right)} + {B}^{2}} \]
      associate-*r* [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(\left(-C\right) \cdot 4\right) \cdot A} + {B}^{2}} \]
      *-commutative [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{A \cdot \left(\left(-C\right) \cdot 4\right)} + {B}^{2}} \]
      fma-def [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{fma}\left(A, \left(-C\right) \cdot 4, {B}^{2}\right)}} \]
      distribute-lft-neg-in [<=] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, \color{blue}{-C \cdot 4}, {B}^{2}\right)} \]
      distribute-rgt-neg-in [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, \color{blue}{C \cdot \left(-4\right)}, {B}^{2}\right)} \]
      metadata-eval [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot \color{blue}{-4}, {B}^{2}\right)} \]
      unpow2 [=>] 52.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 46.1

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

   4. Simplified 46.1

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

      [Start] 46.1	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 46.1	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 46.1	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 46.1

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

   6. Simplified 45.0

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

      [Start] 46.1	\[ \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 46.1	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-inverses [=>] 45.0	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \color{blue}{1} \]
      associate-/r/ [<=] 45.0	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{1}}} \]
      /-rgt-identity [=>] 45.0	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-commutative [=>] 45.0	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in B around -inf 31.4

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

   8. Simplified 31.4

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

      [Start] 31.4	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{-1 \cdot B} \]
      mul-1-neg [=>] 31.4	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{-B}} \]

   if -1.07999999999999995e278 < B < -3.1000000000000002e204

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [=>] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C + \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-in [=>] 64.0	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot C + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 64.0	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot C\right)} + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [<=] 64.0	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)} + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) + \color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-in [<=] 64.0	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) + \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-in [<=] 64.0	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(C + \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [<=] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \color{blue}{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [<=] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\color{blue}{\left(A + C\right)} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)} \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(-\color{blue}{A \cdot 4}\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot \left(-4\right)\right)} \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(\left(-4\right) \cdot C\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(\color{blue}{-4} \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(\left(C - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) + A\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l- [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(C - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - A\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\color{blue}{\mathsf{hypot}\left(B, A - C\right)} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)}} \]
      distribute-lft-neg-in [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)} \]
      *-commutative [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \left(-\color{blue}{A \cdot 4}\right) \cdot C\right)} \]
      distribute-rgt-neg-in [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot \left(-4\right)\right)} \cdot C\right)} \]
      associate-*l* [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(\left(-4\right) \cdot C\right)}\right)} \]
      metadata-eval [=>] 64.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(\color{blue}{-4} \cdot C\right)\right)} \]

   3. Taylor expanded in B around -inf 64.0

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot {B}^{3}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Taylor expanded in A around 0 30.6

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

   5. Simplified 30.6

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

      [Start] 30.6	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      associate-*r* [=>] 30.6	\[ \color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
      *-commutative [=>] 30.6	\[ \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)} \]
      mul-1-neg [=>] 30.6	\[ \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(-\sqrt{2}\right)} \]

   if -3.39999999999999989e40 < B < -8.8000000000000001e-263

   1. Initial program 48.2

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

   2. Simplified 43.0

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

      [Start] 48.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 48.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [=>] 47.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C + \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-in [=>] 47.8	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot C + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 47.8	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot C\right)} + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [<=] 47.8	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)} + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 47.8	\[ \frac{-\sqrt{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) + \color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-in [<=] 47.8	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) + \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 47.8	\[ \frac{-\sqrt{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-in [<=] 47.8	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(C + \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [<=] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \color{blue}{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [<=] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\color{blue}{\left(A + C\right)} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)} \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(-\color{blue}{A \cdot 4}\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot \left(-4\right)\right)} \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(\left(-4\right) \cdot C\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 48.2	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(\color{blue}{-4} \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [=>] 47.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 47.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(\left(C - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) + A\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l- [=>] 47.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(C - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - A\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 47.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 47.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 47.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\color{blue}{\mathsf{hypot}\left(B, A - C\right)} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)}} \]
      distribute-lft-neg-in [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)} \]
      *-commutative [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \left(-\color{blue}{A \cdot 4}\right) \cdot C\right)} \]
      distribute-rgt-neg-in [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot \left(-4\right)\right)} \cdot C\right)} \]
      associate-*l* [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(\left(-4\right) \cdot C\right)}\right)} \]
      metadata-eval [=>] 43.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(\color{blue}{-4} \cdot C\right)\right)} \]

   if -8.8000000000000001e-263 < B < 7.19999999999999995e-97

   1. Initial program 52.3

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

   2. Simplified 46.7

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

      [Start] 52.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 52.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\left(C + A\right)} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [=>] 51.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(C + \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-in [=>] 51.6	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot C + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 51.6	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot C\right)} + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [<=] 51.6	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)} + \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 51.6	\[ \frac{-\sqrt{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) + \color{blue}{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-in [<=] 51.6	\[ \frac{-\sqrt{\color{blue}{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) + \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 51.6	\[ \frac{-\sqrt{2 \cdot \left(C \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) + \color{blue}{\left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-in [<=] 51.6	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(C + \left(A - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [<=] 52.3	\[ \frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \color{blue}{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [<=] 52.3	\[ \frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(\color{blue}{\left(A + C\right)} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \left(\left(\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)} \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(-\color{blue}{A \cdot 4}\right) \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot \left(-4\right)\right)} \cdot C\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(\left(-4\right) \cdot C\right)}\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 52.5	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(\color{blue}{-4} \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate--l+ [=>] 51.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 51.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(\left(C - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) + A\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+l- [=>] 51.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\left(C - \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - A\right)\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 51.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 51.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 51.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\color{blue}{\mathsf{hypot}\left(B, A - C\right)} - A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\color{blue}{B \cdot B} - \left(4 \cdot A\right) \cdot C} \]
      fma-neg [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -\left(4 \cdot A\right) \cdot C\right)}} \]
      distribute-lft-neg-in [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4 \cdot A\right) \cdot C}\right)} \]
      *-commutative [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \left(-\color{blue}{A \cdot 4}\right) \cdot C\right)} \]
      distribute-rgt-neg-in [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot \left(-4\right)\right)} \cdot C\right)} \]
      associate-*l* [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(\left(-4\right) \cdot C\right)}\right)} \]
      metadata-eval [=>] 46.7	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(\color{blue}{-4} \cdot C\right)\right)} \]

   3. Taylor expanded in B around 0 47.6

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

   4. Simplified 47.6

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

      [Start] 47.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A - C} + 2 \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 47.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A - C}, 2 \cdot C\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 47.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{B \cdot B}}{A - C}, 2 \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 47.2

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

   6. Simplified 47.2

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

      [Start] 47.2	\[ \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right) \cdot \frac{-0.5}{\frac{A - C}{B \cdot B}} + \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      distribute-lft-out [=>] 47.2	\[ \frac{-\sqrt{2 \cdot \color{blue}{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot F\right) \cdot \left(\frac{-0.5}{\frac{A - C}{B \cdot B}} + 2 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 47.2	\[ \frac{-\sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)} \cdot \left(\frac{-0.5}{\frac{A - C}{B \cdot B}} + 2 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 47.2	\[ \frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \color{blue}{\left(C \cdot -4\right)}\right)\right) \cdot \left(\frac{-0.5}{\frac{A - C}{B \cdot B}} + 2 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-/r/ [=>] 47.2	\[ \frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\color{blue}{\frac{-0.5}{A - C} \cdot \left(B \cdot B\right)} + 2 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 7.19999999999999995e-97 < B < 9.80000000000000037e58

   1. Initial program 43.5

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

   2. Simplified 38.4

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

      [Start] 43.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 43.5	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 43.5	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot 2\right)\right)} \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 43.5	\[ \frac{-\sqrt{\color{blue}{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 43.5	\[ \frac{-\sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \color{blue}{\left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 43.5	\[ \frac{-\sqrt{\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 43.5	\[ \frac{-\sqrt{\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 43.5	\[ \frac{-\sqrt{\left(\left(-\color{blue}{C \cdot \left(4 \cdot A\right)}\right) + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 43.5	\[ \frac{-\sqrt{\left(\color{blue}{\left(-C\right) \cdot \left(4 \cdot A\right)} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*r* [=>] 43.5	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-C\right) \cdot 4\right) \cdot A} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 43.5	\[ \frac{-\sqrt{\left(\color{blue}{A \cdot \left(\left(-C\right) \cdot 4\right)} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 43.5	\[ \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(A, \left(-C\right) \cdot 4, {B}^{2}\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [<=] 43.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, \color{blue}{-C \cdot 4}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 43.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, \color{blue}{C \cdot \left(-4\right)}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 43.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot \color{blue}{-4}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 43.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-rgt-identity [<=] 43.5	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\left(A + \color{blue}{\left(C + 0\right)}\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+r- [<=] 43.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\color{blue}{\left(A + \left(\left(C + 0\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-rgt-identity [=>] 43.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(\color{blue}{C} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 43.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 43.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 43.2	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{\left(2 \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      *-commutative [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(-\color{blue}{C \cdot \left(4 \cdot A\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(-C\right) \cdot \left(4 \cdot A\right)} + {B}^{2}} \]
      associate-*r* [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(\left(-C\right) \cdot 4\right) \cdot A} + {B}^{2}} \]
      *-commutative [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{A \cdot \left(\left(-C\right) \cdot 4\right)} + {B}^{2}} \]
      fma-def [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{fma}\left(A, \left(-C\right) \cdot 4, {B}^{2}\right)}} \]
      distribute-lft-neg-in [<=] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, \color{blue}{-C \cdot 4}, {B}^{2}\right)} \]
      distribute-rgt-neg-in [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, \color{blue}{C \cdot \left(-4\right)}, {B}^{2}\right)} \]
      metadata-eval [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot \color{blue}{-4}, {B}^{2}\right)} \]
      unpow2 [=>] 38.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 38.3

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

   4. Simplified 37.8

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

      [Start] 38.3	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 38.3	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 37.8	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 9.80000000000000037e58 < B

   1. Initial program 58.9

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

   2. Simplified 57.4

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

      [Start] 58.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 58.9	\[ \frac{-\sqrt{\color{blue}{\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right)} \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 58.9	\[ \frac{-\sqrt{\color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot 2\right)\right)} \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 58.9	\[ \frac{-\sqrt{\color{blue}{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 58.9	\[ \frac{-\sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \color{blue}{\left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 58.9	\[ \frac{-\sqrt{\color{blue}{\left({B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 58.9	\[ \frac{-\sqrt{\color{blue}{\left(\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 58.9	\[ \frac{-\sqrt{\left(\left(-\color{blue}{C \cdot \left(4 \cdot A\right)}\right) + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [=>] 58.9	\[ \frac{-\sqrt{\left(\color{blue}{\left(-C\right) \cdot \left(4 \cdot A\right)} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*r* [=>] 58.9	\[ \frac{-\sqrt{\left(\color{blue}{\left(\left(-C\right) \cdot 4\right) \cdot A} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 58.9	\[ \frac{-\sqrt{\left(\color{blue}{A \cdot \left(\left(-C\right) \cdot 4\right)} + {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      fma-def [=>] 58.9	\[ \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(A, \left(-C\right) \cdot 4, {B}^{2}\right)} \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-lft-neg-in [<=] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, \color{blue}{-C \cdot 4}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      distribute-rgt-neg-in [=>] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, \color{blue}{C \cdot \left(-4\right)}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot \color{blue}{-4}, {B}^{2}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right) \cdot \left(\left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-rgt-identity [<=] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(\left(A + \color{blue}{\left(C + 0\right)}\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-+r- [<=] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\color{blue}{\left(A + \left(\left(C + 0\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)} \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-rgt-identity [=>] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(\color{blue}{C} - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 58.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      hypot-def [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right) \cdot \left(F \cdot 2\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      *-commutative [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \color{blue}{\left(2 \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      sub-neg [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{{B}^{2} + \left(-\left(4 \cdot A\right) \cdot C\right)}} \]
      +-commutative [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(-\left(4 \cdot A\right) \cdot C\right) + {B}^{2}}} \]
      *-commutative [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\left(-\color{blue}{C \cdot \left(4 \cdot A\right)}\right) + {B}^{2}} \]
      distribute-lft-neg-in [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(-C\right) \cdot \left(4 \cdot A\right)} + {B}^{2}} \]
      associate-*r* [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\left(\left(-C\right) \cdot 4\right) \cdot A} + {B}^{2}} \]
      *-commutative [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{A \cdot \left(\left(-C\right) \cdot 4\right)} + {B}^{2}} \]
      fma-def [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{fma}\left(A, \left(-C\right) \cdot 4, {B}^{2}\right)}} \]
      distribute-lft-neg-in [<=] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, \color{blue}{-C \cdot 4}, {B}^{2}\right)} \]
      distribute-rgt-neg-in [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, \color{blue}{C \cdot \left(-4\right)}, {B}^{2}\right)} \]
      metadata-eval [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot \color{blue}{-4}, {B}^{2}\right)} \]
      unpow2 [=>] 57.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)} \]

   3. Applied egg-rr 51.9

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

   4. Simplified 51.9

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

      [Start] 51.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      +-commutative [=>] 51.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(\color{blue}{\left(C + A\right)} - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate--l+ [=>] 51.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(2 \cdot F\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   5. Applied egg-rr 51.9

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

   6. Simplified 50.5

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

      [Start] 51.9	\[ \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 51.9	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-inverses [=>] 50.5	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \cdot \color{blue}{1} \]
      associate-/r/ [<=] 50.5	\[ \color{blue}{\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{1}}} \]
      /-rgt-identity [=>] 50.5	\[ \frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}} \]
      *-commutative [=>] 50.5	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around 0 31.8

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

   8. Simplified 30.2

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

      [Start] 31.8	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{-2 \cdot \frac{A \cdot C}{B} + B} \]
      fma-def [=>] 31.8	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\color{blue}{\mathsf{fma}\left(-2, \frac{A \cdot C}{B}, B\right)}} \]
      associate-/l* [=>] 30.2	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(-2, \color{blue}{\frac{A}{\frac{B}{C}}}, B\right)} \]
      associate-/r/ [=>] 30.2	\[ \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(-2, \color{blue}{\frac{A}{B} \cdot C}, B\right)} \]

3. Recombined 6 regimes into one program.

4. Final simplification 38.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.08 \cdot 10^{+278}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -3.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{-B}\\ \mathbf{elif}\;B \leq -8.8 \cdot 10^{-263}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(\mathsf{hypot}\left(B, A - C\right) - A\right) - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-97}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\frac{-0.5}{A - C} \cdot \left(B \cdot B\right) + 2 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	38.2
Cost	27984

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - t_0\right)}\\ t_2 := \frac{t_1}{-B}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.3 \cdot 10^{+278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-263}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_3 \cdot \left(F \cdot \left(\left(t_0 - A\right) - C\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-97}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(\frac{-0.5}{A - C} \cdot \left(B \cdot B\right) + 2 \cdot C\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{t_1}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 2
Error	39.3
Cost	27736

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_2 := \frac{t_1}{\sqrt{t_0}}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -6.5 \cdot 10^{+278}:\\ \;\;\;\;\frac{t_1}{-B}\\ \mathbf{elif}\;B \leq -4.8 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-66}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-271}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot 4\right) \cdot \left(F \cdot t_0\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-95}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(\frac{-0.5}{A - C} \cdot \left(B \cdot B\right) + 2 \cdot C\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 3
Error	39.3
Cost	27736

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - t_1\right)}\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_4 := \sqrt{t_3}\\ \mathbf{if}\;B \leq -4.3 \cdot 10^{+278}:\\ \;\;\;\;\frac{t_2}{-B}\\ \mathbf{elif}\;B \leq -4.8 \cdot 10^{+115}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \frac{1}{\frac{1}{A + \left(C - t_1\right)}}}}{t_4}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-272}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot 4\right) \cdot \left(F \cdot t_3\right)}}{t_3}\\ \mathbf{elif}\;B \leq 1.06 \cdot 10^{-98}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\frac{-0.5}{A - C} \cdot \left(B \cdot B\right) + 2 \cdot C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.3 \cdot 10^{+56}:\\ \;\;\;\;\frac{t_2}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 4
Error	39.9
Cost	22036

\[\begin{array}{l} t_0 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_2 := \frac{t_0}{-B}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{+278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -4.3 \cdot 10^{-272}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot 4\right) \cdot \left(F \cdot t_1\right)}}{t_1}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-98}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(\frac{-0.5}{A - C} \cdot \left(B \cdot B\right) + 2 \cdot C\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 5
Error	40.0
Cost	21396

\[\begin{array}{l} t_0 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_2 := \frac{t_0}{-B}\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.25 \cdot 10^{+278}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-270}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot 4\right) \cdot \left(F \cdot t_1\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_3 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 6
Error	40.2
Cost	21204

\[\begin{array}{l} t_0 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{t_0}{-B}\\ t_2 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;B \leq -9.6 \cdot 10^{+277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-269}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 7
Error	40.1
Cost	21204

\[\begin{array}{l} t_0 := -\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{t_0}{-B}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -6.5 \cdot 10^{+277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.45 \cdot 10^{-270}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot 4\right) \cdot \left(F \cdot t_2\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(-2, C \cdot \frac{A}{B}, B\right)}\\ \end{array} \]

Alternative 8
Error	40.5
Cost	14932

\[\begin{array}{l} t_0 := \sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{-t_0}{-B}\\ t_2 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;B \leq -8.6 \cdot 10^{+277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-272}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_0}{B + -2 \cdot \frac{A \cdot C}{B}}\\ \end{array} \]

Alternative 9
Error	40.2
Cost	14868

\[\begin{array}{l} t_0 := \sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{-t_0}{-B}\\ t_2 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{+278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-271}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_0}{B}\\ \end{array} \]

Alternative 10
Error	40.6
Cost	14740

\[\begin{array}{l} t_0 := \sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{-t_0}{-B}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{+278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -5.4 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(-16 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{-t_2}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_0}{B}\\ \end{array} \]

Alternative 11
Error	40.2
Cost	14740

\[\begin{array}{l} t_0 := \sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{-t_0}{-B}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{+278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -1.55 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -9 \cdot 10^{-272}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(A \cdot \left(C + C\right)\right) \cdot \left(F \cdot C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_0}{B}\\ \end{array} \]

Alternative 12
Error	41.7
Cost	14480

\[\begin{array}{l} t_0 := \sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{-t_0}{-B}\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{+278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -1.16 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(-16 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_0}{B}\\ \end{array} \]

Alternative 13
Error	42.4
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := \frac{-t_0}{-B}\\ t_2 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -6.2 \cdot 10^{+278}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.1 \cdot 10^{+204}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_0}{B}\\ \end{array} \]

Alternative 14
Error	43.7
Cost	14024

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{B}\\ \end{array} \]

Alternative 15
Error	46.1
Cost	13640

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-120}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 31000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{-B \cdot F}\\ \end{array} \]

Alternative 16
Error	51.2
Cost	13316

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 27000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B \cdot \frac{B}{A}}}\\ \end{array} \]

Alternative 17
Error	56.2
Cost	8712

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.35 \cdot 10^{-64}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 70000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - \left(C - A\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B \cdot \frac{B}{A}}}\\ \end{array} \]

Alternative 18
Error	57.7
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -2.8 \cdot 10^{+122}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B \cdot \frac{B}{A}}}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(A \cdot \left(-4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	55.8
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(A \cdot \left(-4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	56.4
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -7.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 105000000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B \cdot \frac{B}{A}}}\\ \end{array} \]

Alternative 21
Error	59.7
Cost	7940

\[\begin{array}{l} \mathbf{if}\;A \leq 4.1 \cdot 10^{-236}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B \cdot \frac{B}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(4 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 22
Error	59.4
Cost	7940

\[\begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B \cdot \frac{B}{A}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(A \cdot \left(-4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 23
Error	60.9
Cost	6976

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

Alternative 24
Error	60.7
Cost	6976

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

Alternative 25
Error	62.1
Cost	6848

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

Reproduce

herbie shell --seed 2022356 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))