ABCF->ab-angle a

Average Error: 52.2 → 35.4

Time: 1.2min

Precision: binary64

Cost: 41112

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

Math

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

↓

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := -\sqrt{2 \cdot C}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \sqrt{F} \cdot \sqrt{2 \cdot t_2}\\ t_4 := F \cdot t_2\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{t_3 \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_4} \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq 10^{-100}:\\ \;\;\;\;t_3 \cdot \frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}}{t_2}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot t_0\right)\\ \end{array} \]

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 (- (sqrt F)))
        (t_1 (- (sqrt (* 2.0 C))))
        (t_2 (fma B B (* A (* C -4.0))))
        (t_3 (* (sqrt F) (sqrt (* 2.0 t_2))))
        (t_4 (* F t_2)))
   (if (<= B -9.2e+55)
     (* (sqrt (/ 2.0 (* B B))) (* (sqrt (+ C (hypot B C))) t_0))
     (if (<= B -7.2e-64)
       (/ (* t_3 t_1) t_2)
       (if (<= B -3.9e-113)
         (/ (- (sqrt (* 2.0 (* t_4 (+ C (fma -0.5 (/ (* B B) A) C)))))) t_2)
         (if (<= B 1.95e-178)
           (/ (* (sqrt (* 2.0 t_4)) t_1) t_2)
           (if (<= B 1e-100)
             (* t_3 (/ (- (sqrt (+ (hypot B (- A C)) (+ C A)))) t_2))
             (if (<= B 4.5e-56)
               (/ (* t_3 (- (sqrt (fma 2.0 C (* -0.5 (/ B (/ A B))))))) t_2)
               (* (/ (sqrt 2.0) B) (* (sqrt (+ C (hypot C B))) t_0))))))))))

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 = -sqrt(F);
	double t_1 = -sqrt((2.0 * C));
	double t_2 = fma(B, B, (A * (C * -4.0)));
	double t_3 = sqrt(F) * sqrt((2.0 * t_2));
	double t_4 = F * t_2;
	double tmp;
	if (B <= -9.2e+55) {
		tmp = sqrt((2.0 / (B * B))) * (sqrt((C + hypot(B, C))) * t_0);
	} else if (B <= -7.2e-64) {
		tmp = (t_3 * t_1) / t_2;
	} else if (B <= -3.9e-113) {
		tmp = -sqrt((2.0 * (t_4 * (C + fma(-0.5, ((B * B) / A), C))))) / t_2;
	} else if (B <= 1.95e-178) {
		tmp = (sqrt((2.0 * t_4)) * t_1) / t_2;
	} else if (B <= 1e-100) {
		tmp = t_3 * (-sqrt((hypot(B, (A - C)) + (C + A))) / t_2);
	} else if (B <= 4.5e-56) {
		tmp = (t_3 * -sqrt(fma(2.0, C, (-0.5 * (B / (A / B)))))) / t_2;
	} else {
		tmp = (sqrt(2.0) / B) * (sqrt((C + hypot(C, B))) * t_0);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(-sqrt(F))
	t_1 = Float64(-sqrt(Float64(2.0 * C)))
	t_2 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_3 = Float64(sqrt(F) * sqrt(Float64(2.0 * t_2)))
	t_4 = Float64(F * t_2)
	tmp = 0.0
	if (B <= -9.2e+55)
		tmp = Float64(sqrt(Float64(2.0 / Float64(B * B))) * Float64(sqrt(Float64(C + hypot(B, C))) * t_0));
	elseif (B <= -7.2e-64)
		tmp = Float64(Float64(t_3 * t_1) / t_2);
	elseif (B <= -3.9e-113)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_4 * Float64(C + fma(-0.5, Float64(Float64(B * B) / A), C)))))) / t_2);
	elseif (B <= 1.95e-178)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_4)) * t_1) / t_2);
	elseif (B <= 1e-100)
		tmp = Float64(t_3 * Float64(Float64(-sqrt(Float64(hypot(B, Float64(A - C)) + Float64(C + A)))) / t_2));
	elseif (B <= 4.5e-56)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(fma(2.0, C, Float64(-0.5 * Float64(B / Float64(A / B))))))) / t_2);
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(sqrt(Float64(C + hypot(C, B))) * t_0));
	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[F], $MachinePrecision])}, Block[{t$95$1 = (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(F * t$95$2), $MachinePrecision]}, If[LessEqual[B, -9.2e+55], N[(N[Sqrt[N[(2.0 / N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -7.2e-64], N[(N[(t$95$3 * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B, -3.9e-113], N[((-N[Sqrt[N[(2.0 * N[(t$95$4 * N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 1.95e-178], N[(N[(N[Sqrt[N[(2.0 * t$95$4), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B, 1e-100], N[(t$95$3 * N[((-N[Sqrt[N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 4.5e-56], N[(N[(t$95$3 * (-N[Sqrt[N[(2.0 * C + N[(-0.5 * N[(B / N[(A / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -9.1999999999999995e55

   1. Initial program 57.1

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

   2. Simplified 56.1

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 1 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 29 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 12 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around 0 63.6

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

   4. Applied egg-rr 53.4

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

   5. Applied egg-rr 46.8

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

   if -9.1999999999999995e55 < B < -7.1999999999999996e-64

   1. Initial program 42.6

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

   2. Simplified 36.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 1 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 29 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 12 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 31.6

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

   4. Applied egg-rr 34.9

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

   5. Taylor expanded in A around -inf 43.3

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

   if -7.1999999999999996e-64 < B < -3.8999999999999999e-113

   1. Initial program 48.6

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

   2. Simplified 42.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 1 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 29 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 12 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around -inf 40.3

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

   4. Simplified 40.3

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \color{blue}{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (fma.f64 -1/2 (/.f64 (*.f64 B B) A) C): 0 points increase in error, 0 points decrease in error
      (fma.f64 -1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) A) C): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 B 2) A)) C)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 C (*.f64 -1/2 (/.f64 (pow.f64 B 2) A)))): 0 points increase in error, 0 points decrease in error

   if -3.8999999999999999e-113 < B < 1.95000000000000013e-178

   1. Initial program 52.4

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

   2. Simplified 45.2

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 1 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 29 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 12 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 43.3

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

   4. Taylor expanded in A around -inf 34.3

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

   if 1.95000000000000013e-178 < B < 1e-100

   1. Initial program 51.6

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

   2. Simplified 45.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 1 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 29 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 12 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 42.2

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

   4. Applied egg-rr 42.2

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

   5. Applied egg-rr 44.5

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

   if 1e-100 < B < 4.5000000000000001e-56

   1. Initial program 46.0

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

   2. Simplified 37.5

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 1 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 29 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 12 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 35.2

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

   4. Applied egg-rr 42.9

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

   5. Taylor expanded in A around -inf 42.0

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

   6. Simplified 42.0

      \[\leadsto \frac{-\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (fma.f64 2 C (*.f64 -1/2 (/.f64 B (/.f64 A B)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 C (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 B B) A)))): 26 points increase in error, 8 points decrease in error
      (fma.f64 2 C (*.f64 -1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) A))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 C) (*.f64 -1/2 (/.f64 (pow.f64 B 2) A)))): 0 points increase in error, 0 points decrease in error

   if 4.5000000000000001e-56 < B

   1. Initial program 53.0

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

   2. Simplified 50.1

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 1 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 29 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 12 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around 0 50.4

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

   4. Applied egg-rr 42.7

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot F\right)}^{1.5}\right)}^{0.3333333333333333}}\right) \]

   5. Applied egg-rr 21.9

      \[\leadsto -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)}\right) \]
3. Recombined 7 regimes into one program.

4. Final simplification 35.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \left(-\sqrt{2 \cdot C}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 10^{-100}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \frac{-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	35.5
Cost	41112

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := -\sqrt{2 \cdot C}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \sqrt{F} \cdot \sqrt{2 \cdot t_2}\\ t_4 := F \cdot t_2\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{t_3 \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_4} \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq 10^{-100}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot t_0\right)\\ \end{array} \]

Alternative 2
Error	35.3
Cost	40916

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := -\sqrt{2 \cdot C}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := \sqrt{F} \cdot \sqrt{2 \cdot t_2}\\ t_4 := F \cdot t_2\\ t_5 := \frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{t_3 \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-113}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_4} \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot t_0\right)\\ \end{array} \]

Alternative 3
Error	34.8
Cost	33800

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := -\sqrt{2 \cdot C}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{2 \cdot t_2}\right) \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-113}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_3} \cdot t_1}{t_2}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot t_0\right)\\ \end{array} \]

Alternative 4
Error	35.2
Cost	28576

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := \frac{-1}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \cdot \left(2 \cdot \sqrt{C \cdot F}\right)\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right)\right)\right)}}{t_2}\\ t_5 := \frac{\sqrt{2 \cdot t_3} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_2}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-113}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-139}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -6.2 \cdot 10^{-188}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -1.5 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-147}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot t_0\right)\\ \end{array} \]

Alternative 5
Error	35.4
Cost	27472

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := F \cdot t_1\\ t_3 := \sqrt{\frac{2}{B \cdot B}}\\ t_4 := C + \mathsf{hypot}\left(C, B\right)\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;t_3 \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \cdot \left(2 \cdot \sqrt{C \cdot F}\right)\\ \mathbf{elif}\;B \leq -9.8 \cdot 10^{-98}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{F \cdot t_4}\right)\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{2 \cdot t_2} \cdot \left(0.25 \cdot \frac{\sqrt{2}}{\frac{A}{\sqrt{\frac{1}{C}}}}\right)\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(C + C\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{t_4} \cdot t_0\right)\\ \end{array} \]

Alternative 6
Error	35.0
Cost	27400

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -59:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_1\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot t_0\right)\\ \end{array} \]

Alternative 7
Error	35.2
Cost	26568

\[\begin{array}{l} t_0 := C + \mathsf{hypot}\left(C, B\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(-\sqrt{F \cdot t_0}\right)\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + C\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{t_0} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 8
Error	34.4
Cost	26568

\[\begin{array}{l} t_0 := -\sqrt{F}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_0\right)\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + C\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot t_0\right)\\ \end{array} \]

Alternative 9
Error	36.8
Cost	21000

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := -\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot t_1\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 10^{+215}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 10
Error	41.6
Cost	20424

\[\begin{array}{l} \mathbf{if}\;B \leq -9.2 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\right)\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \cdot \left(2 \cdot \sqrt{C \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 11
Error	46.9
Cost	20300

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -4 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(-4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C - B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 10^{+215}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 12
Error	45.7
Cost	20300

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := -\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot t_1\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(-4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C - B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 10^{+215}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 13
Error	47.5
Cost	19976

\[\begin{array}{l} \mathbf{if}\;B \leq -4 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-207}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(-4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C - B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]

Alternative 14
Error	49.9
Cost	14472

\[\begin{array}{l} \mathbf{if}\;B \leq -4 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(-4 \cdot \left(C \cdot A\right)\right)\right) \cdot \left(C - B\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(B + C\right)}\right)\\ \end{array} \]

Alternative 15
Error	50.9
Cost	13700

\[\begin{array}{l} \mathbf{if}\;B \leq -1.4 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{\frac{2}{B \cdot B}} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot F\right)}}{B}\\ \end{array} \]

Alternative 16
Error	55.3
Cost	6912

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

Alternative 17
Error	61.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))