ABCF->ab-angle a

Average Error: 52.1 → 29.0

Time: 56.6s

Precision: binary64

Cost: 41180

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_3 := \sqrt{t_2 \cdot \left(2 \cdot F\right)}\\ t_4 := \frac{t_3 \cdot \left(-\sqrt{t_1}\right)}{t_2}\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ t_6 := \sqrt{C + \left(A + t_5\right)}\\ t_7 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{t_5 + \left(A + C\right)} \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_7\right) \cdot t_1\right)}}{t_7}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{t_6 \cdot \left(-t_3\right)}{t_2}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-127}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-51}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 13.5:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{t_6 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2 \cdot t_2}\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

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 2.0) B))
        (t_1 (+ C (+ C (* -0.5 (/ (* B B) A)))))
        (t_2 (fma C (* A -4.0) (* B B)))
        (t_3 (sqrt (* t_2 (* 2.0 F))))
        (t_4 (/ (* t_3 (- (sqrt t_1))) t_2))
        (t_5 (hypot B (- A C)))
        (t_6 (sqrt (+ C (+ A t_5))))
        (t_7 (fma B B (* A (* C -4.0)))))
   (if (<= B -3.05e+88)
     (* (sqrt (+ t_5 (+ A C))) (* t_0 (sqrt F)))
     (if (<= B -1.1e+74)
       (/ (- (sqrt (* 2.0 (* (* F t_7) t_1)))) t_7)
       (if (<= B -8.2e-86)
         (/ (* t_6 (- t_3)) t_2)
         (if (<= B 3.1e-127)
           t_4
           (if (<= B 1.95e-51)
             (/ (- (sqrt (* t_2 (* (* 2.0 F) (* C 2.0))))) t_2)
             (if (<= B 13.5)
               t_4
               (if (<= B 6.4e+131)
                 (/ (* t_6 (* (sqrt F) (- (sqrt (* 2.0 t_2))))) t_2)
                 (* t_0 (* (sqrt (+ B C)) (- (sqrt F)))))))))))))

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(2.0) / B;
	double t_1 = C + (C + (-0.5 * ((B * B) / A)));
	double t_2 = fma(C, (A * -4.0), (B * B));
	double t_3 = sqrt((t_2 * (2.0 * F)));
	double t_4 = (t_3 * -sqrt(t_1)) / t_2;
	double t_5 = hypot(B, (A - C));
	double t_6 = sqrt((C + (A + t_5)));
	double t_7 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= -3.05e+88) {
		tmp = sqrt((t_5 + (A + C))) * (t_0 * sqrt(F));
	} else if (B <= -1.1e+74) {
		tmp = -sqrt((2.0 * ((F * t_7) * t_1))) / t_7;
	} else if (B <= -8.2e-86) {
		tmp = (t_6 * -t_3) / t_2;
	} else if (B <= 3.1e-127) {
		tmp = t_4;
	} else if (B <= 1.95e-51) {
		tmp = -sqrt((t_2 * ((2.0 * F) * (C * 2.0)))) / t_2;
	} else if (B <= 13.5) {
		tmp = t_4;
	} else if (B <= 6.4e+131) {
		tmp = (t_6 * (sqrt(F) * -sqrt((2.0 * t_2)))) / t_2;
	} else {
		tmp = t_0 * (sqrt((B + C)) * -sqrt(F));
	}
	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(2.0) / B)
	t_1 = Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))
	t_2 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_3 = sqrt(Float64(t_2 * Float64(2.0 * F)))
	t_4 = Float64(Float64(t_3 * Float64(-sqrt(t_1))) / t_2)
	t_5 = hypot(B, Float64(A - C))
	t_6 = sqrt(Float64(C + Float64(A + t_5)))
	t_7 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= -3.05e+88)
		tmp = Float64(sqrt(Float64(t_5 + Float64(A + C))) * Float64(t_0 * sqrt(F)));
	elseif (B <= -1.1e+74)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_7) * t_1)))) / t_7);
	elseif (B <= -8.2e-86)
		tmp = Float64(Float64(t_6 * Float64(-t_3)) / t_2);
	elseif (B <= 3.1e-127)
		tmp = t_4;
	elseif (B <= 1.95e-51)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * Float64(Float64(2.0 * F) * Float64(C * 2.0))))) / t_2);
	elseif (B <= 13.5)
		tmp = t_4;
	elseif (B <= 6.4e+131)
		tmp = Float64(Float64(t_6 * Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 * t_2))))) / t_2);
	else
		tmp = Float64(t_0 * Float64(sqrt(Float64(B + C)) * Float64(-sqrt(F))));
	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[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(t$95$2 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 * (-N[Sqrt[t$95$1], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(C + N[(A + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -3.05e+88], N[(N[Sqrt[N[(t$95$5 + N[(A + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -1.1e+74], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$7), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$7), $MachinePrecision], If[LessEqual[B, -8.2e-86], N[(N[(t$95$6 * (-t$95$3)), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[B, 3.1e-127], t$95$4, If[LessEqual[B, 1.95e-51], N[((-N[Sqrt[N[(t$95$2 * N[(N[(2.0 * F), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 13.5], t$95$4, If[LessEqual[B, 6.4e+131], N[(N[(t$95$6 * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[N[(B + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -3.0499999999999999e88

   1. Initial program 59.9

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

   2. Simplified 59.2

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 F 2)) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 47 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 3 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B))): 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 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B))): 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 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B))): 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 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 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-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2)))): 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<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2))): 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<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.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<= cancel-sign-sub-inv_binary64 (-.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 (-.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 56.8

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

   4. Simplified 56.8

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 56.7

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

   6. Simplified 56.7

      \[\leadsto \color{blue}{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \frac{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 F (fma.f64 C (*.f64 A -4) (*.f64 B B)))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (Rewrite<= +-commutative_binary64 (+.f64 C A)))) (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 F (fma.f64 C (*.f64 A -4) (*.f64 B B)))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 F) (fma.f64 C (*.f64 A -4) (*.f64 B B)))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 3 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))) 1)) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))) (/.f64 1 (fma.f64 C (*.f64 A -4) (*.f64 B B)))))): 16 points increase in error, 4 points decrease in error

   7. Taylor expanded in B around -inf 21.7

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

   if -3.0499999999999999e88 < B < -1.1000000000000001e74

   1. Initial program 47.9

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

   2. Simplified 43.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 37 points increase in error, 1 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)))): 19 points increase in error, 1 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, 1 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. Taylor expanded in A around -inf 55.7

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

   4. Simplified 55.7

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

   if -1.1000000000000001e74 < B < -8.19999999999999959e-86

   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

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 F 2)) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 47 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 3 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B))): 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 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B))): 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 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B))): 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 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 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-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2)))): 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<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2))): 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<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.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<= cancel-sign-sub-inv_binary64 (-.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 (-.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 32.5

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

   4. Simplified 32.5

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error

   if -8.19999999999999959e-86 < B < 3.1e-127 or 1.9499999999999999e-51 < B < 13.5

   1. Initial program 50.8

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

   2. Simplified 46.1

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 F 2)) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 47 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 3 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B))): 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 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B))): 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 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B))): 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 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 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-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2)))): 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<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2))): 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<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.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<= cancel-sign-sub-inv_binary64 (-.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 (-.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 40.1

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

   4. Simplified 40.1

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in A around -inf 35.9

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

   6. Simplified 35.9

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

   if 3.1e-127 < B < 1.9499999999999999e-51

   1. Initial program 45.6

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

   2. Simplified 40.3

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 F 2)) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 47 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 3 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B))): 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 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B))): 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 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B))): 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 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 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-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2)))): 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<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2))): 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<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.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<= cancel-sign-sub-inv_binary64 (-.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 (-.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 34.4

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

   if 13.5 < B < 6.4000000000000004e131

   1. Initial program 44.0

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

   2. Simplified 40.1

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 F 2)) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 47 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 3 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A -4) (*.f64 B B))): 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 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B))): 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 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B))): 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 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B))): 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 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 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-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2)))): 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<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2))): 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<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.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<= cancel-sign-sub-inv_binary64 (-.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 (-.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 32.1

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

   4. Simplified 32.1

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)) 1/2) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 F)))) (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 26.6

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

   if 6.4000000000000004e131 < B

   1. Initial program 62.6

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

   2. Simplified 62.6

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C))) F) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (*.f64 B B) (pow.f64 (-.f64 A C) 2)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 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 4 (*.f64 A C))) F) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (*.f64 B B) (pow.f64 (-.f64 A C) 2)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) F) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (*.f64 B B) (pow.f64 (-.f64 A C) 2)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 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 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (pow.f64 (-.f64 A C) 2)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 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 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 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)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 1 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))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 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))) (-.f64 (*.f64 B B) (*.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)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.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)) (-.f64 (pow.f64 B 2) (Rewrite<= associate-*l*_binary64 (*.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 B around inf 62.7

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

   4. Taylor expanded in A around 0 32.5

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

   5. Simplified 32.5

      \[\leadsto \color{blue}{\sqrt{F \cdot \left(C + B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (*.f64 F (+.f64 C B))) (neg.f64 (/.f64 (sqrt.f64 2) B))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 C B) F))) (neg.f64 (/.f64 (sqrt.f64 2) B))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (*.f64 (+.f64 C B) F)) (/.f64 (sqrt.f64 2) B)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 (+.f64 C B) 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 B) F))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 14.6

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

4. Final simplification 29.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.05 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-51}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 13.5:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(2 \cdot F\right)} \cdot \left(-\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\right)\right)}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	28.8
Cost	41180

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_3 := \sqrt{t_2 \cdot \left(2 \cdot F\right)}\\ t_4 := \frac{t_3 \cdot \left(-\sqrt{t_1}\right)}{t_2}\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ t_6 := \sqrt{t_5 + \left(A + C\right)}\\ t_7 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+88}:\\ \;\;\;\;t_6 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{+74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_7\right) \cdot t_1\right)}}{t_7}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + t_5\right)} \cdot \left(-t_3\right)}{t_2}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-127}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-51}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.3:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{+131}:\\ \;\;\;\;t_6 \cdot \left(-\frac{\sqrt{t_2} \cdot \sqrt{2 \cdot F}}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 2
Error	28.8
Cost	34252

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_3 := \sqrt{t_2 \cdot \left(2 \cdot F\right)}\\ t_4 := \frac{t_3 \cdot \left(-\sqrt{t_1}\right)}{t_2}\\ t_5 := \mathsf{hypot}\left(B, A - C\right)\\ t_6 := \sqrt{t_5 + \left(A + C\right)}\\ t_7 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_8 := -\sqrt{F}\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{+88}:\\ \;\;\;\;t_6 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{+74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_7\right) \cdot t_1\right)}}{t_7}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + t_5\right)} \cdot \left(-t_3\right)}{t_2}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-127}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 14.5:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.15 \cdot 10^{+186}:\\ \;\;\;\;t_6 \cdot \left(t_0 \cdot t_8\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{B + C} \cdot t_8\right)\\ \end{array} \]

Alternative 3
Error	28.9
Cost	28440

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\\ t_5 := -\sqrt{F}\\ t_6 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}\\ t_7 := \frac{\sqrt{t_2 \cdot \left(2 \cdot F\right)} \cdot \left(-\sqrt{t_4}\right)}{t_2}\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+88}:\\ \;\;\;\;t_6 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{+74}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot t_4\right)}}{t_3}\\ \mathbf{elif}\;B \leq -7.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_1\right)\right)} \cdot \left(-t_6\right)}{B \cdot B + t_1}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-128}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.7:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq 3.85 \cdot 10^{+186}:\\ \;\;\;\;t_6 \cdot \left(t_0 \cdot t_5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{B + C} \cdot t_5\right)\\ \end{array} \]

Alternative 4
Error	31.2
Cost	28048

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := -\frac{\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)\right)}}{t_1}\\ t_3 := \frac{\sqrt{t_1 \cdot \left(2 \cdot F\right)} \cdot \left(-\sqrt{C + C}\right)}{t_1}\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-157}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.06 \cdot 10^{-68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{+32}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 5
Error	31.2
Cost	27532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \left(t_1 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1 \cdot 10^{-101}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(2 \cdot F\right)} \cdot \left(-\sqrt{C + C}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 6
Error	31.8
Cost	26892

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)} \cdot \left(t_0 \cdot \sqrt{F}\right)\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.05 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+33}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 7
Error	38.6
Cost	21380

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+32}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 8
Error	38.8
Cost	20868

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq 1.95 \cdot 10^{+33}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 9
Error	38.8
Cost	20868

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq 1.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 10
Error	42.0
Cost	20236

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.55 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 11
Error	49.1
Cost	15444

\[\begin{array}{l} t_0 := \frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(C, B\right)\right)\right)}}{t_1}\\ t_3 := -\sqrt{2}\\ \mathbf{if}\;F \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-81}:\\ \;\;\;\;\frac{t_3}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.95 \cdot 10^{+253}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	47.9
Cost	15312

\[\begin{array}{l} t_0 := \frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := F \cdot t_1\\ \mathbf{if}\;C \leq -3.3 \cdot 10^{-196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.16 \cdot 10^{-169}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;C \leq 3.3 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 2.45 \cdot 10^{+45}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A, B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 13
Error	47.9
Cost	14864

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{if}\;C \leq -2.4 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \mathbf{elif}\;C \leq 3.3 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 2.45 \cdot 10^{+45}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left|B \cdot \left(B \cdot \left(F \cdot \left(B + \left(A + C\right)\right)\right)\right)\right|}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Error	47.8
Cost	14084

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 1.55 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 370000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]

Alternative 15
Error	48.3
Cost	13572

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 132000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + C\right)}\\ \end{array} \]

Alternative 16
Error	48.5
Cost	13444

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 310000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \end{array} \]

Alternative 17
Error	48.1
Cost	13316

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq 114000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 18
Error	59.2
Cost	8320

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0} \end{array} \]

Alternative 19
Error	54.3
Cost	8320

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0} \end{array} \]

Alternative 20
Error	59.6
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := B + \left(A + C\right)\\ \mathbf{if}\;B \leq 2.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot t_1\right)}}{B \cdot B}\\ \end{array} \]

Alternative 21
Error	60.6
Cost	7808

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

Alternative 22
Error	60.6
Cost	7552

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

Alternative 23
Error	62.9
Cost	7104

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

Alternative 24
Error	63.0
Cost	6976

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

Alternative 25
Error	63.0
Cost	6976

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

Alternative 26
Error	63.4
Cost	6848

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

Reproduce

herbie shell --seed 2022329 
(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))))