ABCF->ab-angle a

Average Error: 52.1 → 41.7

Time: 1.1min

Precision: binary64

Cost: 27140

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := C \cdot \left(-4 \cdot A\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := -4 \cdot \left(A \cdot C\right)\\ t_4 := t_3 + B \cdot B\\ t_5 := \sqrt{C + \left(A + t_0\right)}\\ t_6 := \frac{t_5 \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot t_1}\right)}{t_4}\\ t_7 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{+60}:\\ \;\;\;\;\frac{B \cdot \sqrt{F}}{\mathsf{fma}\left(B, B, t_3\right)} \cdot \sqrt{2 \cdot \left(\left(A + C\right) + t_0\right)}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-111}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -2.5 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{t_5 \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot C\right)\right)}\right)}{t_4}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-154}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+70}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (hypot B (- A C)))
        (t_1 (* C (* -4.0 A)))
        (t_2 (fma B B t_1))
        (t_3 (* -4.0 (* A C)))
        (t_4 (+ t_3 (* B B)))
        (t_5 (sqrt (+ C (+ A t_0))))
        (t_6
         (/
          (* t_5 (- (sqrt (+ (* (* F 2.0) (* B B)) (* (* F 2.0) t_1)))))
          t_4))
        (t_7 (sqrt (/ (- F) C))))
   (if (<= B -1.45e+60)
     (* (/ (* B (sqrt F)) (fma B B t_3)) (sqrt (* 2.0 (+ (+ A C) t_0))))
     (if (<= B -2.3e-62)
       t_6
       (if (<= B -2.1e-111)
         t_7
         (if (<= B -9.5e-186)
           (/ (- (sqrt (* 2.0 (* t_2 (* F (* A 2.0)))))) t_2)
           (if (<= B -2.5e-261)
             (/ (- (sqrt (* 2.0 (* t_2 (* F (* C 2.0)))))) t_2)
             (if (<= B 2.5e-192)
               (/ (* t_5 (- (sqrt (* -8.0 (* A (* F C)))))) t_4)
               (if (<= B 2.3e-154)
                 t_7
                 (if (<= B 2.1e+70)
                   t_6
                   (* (/ (- (sqrt 2.0)) B) (sqrt (* F (+ B A))))))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = C * (-4.0 * A);
	double t_2 = fma(B, B, t_1);
	double t_3 = -4.0 * (A * C);
	double t_4 = t_3 + (B * B);
	double t_5 = sqrt((C + (A + t_0)));
	double t_6 = (t_5 * -sqrt((((F * 2.0) * (B * B)) + ((F * 2.0) * t_1)))) / t_4;
	double t_7 = sqrt((-F / C));
	double tmp;
	if (B <= -1.45e+60) {
		tmp = ((B * sqrt(F)) / fma(B, B, t_3)) * sqrt((2.0 * ((A + C) + t_0)));
	} else if (B <= -2.3e-62) {
		tmp = t_6;
	} else if (B <= -2.1e-111) {
		tmp = t_7;
	} else if (B <= -9.5e-186) {
		tmp = -sqrt((2.0 * (t_2 * (F * (A * 2.0))))) / t_2;
	} else if (B <= -2.5e-261) {
		tmp = -sqrt((2.0 * (t_2 * (F * (C * 2.0))))) / t_2;
	} else if (B <= 2.5e-192) {
		tmp = (t_5 * -sqrt((-8.0 * (A * (F * C))))) / t_4;
	} else if (B <= 2.3e-154) {
		tmp = t_7;
	} else if (B <= 2.1e+70) {
		tmp = t_6;
	} else {
		tmp = (-sqrt(2.0) / B) * sqrt((F * (B + A)));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = Float64(C * Float64(-4.0 * A))
	t_2 = fma(B, B, t_1)
	t_3 = Float64(-4.0 * Float64(A * C))
	t_4 = Float64(t_3 + Float64(B * B))
	t_5 = sqrt(Float64(C + Float64(A + t_0)))
	t_6 = Float64(Float64(t_5 * Float64(-sqrt(Float64(Float64(Float64(F * 2.0) * Float64(B * B)) + Float64(Float64(F * 2.0) * t_1))))) / t_4)
	t_7 = sqrt(Float64(Float64(-F) / C))
	tmp = 0.0
	if (B <= -1.45e+60)
		tmp = Float64(Float64(Float64(B * sqrt(F)) / fma(B, B, t_3)) * sqrt(Float64(2.0 * Float64(Float64(A + C) + t_0))));
	elseif (B <= -2.3e-62)
		tmp = t_6;
	elseif (B <= -2.1e-111)
		tmp = t_7;
	elseif (B <= -9.5e-186)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(A * 2.0)))))) / t_2);
	elseif (B <= -2.5e-261)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * Float64(C * 2.0)))))) / t_2);
	elseif (B <= 2.5e-192)
		tmp = Float64(Float64(t_5 * Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(F * C)))))) / t_4);
	elseif (B <= 2.3e-154)
		tmp = t_7;
	elseif (B <= 2.1e+70)
		tmp = t_6;
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(F * Float64(B + A))));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 * (-N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] + N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -1.45e+60], N[(N[(N[(B * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / N[(B * B + t$95$3), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2.3e-62], t$95$6, If[LessEqual[B, -2.1e-111], t$95$7, If[LessEqual[B, -9.5e-186], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(A * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, -2.5e-261], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 2.5e-192], N[(N[(t$95$5 * (-N[Sqrt[N[(-8.0 * N[(A * N[(F * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[B, 2.3e-154], t$95$7, If[LessEqual[B, 2.1e+70], t$95$6, N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(F * N[(B + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if B < -1.45e60

   1. Initial program 58.7

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

   2. Simplified 57.9

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 12 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C)))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.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 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 8 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 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (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))))))) (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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))))))) (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2)))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (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 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 22 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around 0 58.1

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

   4. Simplified 58.0

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

   5. Applied egg-rr 61.4

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{B \cdot \sqrt{F}}{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{\sqrt{2 \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}}\right)} - 1} \]

   6. Simplified 48.3

      \[\leadsto \color{blue}{\frac{B \cdot \sqrt{F}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}} \]
      Proof
      (*.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (fma.f64 B B (*.f64 -4 (*.f64 A C)))) (sqrt.f64 (*.f64 2 (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 B B) (*.f64 -4 (*.f64 A C))))) (sqrt.f64 (*.f64 2 (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C)))))): 10 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -4 (*.f64 A C)) (*.f64 B B)))) (sqrt.f64 (*.f64 2 (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C)))))): 10 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (+.f64 (*.f64 -4 (*.f64 A C)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))) (sqrt.f64 (*.f64 2 (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (Rewrite=> fma-def_binary64 (fma.f64 -4 (*.f64 A C) (pow.f64 B 2)))) (sqrt.f64 (*.f64 2 (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (fma.f64 -4 (*.f64 A C) (Rewrite=> unpow2_binary64 (*.f64 B B)))) (sqrt.f64 (*.f64 2 (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 10 points decrease in error
      (*.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (fma.f64 -4 (*.f64 A C) (*.f64 B B))) (sqrt.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C)))))): 0 points increase in error, 10 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (*.f64 B (sqrt.f64 F)) (/.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) (sqrt.f64 (*.f64 2 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))))))): 0 points increase in error, 10 points decrease in error
      (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (/.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) (sqrt.f64 (*.f64 2 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))))))))): 10 points increase in error, 0 points decrease in error
      (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (/.f64 (*.f64 B (sqrt.f64 F)) (/.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) (sqrt.f64 (*.f64 2 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C)))))))) 1)): 0 points increase in error, 10 points decrease in error

   if -1.45e60 < B < -2.3e-62 or 2.3e-154 < B < 2.10000000000000008e70

   1. Initial program 43.6

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

   2. Simplified 43.6

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 12 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C)))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.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 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 8 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 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (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))))))) (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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))))))) (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2)))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (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 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 22 points increase in error, 0 points decrease in error

   3. Applied egg-rr 39.0

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

   4. Applied egg-rr 33.6

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

   5. Simplified 33.6

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 C (+.f64 (hypot.f64 B (-.f64 A C)) A))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 C (*.f64 A -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C (hypot.f64 B (-.f64 A C))) A))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 C (*.f64 A -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 8 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 C (*.f64 A -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 C A) -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 8 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 A C)) -4)) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 8 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F) 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F)))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F))) (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 33.6

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

   if -2.3e-62 < B < -2.0999999999999999e-111 or 2.5e-192 < B < 2.3e-154

   1. Initial program 49.3

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

   2. Simplified 49.3

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 12 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C)))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.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 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 8 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 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (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))))))) (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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))))))) (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2)))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (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 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 22 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around inf 55.1

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

   4. Applied egg-rr 61.0

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

   5. Simplified 60.9

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + A\right) + C\right)\right)\right)}{{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{2}}}} \]
      Proof
      (sqrt.f64 (/.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 F (*.f64 2 (+.f64 (+.f64 A A) C)))) (pow.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) 2))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) -4))) (*.f64 F (*.f64 2 (+.f64 (+.f64 A A) C)))) (pow.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) 2))): 6 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (*.f64 F (*.f64 2 (Rewrite<= associate-+r+_binary64 (+.f64 A (+.f64 A C)))))) (pow.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) 2))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (*.f64 F (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 A (+.f64 A C)) 2)))) (pow.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) 2))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F) (*.f64 (+.f64 A (+.f64 A C)) 2))) (pow.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) 2))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (*.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F) (*.f64 (+.f64 A (+.f64 A C)) 2)) (pow.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) -4))) 2))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in A around inf 53.3

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

   7. Simplified 53.3

      \[\leadsto \sqrt{\color{blue}{-\frac{F}{C}}} \]
      Proof
      (sqrt.f64 (neg.f64 (/.f64 F C))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 F C)))): 2 points increase in error, 0 points decrease in error

   if -2.0999999999999999e-111 < B < -9.4999999999999998e-186

   1. Initial program 51.6

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

   2. Simplified 46.6

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 12 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C)))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.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 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 8 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 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (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))))))) (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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))))))) (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2)))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (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 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 22 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around inf 48.2

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

   if -9.4999999999999998e-186 < B < -2.4999999999999999e-261

   1. Initial program 52.3

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

   2. Simplified 47.5

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 12 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C)))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.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 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 8 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 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (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))))))) (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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))))))) (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2)))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (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 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 22 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around -inf 48.1

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

   if -2.4999999999999999e-261 < B < 2.5e-192

   1. Initial program 52.6

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

   2. Simplified 52.6

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 12 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C)))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.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 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 8 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 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (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))))))) (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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))))))) (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2)))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (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 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 22 points increase in error, 0 points decrease in error

   3. Applied egg-rr 48.3

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

   4. Applied egg-rr 43.1

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

   5. Simplified 43.0

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 C (+.f64 (hypot.f64 B (-.f64 A C)) A))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 C (*.f64 A -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C (hypot.f64 B (-.f64 A C))) A))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 C (*.f64 A -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 8 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 C (*.f64 A -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 C A) -4))) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 8 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 A C)) -4)) (*.f64 F 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 8 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F) 2))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F)))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F))) (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in B around 0 45.5

      \[\leadsto \frac{-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Simplified 45.5

      \[\leadsto \frac{-\sqrt{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)} \cdot \sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(F \cdot C\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 C (+.f64 (hypot.f64 B (-.f64 A C)) A))) (sqrt.f64 (*.f64 -8 (*.f64 A (*.f64 F C)))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (sqrt.f64 (+.f64 C (+.f64 (hypot.f64 B (-.f64 A C)) A))) (sqrt.f64 (*.f64 -8 (*.f64 A (Rewrite<= *-commutative_binary64 (*.f64 C F))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 2 points increase in error, 0 points decrease in error

   if 2.10000000000000008e70 < B

   1. Initial program 58.4

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

   2. Simplified 58.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 -4 (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 12 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2)) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C)))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F) (*.f64 2 (+.f64 A (+.f64 C (hypot.f64 B (-.f64 A C)))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (+.f64 A (+.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 22 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (*.f64 2 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 -4 (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.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 -4 (*.f64 A C) (*.f64 B B))): 0 points increase in error, 8 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 -4 (*.f64 A C) (*.f64 B B))): 8 points increase in error, 14 points decrease in error
      (/.f64 (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))))))) (fma.f64 (Rewrite<= metadata-eval (neg.f64 4)) (*.f64 A C) (*.f64 B B))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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))))))) (fma.f64 (neg.f64 4) (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (neg.f64 4) (*.f64 A C)) (pow.f64 B 2)))): 22 points increase in error, 0 points decrease in error
      (/.f64 (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 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 (*.f64 A C)))) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (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 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))) (pow.f64 B 2))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 22 points decrease in error
      (/.f64 (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))))))) (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 22 points increase in error, 0 points decrease in error

   3. Taylor expanded in A around 0 58.7

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

   4. Simplified 58.7

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(\left(A + C\right) + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      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) (*.f64 C C)))))))) (-.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 (*.f64 B B) (*.f64 4 (*.f64 A C))) F) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 C C)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 3 points decrease in error
      (/.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 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 C 2))))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 3 points increase in error, 0 points decrease in error

   5. Taylor expanded in C around 0 35.4

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

   6. Simplified 35.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}} \]
      Proof
      (*.f64 (/.f64 (neg.f64 (sqrt.f64 2)) B) (sqrt.f64 (*.f64 F (+.f64 B A)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (sqrt.f64 2))) B) (sqrt.f64 (*.f64 F (+.f64 B A)))): 6 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (sqrt.f64 2) B))) (sqrt.f64 (*.f64 F (+.f64 B A)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1 (/.f64 (sqrt.f64 2) B)) (sqrt.f64 (*.f64 F (Rewrite<= +-commutative_binary64 (+.f64 A B))))): 0 points increase in error, 6 points decrease in error
      (*.f64 (*.f64 -1 (/.f64 (sqrt.f64 2) B)) (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 A B) F)))): 6 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 (+.f64 A B) F))))): 0 points increase in error, 0 points decrease in error
3. Recombined 7 regimes into one program.

4. Final simplification 41.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.45 \cdot 10^{+60}:\\ \;\;\;\;\frac{B \cdot \sqrt{F}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot \left(C \cdot \left(-4 \cdot A\right)\right)}\right)}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq -2.5 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot C\right)\right)}\right)}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-154}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+70}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot \left(C \cdot \left(-4 \cdot A\right)\right)}\right)}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	41.8
Cost	34776

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{C + \left(A + t_1\right)}\\ t_3 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{B \cdot \sqrt{F}}{\mathsf{fma}\left(B, B, t_3\right)} \cdot \sqrt{2 \cdot \left(\left(A + C\right) + t_1\right)}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot 2\right) \cdot t_0} \cdot \frac{-1}{t_0}\right) \cdot t_2\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq -8.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.22 \cdot 10^{+148}:\\ \;\;\;\;\frac{t_2 \cdot \left(\sqrt{t_0} \cdot \left(-\sqrt{F \cdot 2}\right)\right)}{t_3 + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 2
Error	42.8
Cost	34648

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := C + \left(A + t_0\right)\\ t_2 := C \cdot \left(-4 \cdot A\right)\\ t_3 := \mathsf{fma}\left(B, B, t_2\right)\\ t_4 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -2.1 \cdot 10^{+60}:\\ \;\;\;\;\frac{B \cdot \sqrt{F}}{\mathsf{fma}\left(B, B, t_4\right)} \cdot \sqrt{2 \cdot \left(\left(A + C\right) + t_0\right)}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot t_2}\right)}{t_4 + B \cdot B}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{+147}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_1} \cdot \left(-\sqrt{2 \cdot t_3}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 3
Error	42.9
Cost	34648

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{2 \cdot \left(\left(A + C\right) + t_0\right)}\\ t_2 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ \mathbf{if}\;B \leq -2 \cdot 10^{+61}:\\ \;\;\;\;\frac{B \cdot \sqrt{F}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot t_1\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{F \cdot t_2}\right)}{t_2}\\ \mathbf{elif}\;B \leq -1.65 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \left(A + t_0\right)\right)} \cdot \left(-\sqrt{2 \cdot t_3}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 4
Error	42.8
Cost	34648

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := C + \left(A + t_1\right)\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{B \cdot \sqrt{F}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{2 \cdot \left(\left(A + C\right) + t_1\right)}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;\left(\sqrt{\left(F \cdot 2\right) \cdot t_0} \cdot \frac{-1}{t_0}\right) \cdot \sqrt{t_2}\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-111}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_2} \cdot \left(-\sqrt{2 \cdot t_0}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 5
Error	40.5
Cost	27140

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := C \cdot \left(-4 \cdot A\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \frac{\sqrt{C + \left(A + t_0\right)} \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot t_1}\right)}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{F}}{\frac{t_2}{B}} \cdot \sqrt{2 \cdot \left(A + \left(C + t_0\right)\right)}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.32 \cdot 10^{-173}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.16 \cdot 10^{-195}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-150}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+70}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 6
Error	42.1
Cost	22624

\[\begin{array}{l} t_0 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_2 := \sqrt{\frac{-F}{C}}\\ t_3 := C \cdot \left(-4 \cdot A\right)\\ t_4 := \frac{t_0 \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(B \cdot B\right) + \left(F \cdot 2\right) \cdot t_3}\right)}{t_1}\\ t_5 := \mathsf{fma}\left(B, B, t_3\right)\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{t_0 \cdot \left(\sqrt{F \cdot 2} \cdot \left(B + \frac{A \cdot C}{B} \cdot -2\right)\right)}{t_1}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-193}:\\ \;\;\;\;\frac{t_0 \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot C\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+70}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 7
Error	42.1
Cost	22368

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \sqrt{C + \left(A + t_2\right)}\\ t_4 := \frac{-\sqrt{A + \left(C + t_2\right)} \cdot \sqrt{-2 \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)}}{t_0}\\ t_5 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;B \leq -1.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{t_3 \cdot \left(B \cdot \sqrt{F \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq -3.8 \cdot 10^{-62}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.62 \cdot 10^{-111}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -3.5 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.66 \cdot 10^{-192}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot C\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-147}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{+70}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 8
Error	42.1
Cost	22368

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := \frac{-\sqrt{A + \left(C + t_3\right)} \cdot \sqrt{-2 \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)}}{t_1}\\ t_5 := \sqrt{C + \left(A + t_3\right)}\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{+59}:\\ \;\;\;\;\frac{t_5 \cdot \left(\sqrt{F \cdot 2} \cdot \left(B + \frac{A \cdot C}{B} \cdot -2\right)\right)}{t_1}\\ \mathbf{elif}\;B \leq -2.3 \cdot 10^{-62}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq -1.6 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.05 \cdot 10^{-185}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -2.7 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-192}:\\ \;\;\;\;\frac{t_5 \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot C\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{+70}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 9
Error	45.0
Cost	21592

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ t_2 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;B \leq -6 \cdot 10^{-76}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F \cdot 2} \cdot \left(-B\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 10
Error	43.9
Cost	21592

\[\begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_3 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_4 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -7.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{t_3 \cdot \left(B \cdot t_0\right)}{t_2}\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-186}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{-141}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-129}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_3 \cdot \left(t_0 \cdot \left(-B\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 11
Error	43.0
Cost	21328

\[\begin{array}{l} t_0 := \sqrt{F \cdot 2}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_3 := \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\\ \mathbf{if}\;B \leq -4.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{t_3 \cdot \left(B \cdot t_0\right)}{t_2}\\ \mathbf{elif}\;B \leq -3 \cdot 10^{-218}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{t_3 \cdot \left(-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot C\right)\right)}\right)}{t_2}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{t_3 \cdot \left(t_0 \cdot \left(-B\right)\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 12
Error	46.4
Cost	21264

\[\begin{array}{l} t_0 := \frac{-\sqrt{2}}{B}\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_1}\\ t_3 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{if}\;B \leq -8 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -4.4 \cdot 10^{-200}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 10^{+115}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 13
Error	47.0
Cost	21264

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 14
Error	46.5
Cost	21016

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ t_3 := \frac{-\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -6.5 \cdot 10^{-76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 7.7 \cdot 10^{-255}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;t_3 \cdot \sqrt{F \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 15
Error	46.4
Cost	20884

\[\begin{array}{l} t_0 := \frac{-\sqrt{2}}{B}\\ t_1 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot t_1\right)\right)}}{t_1}\\ t_3 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -6 \cdot 10^{-76}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{+69}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 16
Error	45.8
Cost	15704

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ t_2 := \frac{-\sqrt{-2 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-76}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.56 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 17
Error	48.5
Cost	14276

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{-54}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)}}{B \cdot B}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 18
Error	49.1
Cost	13968

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot t_1\right)}}{t_0}\\ \mathbf{if}\;B \leq -9 \cdot 10^{-54}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.48 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-300}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(B + A\right)}\right)}{B}\\ \end{array} \]

Alternative 19
Error	49.1
Cost	13968

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot t_1\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.56 \cdot 10^{-52}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(B + A\right)}\\ \end{array} \]

Alternative 20
Error	48.8
Cost	13712

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot t_1\right)}}{t_0}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 21
Error	53.6
Cost	8848

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot t_1\right)}}{t_0}\\ \mathbf{if}\;B \leq -1 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -5.7 \cdot 10^{-257}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(C + \left(A + C\right)\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	53.4
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -1.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 23
Error	53.4
Cost	8456

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(B + A\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 24
Error	56.4
Cost	8324

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \end{array} \]

Alternative 25
Error	54.7
Cost	8324

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot 2\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(B + A\right) \cdot \left(F \cdot \left(\left(A \cdot C\right) \cdot 4 - B \cdot B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 26
Error	57.7
Cost	8204

\[\begin{array}{l} t_0 := \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{+64}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot 8\right)\right)}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot t_0\\ \end{array} \]

Alternative 27
Error	56.7
Cost	7112

\[\begin{array}{l} t_0 := \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{+63}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot t_0\\ \end{array} \]

Alternative 28
Error	57.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{F \cdot C}}{B}\\ \end{array} \]

Alternative 29
Error	56.7
Cost	6656

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

Alternative 30
Error	63.2
Cost	6592

\[\sqrt{\frac{F}{C}} \]

Reproduce

herbie shell --seed 2022340 
(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))))