Average Error: 52.4 → 40.8
Time: 48.4s
Precision: binary64
Cost: 27408
\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
↓
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ t_3 := \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_0\right)\right)}}{B}\\ \mathbf{if}\;B \leq -6 \cdot 10^{-41}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-191}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-234}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right) \cdot \left(\sqrt{A \cdot F} \cdot -2\right)}{t_1}\\ \mathbf{elif}\;B \leq -4.7 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(C + A\right) - t_0\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
(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 (fma B B (* A (* C -4.0))))
        (t_2
         (/
          (- (sqrt (* 2.0 (* -4.0 (* (* C A) (* F (+ C C)))))))
          (+ (* B B) (* -4.0 (* C A)))))
        (t_3 (/ (sqrt (* (* 2.0 F) (+ C (- A t_0)))) B)))
   (if (<= B -6e-41)
     t_3
     (if (<= B -4.5e-139)
       t_2
       (if (<= B -6.8e-191)
         t_3
         (if (<= B -1.3e-234)
           (/
            (* (hypot B (sqrt (* C (* A -4.0)))) (* (sqrt (* A F)) -2.0))
            t_1)
           (if (<= B -4.7e-290)
             t_2
             (if (<= B 1.85e-104)
               (/ (- (sqrt (* t_1 (* F (* A 4.0))))) t_1)
               (* (sqrt (* F (- (+ C A) t_0))) (/ (- (sqrt 2.0)) B))))))))))
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 = fma(B, B, (A * (C * -4.0)));
	double t_2 = -sqrt((2.0 * (-4.0 * ((C * A) * (F * (C + C)))))) / ((B * B) + (-4.0 * (C * A)));
	double t_3 = sqrt(((2.0 * F) * (C + (A - t_0)))) / B;
	double tmp;
	if (B <= -6e-41) {
		tmp = t_3;
	} else if (B <= -4.5e-139) {
		tmp = t_2;
	} else if (B <= -6.8e-191) {
		tmp = t_3;
	} else if (B <= -1.3e-234) {
		tmp = (hypot(B, sqrt((C * (A * -4.0)))) * (sqrt((A * F)) * -2.0)) / t_1;
	} else if (B <= -4.7e-290) {
		tmp = t_2;
	} else if (B <= 1.85e-104) {
		tmp = -sqrt((t_1 * (F * (A * 4.0)))) / t_1;
	} else {
		tmp = sqrt((F * ((C + A) - t_0))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

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 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(-sqrt(Float64(2.0 * Float64(-4.0 * Float64(Float64(C * A) * Float64(F * Float64(C + C))))))) / Float64(Float64(B * B) + Float64(-4.0 * Float64(C * A))))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + Float64(A - t_0)))) / B)
	tmp = 0.0
	if (B <= -6e-41)
		tmp = t_3;
	elseif (B <= -4.5e-139)
		tmp = t_2;
	elseif (B <= -6.8e-191)
		tmp = t_3;
	elseif (B <= -1.3e-234)
		tmp = Float64(Float64(hypot(B, sqrt(Float64(C * Float64(A * -4.0)))) * Float64(sqrt(Float64(A * F)) * -2.0)) / t_1);
	elseif (B <= -4.7e-290)
		tmp = t_2;
	elseif (B <= 1.85e-104)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(F * Float64(A * 4.0))))) / t_1);
	else
		tmp = Float64(sqrt(Float64(F * Float64(Float64(C + A) - t_0))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

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[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(2.0 * N[(-4.0 * N[(N[(C * A), $MachinePrecision] * N[(F * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, -6e-41], t$95$3, If[LessEqual[B, -4.5e-139], t$95$2, If[LessEqual[B, -6.8e-191], t$95$3, If[LessEqual[B, -1.3e-234], N[(N[(N[Sqrt[B ^ 2 + N[Sqrt[N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, -4.7e-290], t$95$2, If[LessEqual[B, 1.85e-104], N[((-N[Sqrt[N[(t$95$1 * N[(F * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[Sqrt[N[(F * N[(N[(C + A), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

\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 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\
t_2 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\
t_3 := \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_0\right)\right)}}{B}\\
\mathbf{if}\;B \leq -6 \cdot 10^{-41}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;B \leq -4.5 \cdot 10^{-139}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;B \leq -6.8 \cdot 10^{-191}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;B \leq -1.3 \cdot 10^{-234}:\\
\;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right) \cdot \left(\sqrt{A \cdot F} \cdot -2\right)}{t_1}\\

\mathbf{elif}\;B \leq -4.7 \cdot 10^{-290}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;B \leq 1.85 \cdot 10^{-104}:\\
\;\;\;\;\frac{-\sqrt{t_1 \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{t_1}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{F \cdot \left(\left(C + A\right) - t_0\right)} \cdot \frac{-\sqrt{2}}{B}\\


\end{array}

Error

Derivation

  1. Split input into 5 regimes

  2. if B < -5.99999999999999978e-41 or -4.50000000000000023e-139 < B < -6.79999999999999988e-191

    1. Initial program 53.6

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

    2. Simplified51.0

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (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 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 10 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.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)))): 27 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 18 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 23 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 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 27 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 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 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))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 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 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 18 points increase in error, 9 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 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 9 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))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 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 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr63.1

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

    4. Simplified63.1

      \[\leadsto \color{blue}{\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot C\right) \cdot -4}\right)}}} \]
      Proof
      (/.f64 (sqrt.f64 (*.f64 (+.f64 C (-.f64 A (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 C A) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4)))) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4))))))): 9 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))) (fma.f64 B B (*.f64 A (*.f64 C -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1)) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F)))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1) (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (fma.f64 B B (*.f64 A (*.f64 C -4)))))): 0 points increase in error, 0 points decrease in error

    5. Taylor expanded in B around inf 36.7

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

    if -5.99999999999999978e-41 < B < -4.50000000000000023e-139 or -1.29999999999999995e-234 < B < -4.7000000000000001e-290

    1. Initial program 50.0

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

    2. Simplified49.5

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(A + \left(C - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (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 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 10 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.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)))): 27 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 18 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 23 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 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 27 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 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 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))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 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 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 18 points increase in error, 9 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 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 9 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))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 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 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in B around 0 55.3

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

    4. Simplified54.6

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

    5. Taylor expanded in A around inf 51.2

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

    6. Simplified49.0

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(-4 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 -4 (*.f64 (*.f64 A C) (*.f64 F (+.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 -4 (*.f64 (*.f64 A C) (*.f64 F (+.f64 C (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 C)))))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 6 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 -4 (*.f64 (*.f64 A C) (*.f64 F (+.f64 C (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 C)))))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 6 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 -4 (*.f64 (*.f64 A C) (*.f64 F (Rewrite<= sub-neg_binary64 (-.f64 C (*.f64 -1 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 -4 (*.f64 (*.f64 A C) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 C (*.f64 -1 C)) F))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 0 points increase in error, 6 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 2 (*.f64 -4 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C (*.f64 (-.f64 C (*.f64 -1 C)) F)))))))) (-.f64 (*.f64 B B) (*.f64 4 (*.f64 A C)))): 6 points increase in error, 0 points decrease in error

    if -6.79999999999999988e-191 < B < -1.29999999999999995e-234

    1. Initial program 53.9

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

    2. Simplified48.5

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (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 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 10 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.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)))): 27 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 18 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 23 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 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 27 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 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 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))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 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 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 18 points increase in error, 9 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 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 9 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))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 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 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in A around -inf 50.4

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

    4. Simplified50.4

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(2, A, 0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (/.f64 (sqrt.f64 (*.f64 (+.f64 C (-.f64 A (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 C A) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4)))) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4))))))): 9 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))) (fma.f64 B B (*.f64 A (*.f64 C -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1)) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F)))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1) (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (fma.f64 B B (*.f64 A (*.f64 C -4)))))): 0 points increase in error, 0 points decrease in error

    5. Applied egg-rr56.0

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

    6. Simplified56.0

      \[\leadsto \frac{-\color{blue}{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(-4 \cdot A\right)}\right) \cdot \sqrt{\left(B \cdot \frac{B}{A}\right) \cdot F + F \cdot \left(A \cdot 4\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 B (/.f64 B A)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 C -4) A)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 B (/.f64 B A)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 A (*.f64 C -4))))) (sqrt.f64 (+.f64 (*.f64 (*.f64 B (/.f64 B A)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 B A) B)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 (/.f64 B A) B) 1)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 24 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 (*.f64 (/.f64 B A) B) (Rewrite<= metadata-eval (*.f64 1/2 2))) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 B A) B) 1/2) 2)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 24 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (*.f64 (/.f64 B A) B))) 2) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 1/2 (*.f64 (/.f64 B A) B)) (*.f64 2 F))) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B)))) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (*.f64 A (Rewrite<= metadata-eval (*.f64 2 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A 2) 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 24 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 A)) 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (Rewrite=> associate-*l*_binary64 (*.f64 2 (*.f64 A 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 2 A)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 F 2) (*.f64 2 A))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 F)) (*.f64 2 A)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 (*.f64 2 F) (Rewrite=> *-commutative_binary64 (*.f64 A 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 A 2)) (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (Rewrite<= *-commutative_binary64 (*.f64 2 A))) (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (Rewrite=> distribute-lft-out_binary64 (*.f64 (*.f64 2 F) (+.f64 (*.f64 2 A) (*.f64 1/2 (*.f64 (/.f64 B A) B)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (*.f64 (*.f64 2 F) (Rewrite<= fma-udef_binary64 (fma.f64 2 A (*.f64 1/2 (*.f64 (/.f64 B A) B)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 F (fma.f64 2 A (*.f64 1/2 (*.f64 (/.f64 B A) B))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 (*.f64 2 (*.f64 F (fma.f64 2 A (*.f64 1/2 (*.f64 (/.f64 B A) B)))))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error

    7. Taylor expanded in B around 0 54.4

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

    8. Simplified54.4

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

    if -4.7000000000000001e-290 < B < 1.85e-104

    1. Initial program 51.8

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

    2. Simplified46.0

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (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 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 10 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.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)))): 27 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 18 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 23 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 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 27 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 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 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))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 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 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 18 points increase in error, 9 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 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 9 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))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 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 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in A around -inf 50.3

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

    4. Simplified50.3

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(2, A, 0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (/.f64 (sqrt.f64 (*.f64 (+.f64 C (-.f64 A (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 C A) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4)))) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4))))))): 9 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))) (fma.f64 B B (*.f64 A (*.f64 C -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1)) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F)))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1) (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (fma.f64 B B (*.f64 A (*.f64 C -4)))))): 0 points increase in error, 0 points decrease in error

    5. Taylor expanded in A around inf 49.7

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

    6. Simplified49.7

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \color{blue}{\left(\left(4 \cdot A\right) \cdot F\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 C (*.f64 -4 A)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 B (/.f64 B A)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 C -4) A)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 B (/.f64 B A)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 A (*.f64 C -4))))) (sqrt.f64 (+.f64 (*.f64 (*.f64 B (/.f64 B A)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 B A) B)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 (/.f64 B A) B) 1)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 24 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 (*.f64 (/.f64 B A) B) (Rewrite<= metadata-eval (*.f64 1/2 2))) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 B A) B) 1/2) 2)) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 24 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/2 (*.f64 (/.f64 B A) B))) 2) F) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 1/2 (*.f64 (/.f64 B A) B)) (*.f64 2 F))) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B)))) (*.f64 F (*.f64 A 4)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (*.f64 A (Rewrite<= metadata-eval (*.f64 2 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 24 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A 2) 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 24 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 A)) 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (Rewrite=> associate-*l*_binary64 (*.f64 2 (*.f64 A 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 F (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 2 A)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 F 2) (*.f64 2 A))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 F)) (*.f64 2 A)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))) (*.f64 (*.f64 2 F) (Rewrite=> *-commutative_binary64 (*.f64 A 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (*.f64 2 F) (*.f64 A 2)) (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (+.f64 (*.f64 (*.f64 2 F) (Rewrite<= *-commutative_binary64 (*.f64 2 A))) (*.f64 (*.f64 2 F) (*.f64 1/2 (*.f64 (/.f64 B A) B))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (Rewrite=> distribute-lft-out_binary64 (*.f64 (*.f64 2 F) (+.f64 (*.f64 2 A) (*.f64 1/2 (*.f64 (/.f64 B A) B)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (*.f64 (*.f64 2 F) (Rewrite<= fma-udef_binary64 (fma.f64 2 A (*.f64 1/2 (*.f64 (/.f64 B A) B)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 F (fma.f64 2 A (*.f64 1/2 (*.f64 (/.f64 B A) B))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 (*.f64 2 (*.f64 F (fma.f64 2 A (*.f64 1/2 (*.f64 (/.f64 B A) B)))))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error

    if 1.85e-104 < B

    1. Initial program 52.4

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

    2. Simplified49.3

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (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 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 18 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 10 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 10 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 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 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (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 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.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)))): 27 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 18 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 23 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 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 27 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 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 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))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 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 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 18 points increase in error, 9 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 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 9 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))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 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 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

    3. Applied egg-rr46.4

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

    4. Simplified46.4

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

    5. Applied egg-rr46.7

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

    6. Simplified46.7

      \[\leadsto \color{blue}{\sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (*.f64 F (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))) (neg.f64 (/.f64 (sqrt.f64 (*.f64 2 (fma.f64 B B (*.f64 A (*.f64 C -4))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 F (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))) (neg.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (sqrt.f64 (*.f64 2 (fma.f64 B B (*.f64 A (*.f64 C -4))))) 1)) (fma.f64 B B (*.f64 A (*.f64 C -4)))))): 0 points increase in error, 4 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 F (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (sqrt.f64 (*.f64 2 (fma.f64 B B (*.f64 A (*.f64 C -4))))) (/.f64 1 (fma.f64 B B (*.f64 A (*.f64 C -4)))))))): 0 points increase in error, 4 points decrease in error
      (*.f64 (sqrt.f64 (*.f64 F (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))) (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 2 (fma.f64 B B (*.f64 A (*.f64 C -4)))))) (/.f64 1 (fma.f64 B B (*.f64 A (*.f64 C -4))))))): 4 points increase in error, 0 points decrease in error

    7. Taylor expanded in B around inf 35.9

      \[\leadsto \sqrt{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \]
  3. Recombined 5 regimes into one program.

  4. Final simplification40.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq -6.8 \cdot 10^{-191}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B}\\ \mathbf{elif}\;B \leq -1.3 \cdot 10^{-234}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{C \cdot \left(A \cdot -4\right)}\right) \cdot \left(\sqrt{A \cdot F} \cdot -2\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq -4.7 \cdot 10^{-290}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-104}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(C + A\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error39.2
Cost27984
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_0\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{-39}:\\ \;\;\;\;\frac{t_1}{B}\\ \mathbf{elif}\;B \leq -3.9 \cdot 10^{-291}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A - C}, C \cdot 2\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-119}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(\left(t_0 - A\right) - C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{-B}\\ \end{array} \]
Alternative 2
Error37.6
Cost27720
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_0\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -6.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{t_1}{B}\\ \mathbf{elif}\;B \leq 6.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_2 \cdot \left(F \cdot \left(\left(t_0 - A\right) - C\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{-B}\\ \end{array} \]
Alternative 3
Error37.7
Cost27720
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_0\right)\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -1.7 \cdot 10^{+44}:\\ \;\;\;\;\frac{t_1}{B}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{+35}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(\left(t_0 - A\right) - C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{-B}\\ \end{array} \]
Alternative 4
Error40.6
Cost21136
\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - t_0\right)\right)}}{B}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.9 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-190}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\left(C + A\right) - t_0\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
Alternative 5
Error41.5
Cost14684
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ t_2 := \sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ t_3 := \frac{t_2}{B}\\ \mathbf{if}\;B \leq -8 \cdot 10^{-41}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -7.8 \cdot 10^{-191}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -6 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 9.8 \cdot 10^{-108}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{-B}\\ \end{array} \]
Alternative 6
Error42.9
Cost14364
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ t_2 := \frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{B}\\ \mathbf{if}\;B \leq -7.8 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.6 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -5.3 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.25 \cdot 10^{-253}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]
Alternative 7
Error43.4
Cost13968
\[\begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{if}\;B \leq -1 \cdot 10^{-53}:\\ \;\;\;\;t_0 \cdot \sqrt{\frac{F}{B}}\\ \mathbf{elif}\;B \leq 1.06 \cdot 10^{-226}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-107}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{B} \cdot \sqrt{F \cdot \left(C - B\right)}\\ \end{array} \]
Alternative 8
Error48.3
Cost13316
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-53}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-108}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(A + \left(C + \left(-0.5 \cdot \left(A \cdot \frac{A}{B}\right) - B\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 9
Error52.3
Cost9608
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -0.32:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + \left(C + \left(A - C\right)\right)\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.5 \cdot 10^{-273}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(4 \cdot \left(C \cdot A\right) - B \cdot B\right)\right) \cdot \left(\left(\left(B - 0.5 \cdot \left(2 \cdot \frac{C \cdot A}{B} - \frac{A \cdot A}{B}\right)\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 10
Error52.2
Cost9096
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -0.014:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + \left(C + \left(A - C\right)\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq 7.2 \cdot 10^{-274}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + \left(C + \left(-0.5 \cdot \left(A \cdot \frac{A}{B}\right) - B\right)\right)\right) \cdot t_1\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 11
Error52.4
Cost8712
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -0.014:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + \left(C + \left(A - C\right)\right)\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.9 \cdot 10^{-273}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(A + \left(C + \left(-0.5 \cdot \left(A \cdot \frac{A}{B}\right) - B\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 12
Error52.8
Cost8584
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -1200:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.2 \cdot 10^{-277}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A - \left(B - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 13
Error52.6
Cost8584
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -1.85 \cdot 10^{+50}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.8 \cdot 10^{-273}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A - \left(B - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 14
Error53.0
Cost8332
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \mathbf{if}\;C \leq -8.6 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 3.05 \cdot 10^{-207}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(B - C\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 15
Error53.1
Cost8324
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -0.014:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(\left(\left(C - A\right) - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.65 \cdot 10^{-273}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B - C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 16
Error53.7
Cost8200
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{elif}\;A \leq 6 \cdot 10^{-260}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B - C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 17
Error53.2
Cost8200
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq -0.08:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.28 \cdot 10^{-256}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B - C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot \left(F \cdot \left(C + C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 18
Error55.4
Cost8072
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;C \leq -1.1 \cdot 10^{-91}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 4.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(B \cdot \left(B \cdot F\right)\right) \cdot \left(B - C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right) \cdot 8\right)}}{t_0}\\ \end{array} \]
Alternative 19
Error56.4
Cost7940
\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;A \leq 3.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]
Alternative 20
Error58.0
Cost7808
\[\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)} \]
Alternative 21
Error62.4
Cost6656
\[-\sqrt{\frac{F}{A}} \]

Error

Reproduce

herbie shell --seed 2022341 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))