ABCF->ab-angle a

Average Error: 52.2 → 38.5

Time: 1.6min

Precision: binary64

Cost: 41312

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(B \cdot \sqrt{F}\right)}}\\ t_2 := \sqrt{\frac{-F}{A}}\\ t_3 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ t_4 := \frac{t_3 \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{if}\;B \leq -850000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-122}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{2 \cdot t_0}\right) \cdot \left(-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0))))
        (t_1
         (/
          (sqrt 2.0)
          (/
           (fma B B (* C (* A -4.0)))
           (* (sqrt (+ C (hypot B C))) (* B (sqrt F))))))
        (t_2 (sqrt (/ (- F) A)))
        (t_3 (sqrt (* 2.0 (* F t_0))))
        (t_4 (/ (* t_3 (- (sqrt (* C 2.0)))) t_0)))
   (if (<= B -850000.0)
     t_1
     (if (<= B -2.7e-6)
       t_4
       (if (<= B -5e-72)
         t_1
         (if (<= B -1.75e-103)
           (/ (* t_3 (- (sqrt (fma 2.0 C (* -0.5 (/ B (/ A B))))))) t_0)
           (if (<= B -5.5e-245)
             t_2
             (if (<= B 1.75e-246)
               t_4
               (if (<= B 7.6e-122)
                 t_2
                 (if (<= B 1.25e+136)
                   (/
                    (*
                     (* (sqrt F) (sqrt (* 2.0 t_0)))
                     (- (sqrt (+ (hypot B (- A C)) (+ C A)))))
                    t_0)
                   (*
                    (sqrt (* F (+ C (hypot C B))))
                    (- (/ (sqrt 2.0) B)))))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = sqrt(2.0) / (fma(B, B, (C * (A * -4.0))) / (sqrt((C + hypot(B, C))) * (B * sqrt(F))));
	double t_2 = sqrt((-F / A));
	double t_3 = sqrt((2.0 * (F * t_0)));
	double t_4 = (t_3 * -sqrt((C * 2.0))) / t_0;
	double tmp;
	if (B <= -850000.0) {
		tmp = t_1;
	} else if (B <= -2.7e-6) {
		tmp = t_4;
	} else if (B <= -5e-72) {
		tmp = t_1;
	} else if (B <= -1.75e-103) {
		tmp = (t_3 * -sqrt(fma(2.0, C, (-0.5 * (B / (A / B)))))) / t_0;
	} else if (B <= -5.5e-245) {
		tmp = t_2;
	} else if (B <= 1.75e-246) {
		tmp = t_4;
	} else if (B <= 7.6e-122) {
		tmp = t_2;
	} else if (B <= 1.25e+136) {
		tmp = ((sqrt(F) * sqrt((2.0 * t_0))) * -sqrt((hypot(B, (A - C)) + (C + A)))) / t_0;
	} else {
		tmp = sqrt((F * (C + hypot(C, B)))) * -(sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(sqrt(2.0) / Float64(fma(B, B, Float64(C * Float64(A * -4.0))) / Float64(sqrt(Float64(C + hypot(B, C))) * Float64(B * sqrt(F)))))
	t_2 = sqrt(Float64(Float64(-F) / A))
	t_3 = sqrt(Float64(2.0 * Float64(F * t_0)))
	t_4 = Float64(Float64(t_3 * Float64(-sqrt(Float64(C * 2.0)))) / t_0)
	tmp = 0.0
	if (B <= -850000.0)
		tmp = t_1;
	elseif (B <= -2.7e-6)
		tmp = t_4;
	elseif (B <= -5e-72)
		tmp = t_1;
	elseif (B <= -1.75e-103)
		tmp = Float64(Float64(t_3 * Float64(-sqrt(fma(2.0, C, Float64(-0.5 * Float64(B / Float64(A / B))))))) / t_0);
	elseif (B <= -5.5e-245)
		tmp = t_2;
	elseif (B <= 1.75e-246)
		tmp = t_4;
	elseif (B <= 7.6e-122)
		tmp = t_2;
	elseif (B <= 1.25e+136)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(2.0 * t_0))) * Float64(-sqrt(Float64(hypot(B, Float64(A - C)) + Float64(C + A))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(C, B)))) * Float64(-Float64(sqrt(2.0) / B)));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(B * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 * (-N[Sqrt[N[(C * 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[B, -850000.0], t$95$1, If[LessEqual[B, -2.7e-6], t$95$4, If[LessEqual[B, -5e-72], t$95$1, If[LessEqual[B, -1.75e-103], 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$0), $MachinePrecision], If[LessEqual[B, -5.5e-245], t$95$2, If[LessEqual[B, 1.75e-246], t$95$4, If[LessEqual[B, 7.6e-122], t$95$2, If[LessEqual[B, 1.25e+136], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -8.5e5 or -2.69999999999999998e-6 < B < -4.9999999999999996e-72

   1. Initial program 54.1

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

   2. Applied egg-rr 51.7

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

   3. Taylor expanded in A around 0 54.6

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

   4. Simplified 53.7

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}} \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 F (*.f64 B B)) (+.f64 C (hypot.f64 C B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 F (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (+.f64 C (hypot.f64 C B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 F (pow.f64 B 2)) (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 C C) (*.f64 B B))))))): 30 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 F (pow.f64 B 2)) (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 C 2)) (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 F (pow.f64 B 2)) (+.f64 C (sqrt.f64 (+.f64 (pow.f64 C 2) (Rewrite<= unpow2_binary64 (pow.f64 B 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 F (pow.f64 B 2)) (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (pow.f64 C 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 C 2)))) (*.f64 F (pow.f64 B 2))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 47.0

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

   if -8.5e5 < B < -2.69999999999999998e-6 or -5.49999999999999962e-245 < B < 1.7500000000000001e-246

   1. Initial program 51.5

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

   2. Simplified 44.7

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 0 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)))): 21 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)))): 15 points increase in error, 4 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, 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)) (-.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 41.7

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

   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 -4.9999999999999996e-72 < B < -1.75000000000000008e-103

   1. Initial program 47.0

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

   2. Simplified 40.6

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 0 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)))): 21 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)))): 15 points increase in error, 4 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, 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)) (-.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 36.8

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

   4. Taylor expanded in A around -inf 34.4

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

   5. Simplified 34.4

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      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, 7 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 -1.75000000000000008e-103 < B < -5.49999999999999962e-245 or 1.7500000000000001e-246 < B < 7.6000000000000002e-122

   1. Initial program 51.9

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

   2. Simplified 45.3

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 0 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)))): 21 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)))): 15 points increase in error, 4 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, 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)) (-.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 42.4

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

   4. Taylor expanded in A around -inf 35.5

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

   5. Simplified 35.5

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      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, 7 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

   6. Applied egg-rr 55.4

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

   7. Taylor expanded in B around 0 41.0

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]

   8. Simplified 41.0

      \[\leadsto \sqrt{\color{blue}{\frac{-F}{A}}} \]
      Proof
      (/.f64 (neg.f64 F) A): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 F)) A): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 F A))): 0 points increase in error, 0 points decrease in error

   if 7.6000000000000002e-122 < B < 1.25e136

   1. Initial program 43.7

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

   2. Simplified 38.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 0 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)))): 21 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)))): 15 points increase in error, 4 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, 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)) (-.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 33.8

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

   4. Applied egg-rr 32.9

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

   if 1.25e136 < B

   1. Initial program 62.8

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

   2. Simplified 62.6

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 0 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)))): 21 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)))): 15 points increase in error, 4 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, 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)) (-.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 61.5

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

   4. Taylor expanded in A around 0 60.9

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

   5. Simplified 29.1

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\right)} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (+.f64 C (hypot.f64 C B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 C C) (*.f64 B B))))))))): 67 points increase in error, 8 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 C 2)) (*.f64 B B)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (+.f64 C (sqrt.f64 (+.f64 (pow.f64 C 2) (Rewrite<= unpow2_binary64 (pow.f64 B 2))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (pow.f64 C 2))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 C 2)))) F))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 C 2)))) F))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 C 2)))) F))))): 0 points increase in error, 0 points decrease in error
3. Recombined 6 regimes into one program.

4. Final simplification 38.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -850000:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(B \cdot \sqrt{F}\right)}}\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(B \cdot \sqrt{F}\right)}}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+136}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \cdot \left(-\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	38.1
Cost	74904

\[\begin{array}{l} t_0 := C + \mathsf{hypot}\left(C, B\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot {\left({\left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{0.25} \cdot {F}^{0.25}\right)}^{2}\right) \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ t_3 := \sqrt{2 \cdot \left(F \cdot t_1\right)}\\ t_4 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B \leq -2.05 \cdot 10^{-70}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{C \cdot 2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-122}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+127}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \sqrt{2} \cdot \left(\left(B \cdot \left(A \cdot \left(4 \cdot \frac{t_0}{\frac{{B}^{4}}{C \cdot F}} - \frac{F}{B} \cdot \frac{C \cdot \sqrt{\frac{1}{\mathsf{fma}\left(C, C, B \cdot B\right)}} + -1}{B}\right)\right)\right) \cdot \sqrt{\frac{\frac{1}{t_0}}{F}}\right), \sqrt{F \cdot t_0} \cdot \left(-\frac{\sqrt{2}}{B}\right)\right)\\ \end{array} \]

Alternative 2
Error	38.1
Cost	47768

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot {\left({\left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)}^{0.25} \cdot {F}^{0.25}\right)}^{2}\right) \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ t_2 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ t_3 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B \leq -2.05 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7.6 \cdot 10^{-122}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 3
Error	39.5
Cost	34448

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(B \cdot \sqrt{F}\right)\\ t_2 := \sqrt{\frac{-F}{A}}\\ t_3 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_5 := \frac{\sqrt{2}}{\frac{t_4}{t_1}}\\ t_6 := \frac{t_3 \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{if}\;B \leq -850000:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-103}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 10^{+150}:\\ \;\;\;\;\frac{1}{-t_4} \cdot \left(\sqrt{2} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 4
Error	39.1
Cost	34136

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(B \cdot \sqrt{F}\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := \frac{\sqrt{2}}{\frac{t_2}{t_1}}\\ t_4 := \frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{if}\;B \leq -850000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-93}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 10^{+150}:\\ \;\;\;\;\frac{1}{-t_2} \cdot \left(\sqrt{2} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 5
Error	40.4
Cost	33548

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := B \cdot \sqrt{F}\\ t_2 := \frac{\sqrt{2}}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot t_1}}\\ t_3 := \frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{if}\;B \leq -850000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(\sqrt{2} \cdot t_1\right) \cdot \left(\sqrt{B} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 6
Error	41.9
Cost	27784

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := F \cdot t_1\\ \mathbf{if}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{B \cdot \left(2 \cdot \left(B \cdot F\right)\right)}\right)\\ \mathbf{elif}\;B \leq -2.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \mathsf{fma}\left(2, C, \frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_2} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right) \cdot \left(\sqrt{B} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 7
Error	41.2
Cost	27400

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{B \cdot \left(2 \cdot \left(B \cdot F\right)\right)}\right)\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right) \cdot \left(\sqrt{B} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 8
Error	41.6
Cost	27348

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{-B} \cdot \left(B \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right) \cdot \left(\sqrt{B} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 9
Error	41.6
Cost	27348

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\\ \mathbf{if}\;B \leq -1.25 \cdot 10^{+47}:\\ \;\;\;\;\frac{t_1 \cdot \sqrt{-B}}{t_0}\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{B} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 10
Error	42.6
Cost	27216

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(B \cdot \left(2 \cdot \left(B \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right) \cdot \left(\sqrt{B} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 11
Error	41.5
Cost	27216

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{B \cdot \left(2 \cdot \left(B \cdot F\right)\right)}\right)\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)\right) \cdot \left(\sqrt{B} \cdot \frac{-1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 12
Error	43.1
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(B \cdot \left(2 \cdot \left(B \cdot F\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 13
Error	44.7
Cost	20676

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(B \cdot \left(2 \cdot \left(B \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{-t_0} \cdot \sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 14
Error	45.5
Cost	20564

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B \leq -2.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{-B}\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)}\\ \end{array} \]

Alternative 15
Error	45.0
Cost	20564

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B \leq -2.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{-B}\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 16
Error	44.9
Cost	20564

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)} \cdot \left(-\sqrt{-B}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+155}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 17
Error	45.6
Cost	14476

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ \mathbf{if}\;B \leq -2.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{-B}\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-306}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 18
Error	45.1
Cost	13892

\[\begin{array}{l} \mathbf{if}\;B \leq -2.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(B \cdot B\right)\right)} \cdot \left(-\sqrt{-B}\right)}{B \cdot B}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 19
Error	46.0
Cost	13316

\[\begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 20
Error	51.2
Cost	6656

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

Alternative 21
Error	61.5
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022306 
(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))))