ABCF->ab-angle a

Average Error: 51.9 → 33.6

Time: 1.1min

Precision: binary64

Cost: 209168

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

Math

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

↓

\[\begin{array}{l} t_0 := C \cdot \left(A \cdot -4\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := {B}^{2} + t_0\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_3}\\ t_5 := A \cdot \left(C \cdot -4\right)\\ t_6 := \mathsf{hypot}\left(B, \sqrt{t_5}\right)\\ t_7 := \mathsf{fma}\left(B, B, t_5\right)\\ t_8 := \frac{\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, t_0\right)} \cdot \left(-\sqrt{F}\right)\right)}{t_7}\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{+245}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;t_4 \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A + t_2\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;t_8\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right) \cdot \left(-C\right)}{t_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + t_2}}{t_6} \cdot \frac{\sqrt{2 \cdot F}}{\frac{t_6}{B}}\\ \end{array} \]

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 (* C (* A -4.0)))
        (t_1 (fma C (* A -4.0) (* B B)))
        (t_2 (hypot B (- A C)))
        (t_3 (+ (pow B 2.0) t_0))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_3))
        (t_5 (* A (* C -4.0)))
        (t_6 (hypot B (sqrt t_5)))
        (t_7 (fma B B t_5))
        (t_8
         (/
          (*
           (sqrt (+ C (+ C (* -0.5 (/ (* B B) A)))))
           (* (sqrt (* 2.0 (fma B B t_0))) (- (sqrt F))))
          t_7)))
   (if (<= t_4 -2e+245)
     t_8
     (if (<= t_4 -2e-206)
       (/ (* (sqrt t_1) (- (sqrt (* 2.0 (* F (+ C (+ A t_2))))))) t_1)
       (if (<= t_4 0.0)
         t_8
         (if (<= t_4 INFINITY)
           (/ (* (* (sqrt (* F -16.0)) (sqrt A)) (- C)) t_7)
           (*
            (/ (sqrt (+ (+ A C) t_2)) t_6)
            (/ (sqrt (* 2.0 F)) (/ t_6 B)))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = C * (A * -4.0);
	double t_1 = fma(C, (A * -4.0), (B * B));
	double t_2 = hypot(B, (A - C));
	double t_3 = pow(B, 2.0) + t_0;
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double t_5 = A * (C * -4.0);
	double t_6 = hypot(B, sqrt(t_5));
	double t_7 = fma(B, B, t_5);
	double t_8 = (sqrt((C + (C + (-0.5 * ((B * B) / A))))) * (sqrt((2.0 * fma(B, B, t_0))) * -sqrt(F))) / t_7;
	double tmp;
	if (t_4 <= -2e+245) {
		tmp = t_8;
	} else if (t_4 <= -2e-206) {
		tmp = (sqrt(t_1) * -sqrt((2.0 * (F * (C + (A + t_2)))))) / t_1;
	} else if (t_4 <= 0.0) {
		tmp = t_8;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = ((sqrt((F * -16.0)) * sqrt(A)) * -C) / t_7;
	} else {
		tmp = (sqrt(((A + C) + t_2)) / t_6) * (sqrt((2.0 * F)) / (t_6 / B));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(C * Float64(A * -4.0))
	t_1 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_2 = hypot(B, Float64(A - C))
	t_3 = Float64((B ^ 2.0) + t_0)
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_3)
	t_5 = Float64(A * Float64(C * -4.0))
	t_6 = hypot(B, sqrt(t_5))
	t_7 = fma(B, B, t_5)
	t_8 = Float64(Float64(sqrt(Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A))))) * Float64(sqrt(Float64(2.0 * fma(B, B, t_0))) * Float64(-sqrt(F)))) / t_7)
	tmp = 0.0
	if (t_4 <= -2e+245)
		tmp = t_8;
	elseif (t_4 <= -2e-206)
		tmp = Float64(Float64(sqrt(t_1) * Float64(-sqrt(Float64(2.0 * Float64(F * Float64(C + Float64(A + t_2))))))) / t_1);
	elseif (t_4 <= 0.0)
		tmp = t_8;
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(F * -16.0)) * sqrt(A)) * Float64(-C)) / t_7);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(A + C) + t_2)) / t_6) * Float64(sqrt(Float64(2.0 * F)) / Float64(t_6 / B)));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[B ^ 2 + N[Sqrt[t$95$5], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$7 = N[(B * B + t$95$5), $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[Sqrt[N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+245], t$95$8, If[LessEqual[t$95$4, -2e-206], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * N[(C + N[(A + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.0], t$95$8, If[LessEqual[t$95$4, Infinity], N[(N[(N[(N[Sqrt[N[(F * -16.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[A], $MachinePrecision]), $MachinePrecision] * (-C)), $MachinePrecision] / t$95$7), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / N[(t$95$6 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.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 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2.00000000000000009e245 or -2.00000000000000006e-206 < (/.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 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -0.0

   1. Initial program 62.5

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

   2. Simplified 54.3

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 37 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 2 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 47.6

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

   4. Simplified 45.4

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 43 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (hypot.f64 B (-.f64 A C)))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C)))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (Rewrite<= *-commutative_binary64 (*.f64 -4 A)) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 -4 A) C (Rewrite<= unpow2_binary64 (pow.f64 B 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 -4 A) C) (pow.f64 B 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 A C))) (pow.f64 B 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 A C) -4)) (pow.f64 B 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4))) (pow.f64 B 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 A (*.f64 C -4)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite=> unpow2_binary64 (*.f64 B B)) (*.f64 A (*.f64 C -4))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (Rewrite<= fma-udef_binary64 (fma.f64 B B (*.f64 A (*.f64 C -4)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (fma.f64 B B (*.f64 A (*.f64 C -4)))) F)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F))))): 0 points increase in error, 1 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F)) 1/2) (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F)))) (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 35.9

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

   6. Taylor expanded in A around -inf 27.4

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

   7. Simplified 27.4

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

   if -2.00000000000000009e245 < (/.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 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2.00000000000000006e-206

   1. Initial program 1.6

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

   2. Simplified 2.1

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 F 2)) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 48 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 2 points increase in error, 5 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F))) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 1.8

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

   if -0.0 < (/.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 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 37.5

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

   2. Simplified 24.0

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 37 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 2 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in C around inf 16.3

      \[\leadsto \frac{-\color{blue}{C \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   4. Simplified 16.5

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

   5. Applied egg-rr 8.6

      \[\leadsto \frac{-C \cdot \color{blue}{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if +inf.0 < (/.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 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 64.0

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

   2. Simplified 63.0

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 37 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 C (+.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 2 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 64.0

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

   4. Simplified 64.0

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C A) (hypot.f64 B (-.f64 A C))))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 43 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (hypot.f64 B (-.f64 A C)))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C)))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 A -4) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (Rewrite<= *-commutative_binary64 (*.f64 -4 A)) C (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (fma.f64 (*.f64 -4 A) C (Rewrite<= unpow2_binary64 (pow.f64 B 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 -4 A) C) (pow.f64 B 2))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 A C))) (pow.f64 B 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 A C) -4)) (pow.f64 B 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4))) (pow.f64 B 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 A (*.f64 C -4)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (+.f64 (Rewrite=> unpow2_binary64 (*.f64 B B)) (*.f64 A (*.f64 C -4))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (*.f64 F (*.f64 2 (Rewrite<= fma-udef_binary64 (fma.f64 B B (*.f64 A (*.f64 C -4)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (fma.f64 B B (*.f64 A (*.f64 C -4)))) F)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F))))): 0 points increase in error, 1 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F)) 1/2) (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) F)))) (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in B around inf 63.0

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

   6. Applied egg-rr 56.9

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

   7. Simplified 54.6

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\sqrt{2 \cdot F}}{\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{B}}} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 A C))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))) (/.f64 (sqrt.f64 (*.f64 2 F)) (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) B))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (Rewrite<= +-commutative_binary64 (+.f64 C A)))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))) (/.f64 (sqrt.f64 (*.f64 2 F)) (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) B))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (hypot.f64 B (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 C -4) A))))) (/.f64 (sqrt.f64 (*.f64 2 F)) (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) B))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 C (*.f64 -4 A)))))) (/.f64 (sqrt.f64 (*.f64 2 F)) (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) B))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A))))) (/.f64 (sqrt.f64 (*.f64 2 F)) (/.f64 (hypot.f64 B (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 C -4) A)))) B))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A))))) (/.f64 (sqrt.f64 (*.f64 2 F)) (/.f64 (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 C (*.f64 -4 A))))) B))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A))))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (*.f64 2 F)) B) (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A))))))): 14 points increase in error, 5 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 (+.f64 (hypot.f64 B (-.f64 A C)) (+.f64 C A))) (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A))))) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 B (sqrt.f64 (*.f64 2 F)))) (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A)))))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 33.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -2 \cdot 10^{+245}:\\ \;\;\;\;\frac{\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq 0:\\ \;\;\;\;\frac{\sqrt{C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} + C \cdot \left(A \cdot -4\right)} \leq \infty:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right) \cdot \left(-C\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \frac{\sqrt{2 \cdot F}}{\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)}{B}}\\ \end{array} \]

Alternatives

Alternative 1
Error	35.0
Cost	183120

\[\begin{array}{l} t_0 := C \cdot \left(A \cdot -4\right)\\ t_1 := \frac{B \cdot B}{A}\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_3 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_4 := {B}^{2} + t_0\\ t_5 := \frac{-\sqrt{\left(2 \cdot \left(t_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_4}\\ t_6 := \frac{\sqrt{C + \left(C + -0.5 \cdot t_1\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, t_0\right)} \cdot \left(-\sqrt{F}\right)\right)}{t_3}\\ \mathbf{if}\;t_5 \leq -2 \cdot 10^{+245}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t_5 \leq -2 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{t_2} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\right)}{t_2}\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right) \cdot \left(-C\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{C + \mathsf{fma}\left(-0.5, t_1, C\right)}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{2 \cdot F}}}\\ \end{array} \]

Alternative 2
Error	37.4
Cost	34512

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_3 := C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\\ t_4 := \frac{\sqrt{t_3} \cdot \left(\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right) \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{t_2}\\ \mathbf{if}\;A \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot t_2\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;A \leq -2.35 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{t_0} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_1\right)}\right)}{t_0}\\ \mathbf{elif}\;A \leq -8.4 \cdot 10^{-109}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)\right)}\right)}{t_2}\\ \end{array} \]

Alternative 3
Error	37.4
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -2.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot F\right)}}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)\right)}\right)}{t_0}\\ \end{array} \]

Alternative 4
Error	41.5
Cost	28112

\[\begin{array}{l} t_0 := \frac{B \cdot B}{A}\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;C \leq -4.4 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{C + \mathsf{fma}\left(-0.5, t_0, C\right)}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{2 \cdot F}}}\\ \mathbf{elif}\;C \leq -5.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \left(C - t_0 \cdot 0.5\right)\right) \cdot \left(F \cdot t_2\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 5.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{t_1}\right) \cdot \left(-\sqrt{F \cdot \left(4 \cdot C - B \cdot \frac{B}{A}\right)}\right)}{t_3}\\ \mathbf{elif}\;C \leq 0.21:\\ \;\;\;\;\sqrt{t_3 \cdot \left(2 \cdot \left(F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)} \cdot \frac{1}{-t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)\right)} \cdot \left(-\sqrt{C + C}\right)}{t_2}\\ \end{array} \]

Alternative 5
Error	41.5
Cost	27984

\[\begin{array}{l} t_0 := \frac{B \cdot B}{A}\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B, B, t_1\right)\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;C \leq -6 \cdot 10^{-132}:\\ \;\;\;\;\sqrt{\frac{C + \mathsf{fma}\left(-0.5, t_0, C\right)}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{2 \cdot F}}}\\ \mathbf{elif}\;C \leq -5.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \left(C - t_0 \cdot 0.5\right)\right) \cdot \left(F \cdot t_2\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 2.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{t_1}\right) \cdot \left(-\sqrt{F \cdot \left(4 \cdot C - B \cdot \frac{B}{A}\right)}\right)}{t_3}\\ \mathbf{elif}\;C \leq 0.9:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)\right)} \cdot \left(-\sqrt{C + C}\right)}{t_2}\\ \end{array} \]

Alternative 6
Error	37.5
Cost	27848

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -2.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot F\right)}}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 7
Error	40.6
Cost	27468

\[\begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ t_1 := \sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1 \cdot 10^{-48}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \frac{B}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\\ \mathbf{elif}\;B \leq 1.22 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(F \cdot \left(4 \cdot C - \frac{B \cdot B}{A}\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t_0 \cdot \left(-B\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \end{array} \]

Alternative 8
Error	41.2
Cost	27212

\[\begin{array}{l} t_0 := \sqrt{2 \cdot F}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \left(t_0 \cdot \frac{B}{t_1}\right)\\ \mathbf{elif}\;B \leq 6.3 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(F \cdot \left(4 \cdot C - \frac{B \cdot B}{A}\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \left(\left(-B\right) \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)}{t_1}\\ \end{array} \]

Alternative 9
Error	46.9
Cost	22444

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := \frac{-\sqrt{t_1 \cdot \left(F \cdot \left(4 \cdot C - \frac{B \cdot B}{A}\right)\right)}}{t_1}\\ t_3 := C \cdot \frac{-\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{if}\;F \leq -3.2 \cdot 10^{-268}:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right) \cdot \left(-C\right)}{t_0}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{-160}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 135000000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+128}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+148}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 1.12 \cdot 10^{+170}:\\ \;\;\;\;\frac{C \cdot \left(\sqrt{A \cdot -16} \cdot \left(-\sqrt{F}\right)\right)}{t_0}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot \left(4 \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+187}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+194}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(B \cdot \left(F \cdot -2\right)\right)}}{t_1}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{+301}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 10
Error	41.5
Cost	21516

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;A \leq -3.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(F \cdot \left(4 \cdot C - \frac{B \cdot B}{A}\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.25 \cdot 10^{-63}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot F\right)}}{\frac{t_0}{C}}\\ \mathbf{elif}\;A \leq 4.1 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot t_0\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right) \cdot \left(-C\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \end{array} \]

Alternative 11
Error	41.7
Cost	21516

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -6.6 \cdot 10^{+81}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right) \cdot \left(F \cdot t_1\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.25 \cdot 10^{-63}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot F\right)}}{\frac{t_0}{C}}\\ \mathbf{elif}\;A \leq 3.2 \cdot 10^{-245}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot t_0\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right) \cdot \left(-C\right)}{t_1}\\ \end{array} \]

Alternative 12
Error	42.7
Cost	21068

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-48}:\\ \;\;\;\;B \cdot \frac{t_1}{t_0}\\ \mathbf{elif}\;B \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(C \cdot \left(4 \cdot F\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{B \cdot \left(-t_1\right)}{t_0}\\ \end{array} \]

Alternative 13
Error	42.7
Cost	21068

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_2 := \left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{B \cdot \sqrt{2 \cdot \left(F \cdot t_2\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot \left(4 \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{B \cdot \left(-\sqrt{t_2 \cdot \left(2 \cdot F\right)}\right)}{t_0}\\ \end{array} \]

Alternative 14
Error	43.0
Cost	21004

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-49}:\\ \;\;\;\;B \cdot \frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(2 \cdot F\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-221}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(C \cdot \left(4 \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{2 \cdot F} \cdot \left(-B\right)\right) \cdot \sqrt{B + C}}{t_0}\\ \end{array} \]

Alternative 15
Error	45.6
Cost	20752

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -3.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{C \cdot \left(\sqrt{A \cdot -16} \cdot \left(-\sqrt{F}\right)\right)}{t_0}\\ \mathbf{elif}\;A \leq -2.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A \cdot \left(\left(C \cdot \left(C \cdot F\right)\right) \cdot -8\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -1.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot F\right)}}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{C}}\\ \mathbf{elif}\;A \leq 1.46 \cdot 10^{-256}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot -16} \cdot \sqrt{A}\right) \cdot \left(-C\right)}{t_0}\\ \end{array} \]

Alternative 16
Error	44.6
Cost	20488

\[\begin{array}{l} \mathbf{if}\;C \leq -2.5 \cdot 10^{-292}:\\ \;\;\;\;\sqrt{\frac{F \cdot \mathsf{fma}\left(C, 4, -\frac{B \cdot B}{A}\right)}{\mathsf{fma}\left(C \cdot -4, A, B \cdot B\right)}}\\ \mathbf{elif}\;C \leq 3.1 \cdot 10^{-20}:\\ \;\;\;\;\frac{C \cdot \left(\sqrt{A \cdot -16} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq 3.3 \cdot 10^{+38}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \end{array} \]

Alternative 17
Error	45.8
Cost	20356

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \sqrt{-16 \cdot \left(A \cdot F\right)}\\ \mathbf{if}\;A \leq -2.95 \cdot 10^{+122}:\\ \;\;\;\;\frac{C \cdot \left(\sqrt{A \cdot -16} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq -9.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A \cdot \left(\left(C \cdot \left(C \cdot F\right)\right) \cdot -8\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-t_2}{\frac{t_0}{C}}\\ \mathbf{elif}\;A \leq 4.8 \cdot 10^{-221}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{-C}{t_0}\\ \end{array} \]

Alternative 18
Error	45.5
Cost	14536

\[\begin{array}{l} t_0 := \frac{-\sqrt{2 \cdot \left(\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	47.4
Cost	14088

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -3.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right) \cdot \left(\left(B + \frac{0.5}{\frac{B}{A \cdot A}}\right) - \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;C \cdot \frac{-\sqrt{A \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	46.9
Cost	14088

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right) \cdot \left(\left(B + \frac{0.5}{\frac{B}{A \cdot A}}\right) - \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;\sqrt{-16 \cdot \left(A \cdot F\right)} \cdot \frac{-C}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 21
Error	47.5
Cost	8964

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right) \cdot \left(\left(B + \frac{0.5}{\frac{B}{A \cdot A}}\right) - \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(-C\right) \cdot \sqrt{A \cdot \left(F \cdot -16\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	47.6
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B\right) \cdot \left(F \cdot \left(C + \left(\frac{-0.5 \cdot \left(C \cdot C\right)}{B} - B\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(-C\right) \cdot \sqrt{A \cdot \left(F \cdot -16\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 23
Error	47.5
Cost	8584

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\left(-C\right) \cdot \sqrt{A \cdot \left(F \cdot -16\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 24
Error	46.1
Cost	7940

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;C \leq 8 \cdot 10^{-283}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(A \cdot \left(\left(C \cdot \left(C \cdot F\right)\right) \cdot -8\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-C\right) \cdot \sqrt{A \cdot \left(F \cdot -16\right)}}{t_0}\\ \end{array} \]

Alternative 25
Error	63.0
Cost	7552

\[\frac{\sqrt{C \cdot \left(F \cdot -16\right)} \cdot \left(-A\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]

Alternative 26
Error	49.1
Cost	7552

\[\frac{\left(-C\right) \cdot \sqrt{A \cdot \left(F \cdot -16\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)} \]

Alternative 27
Error	63.2
Cost	7296

\[\frac{-\sqrt{C \cdot \left(\left(F \cdot -16\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B} \]

Alternative 28
Error	63.0
Cost	7296

\[\frac{\sqrt{C \cdot \left(F \cdot -16\right)} \cdot \left(-A\right)}{A \cdot \left(C \cdot -4\right)} \]

Alternative 29
Error	63.2
Cost	7104

\[\sqrt{C \cdot \left(F \cdot -16\right)} \cdot \frac{A}{B \cdot B} \]

Alternative 30
Error	63.1
Cost	7104

\[\frac{\sqrt{C \cdot \left(F \cdot -16\right)}}{B} \cdot \frac{A}{B} \]

Alternative 31
Error	63.1
Cost	7104

\[\frac{\frac{A \cdot \sqrt{C \cdot \left(F \cdot -16\right)}}{B}}{B} \]

Reproduce

herbie shell --seed 2022335 
(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))))