ABCF->ab-angle a

Average Error: 52.3 → 39.5

Time: 39.6s

Precision: binary64

Cost: 80900

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_1} \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{\sqrt{t_0} \cdot \left(-\sqrt{2 \cdot F}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{2}{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)} \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma C (* A -4.0) (* B B))) (t_1 (- (pow B 2.0) (* (* 4.0 A) C))))
   (if (<=
        (/
         (-
          (sqrt
           (*
            (* 2.0 (* t_1 F))
            (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
         t_1)
        -1e-213)
     (*
      (sqrt (+ C (+ A (hypot (- A C) B))))
      (/ (* (sqrt t_0) (- (sqrt (* 2.0 F)))) t_0))
     (sqrt
      (*
       F
       (*
        (/ 2.0 (fma (* A -4.0) C (* B B)))
        (+ C (+ A (hypot B (- A C))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = fma(C, (A * -4.0), (B * B));
	double t_1 = pow(B, 2.0) - ((4.0 * A) * C);
	double tmp;
	if ((-sqrt(((2.0 * (t_1 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_1) <= -1e-213) {
		tmp = sqrt((C + (A + hypot((A - C), B)))) * ((sqrt(t_0) * -sqrt((2.0 * F))) / t_0);
	} else {
		tmp = sqrt((F * ((2.0 / fma((A * -4.0), C, (B * B))) * (C + (A + hypot(B, (A - C)))))));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = fma(C, Float64(A * -4.0), Float64(B * B))
	t_1 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_1) <= -1e-213)
		tmp = Float64(sqrt(Float64(C + Float64(A + hypot(Float64(A - C), B)))) * Float64(Float64(sqrt(t_0) * Float64(-sqrt(Float64(2.0 * F)))) / t_0));
	else
		tmp = sqrt(Float64(F * Float64(Float64(2.0 / fma(Float64(A * -4.0), C, Float64(B * B))) * Float64(C + Float64(A + hypot(B, Float64(A - C)))))));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], -1e-213], N[(N[Sqrt[N[(C + N[(A + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(F * N[(N[(2.0 / N[(N[(A * -4.0), $MachinePrecision] * C + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -9.9999999999999995e-214

   1. Initial program 37.3

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

   2. Simplified 37.3

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

   3. Applied egg-rr 32.6

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

   4. Applied egg-rr 24.1

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

   5. Simplified 22.9

      \[\leadsto \frac{-\color{blue}{\sqrt{C + \left(\mathsf{hypot}\left(A - C, B\right) + A\right)} \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 (hypot.f64 (-.f64 A C) B) A))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 F 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 C (hypot.f64 (-.f64 A C) B)) A))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 F 2)))): 0 points increase in error, 28 points decrease in error
      (*.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 A (+.f64 C (hypot.f64 (-.f64 A C) B))))) (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 F 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 (-.f64 A C) B)))) (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) -4))) (*.f64 F 2)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 (-.f64 A C) B)))) (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F) 2)))): 1 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 (-.f64 A C) B)))) (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 (-.f64 A C) B)))) (Rewrite<= unpow1/2_binary64 (pow.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F)) 1/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F)) 1/2) (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 (-.f64 A C) B)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> unpow1/2_binary64 (sqrt.f64 (*.f64 2 (*.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) F)))) (sqrt.f64 (+.f64 A (+.f64 C (hypot.f64 (-.f64 A C) B))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 23.0

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

   7. Simplified 22.9

      \[\leadsto \color{blue}{\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{-\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 C (*.f64 A -4) (*.f64 B B))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 C (Rewrite<= *-commutative_binary64 (*.f64 -4 A)) (*.f64 B B))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (*.f64 -4 A)) (*.f64 B B)))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 B B) (*.f64 C (*.f64 -4 A))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (Rewrite<= fma-udef_binary64 (fma.f64 B B (*.f64 C (*.f64 -4 A))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 B B (*.f64 C (*.f64 -4 A)))))) 1)) (fma.f64 C (*.f64 A -4) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 B B (*.f64 C (*.f64 -4 A)))))) 1) (fma.f64 C (Rewrite<= *-commutative_binary64 (*.f64 -4 A)) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 B B (*.f64 C (*.f64 -4 A)))))) 1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (*.f64 -4 A)) (*.f64 B B))))): 1 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 B B (*.f64 C (*.f64 -4 A)))))) 1) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 B B) (*.f64 C (*.f64 -4 A)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (/.f64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 B B (*.f64 C (*.f64 -4 A)))))) 1) (Rewrite<= fma-udef_binary64 (fma.f64 B B (*.f64 C (*.f64 -4 A)))))): 0 points increase in error, 2 points decrease in error
      (*.f64 (sqrt.f64 (+.f64 C (+.f64 A (hypot.f64 (-.f64 A C) B)))) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F 2) (fma.f64 B B (*.f64 C (*.f64 -4 A)))))) (/.f64 1 (fma.f64 B B (*.f64 C (*.f64 -4 A))))))): 14 points increase in error, 14 points decrease in error

   8. Applied egg-rr 13.5

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

   if -9.9999999999999995e-214 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 60.0

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

   2. Simplified 57.4

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2)) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (*.f64 (*.f64 2 F) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 F 2)) (+.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 45 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (*.f64 F 2) (+.f64 (+.f64 A C) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 10 points increase in error, 5 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F))) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (Rewrite<= --rgt-identity_binary64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 C (*.f64 A -4) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (*.f64 A (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) (*.f64 B B))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 C (neg.f64 (*.f64 4 A)) (Rewrite<= unpow2_binary64 (pow.f64 B 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 C (neg.f64 (*.f64 4 A))) (pow.f64 B 2)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (*.f64 4 A)) C)) (pow.f64 B 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 59.2

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

   4. Simplified 58.4

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{{\left(\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}^{2}} \cdot \left(2 \cdot \left(F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}} \]
      Proof
      (sqrt.f64 (*.f64 (/.f64 (fma.f64 -4 (*.f64 C A) (*.f64 B B)) (pow.f64 (fma.f64 -4 (*.f64 C A) (*.f64 B B)) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -4 (*.f64 C A)) (*.f64 B B))) (pow.f64 (fma.f64 -4 (*.f64 C A) (*.f64 B B)) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 C A) -4)) (*.f64 B B)) (pow.f64 (fma.f64 -4 (*.f64 C A) (*.f64 B B)) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (+.f64 (Rewrite=> associate-*l*_binary64 (*.f64 C (*.f64 A -4))) (*.f64 B B)) (pow.f64 (fma.f64 -4 (*.f64 C A) (*.f64 B B)) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (Rewrite<= fma-udef_binary64 (fma.f64 C (*.f64 A -4) (*.f64 B B))) (pow.f64 (fma.f64 -4 (*.f64 C A) (*.f64 B B)) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -4 (*.f64 C A)) (*.f64 B B))) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (pow.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 C A) -4)) (*.f64 B B)) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (pow.f64 (+.f64 (Rewrite=> associate-*l*_binary64 (*.f64 C (*.f64 A -4))) (*.f64 B B)) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (*.f64 (/.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (pow.f64 (Rewrite<= fma-udef_binary64 (fma.f64 C (*.f64 A -4) (*.f64 B B))) 2)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (/.f64 (pow.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) 2) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))))))): 11 points increase in error, 5 points decrease in error
      (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) (*.f64 2 (*.f64 F (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C))))))) (pow.f64 (fma.f64 C (*.f64 A -4) (*.f64 B B)) 2)))): 27 points increase in error, 13 points decrease in error

   5. Applied egg-rr 54.6

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

   6. Simplified 52.9

      \[\leadsto \sqrt{\color{blue}{F \cdot \left(\frac{2}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}} \]
      Proof
      (*.f64 F (*.f64 (/.f64 2 (fma.f64 (*.f64 -4 A) C (*.f64 B B))) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 F (*.f64 (/.f64 2 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 -4 A) C) (*.f64 B B)))) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 5 points increase in error, 0 points decrease in error
      (*.f64 F (*.f64 (/.f64 2 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 A C))) (*.f64 B B))) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 F (*.f64 (/.f64 2 (+.f64 (*.f64 -4 (Rewrite<= *-commutative_binary64 (*.f64 C A))) (*.f64 B B))) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 F (*.f64 (/.f64 2 (Rewrite<= fma-udef_binary64 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))) (+.f64 C (+.f64 A (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 F (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 C (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))) (*.f64 (+.f64 A (hypot.f64 B (-.f64 A C))) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 F (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B))) C)) (*.f64 (+.f64 A (hypot.f64 B (-.f64 A C))) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 F (*.f64 (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B))) C)) (*.f64 F (*.f64 (+.f64 A (hypot.f64 B (-.f64 A C))) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B))) C) F)) (*.f64 F (*.f64 (+.f64 A (hypot.f64 B (-.f64 A C))) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B))) (*.f64 C F))) (*.f64 F (*.f64 (+.f64 A (hypot.f64 B (-.f64 A C))) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))))): 48 points increase in error, 20 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 C F) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B))))) (*.f64 F (*.f64 (+.f64 A (hypot.f64 B (-.f64 A C))) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (*.f64 C F) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 F (+.f64 A (hypot.f64 B (-.f64 A C)))) (/.f64 2 (fma.f64 -4 (*.f64 C A) (*.f64 B B)))))): 26 points increase in error, 10 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 39.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1 \cdot 10^{-213}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \left(-\sqrt{2 \cdot F}\right)}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{2}{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)} \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	39.8
Cost	40388

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \sqrt{F \cdot \frac{2}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{\sqrt{\mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)} \cdot \left(-\sqrt{F}\right)}{t_0}\\ \end{array} \]

Alternative 2
Error	43.6
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;A \leq -3.1 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{F} \cdot \sqrt{\left(C + \left(A + t_0\right)\right) \cdot \frac{2}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\\ \mathbf{elif}\;A \leq 3.9 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{-\sqrt{t_1 \cdot \left(2 \cdot F\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{2 \cdot \left(\left(A + C\right) + t_0\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\\ \end{array} \]

Alternative 3
Error	42.3
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \left(A + C\right) + t_1\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{t_2} \cdot \sqrt{F \cdot \frac{2}{t_0}}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{+222}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \left(A + t_1\right)\right)\right)} \cdot \left(-\sqrt{t_0}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F} \cdot \sqrt{t_2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \end{array} \]

Alternative 4
Error	43.6
Cost	27784

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;A \leq -3.2 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{F} \cdot \sqrt{\left(C + \left(A + t_0\right)\right) \cdot \frac{2}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\\ \mathbf{elif}\;A \leq 3.9 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \frac{\sqrt{F \cdot \mathsf{fma}\left(2, B \cdot B, -8 \cdot \left(A \cdot C\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{2 \cdot \left(\left(A + C\right) + t_0\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\\ \end{array} \]

Alternative 5
Error	43.6
Cost	27012

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;A \leq -1.1 \cdot 10^{+65}:\\ \;\;\;\;\sqrt{F} \cdot \sqrt{\left(C + \left(A + t_0\right)\right) \cdot \frac{2}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}\\ \mathbf{elif}\;A \leq 3.9 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B\right) \cdot \left(2 \cdot F\right) + \left(\left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot -2\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{2 \cdot \left(\left(A + C\right) + t_0\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\\ \end{array} \]

Alternative 6
Error	43.8
Cost	21832

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ \mathbf{if}\;A \leq -3.2 \cdot 10^{+66}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{2}{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)} \cdot \left(C + \left(A + t_0\right)\right)\right)}\\ \mathbf{elif}\;A \leq 3.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B\right) \cdot \left(2 \cdot F\right) + \left(\left(4 \cdot A\right) \cdot C\right) \cdot \left(F \cdot -2\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(A - C, B\right)\right)}\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{2 \cdot \left(\left(A + C\right) + t_0\right)}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}\\ \end{array} \]

Alternative 7
Error	47.4
Cost	21252

\[\begin{array}{l} t_0 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ \mathbf{if}\;A \leq 4.5 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sqrt{t_0 \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(\frac{2}{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)} \cdot t_0\right)}\\ \end{array} \]

Alternative 8
Error	48.1
Cost	20744

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{F \cdot \left(\frac{2}{\mathsf{fma}\left(A \cdot -4, C, B \cdot B\right)} \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}\\ \mathbf{if}\;A \leq -1.65 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	49.1
Cost	14912

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

Reproduce

herbie shell --seed 2022334 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))