ABCF->ab-angle b

Average Error: 52.6 → 34.4

Time: 35.2s

Precision: binary64

Cost: 34448

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

Math

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

↓

\[\begin{array}{l} t_0 := A \cdot \left(-4 \cdot C\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{-{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{0.25} \cdot {\left(\left(F \cdot 2\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}^{0.25}\right)}^{2}}{t_1}\\ t_3 := 2 \cdot \left(A \cdot \left(F + F\right)\right)\\ \mathbf{if}\;B \leq -1.8 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(A + A\right)}}{t_1}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot B\right) \cdot t_3 + t_3 \cdot t_0}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

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 (* A (* -4.0 C)))
        (t_1 (+ (* B B) (* -4.0 (* A C))))
        (t_2
         (/
          (-
           (pow
            (*
             (pow (fma B B (* C (* -4.0 A))) 0.25)
             (pow (* (* F 2.0) (- A (hypot A B))) 0.25))
            2.0))
          t_1))
        (t_3 (* 2.0 (* A (+ F F)))))
   (if (<= B -1.8e+134)
     (* (sqrt (/ F B)) (- (sqrt 2.0)))
     (if (<= B -1.8e-41)
       t_2
       (if (<= B 3.5e-14)
         (/ (- (sqrt (* (* 2.0 (* F t_1)) (+ A A)))) t_1)
         (if (<= B 4.4e+47)
           t_2
           (if (<= B 4e+87)
             (/ (- (sqrt (+ (* (* B B) t_3) (* t_3 t_0)))) (fma B B t_0))
             (* (- (/ (sqrt 2.0) B)) (sqrt (* F (- A (hypot B A))))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = A * (-4.0 * C);
	double t_1 = (B * B) + (-4.0 * (A * C));
	double t_2 = -pow((pow(fma(B, B, (C * (-4.0 * A))), 0.25) * pow(((F * 2.0) * (A - hypot(A, B))), 0.25)), 2.0) / t_1;
	double t_3 = 2.0 * (A * (F + F));
	double tmp;
	if (B <= -1.8e+134) {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	} else if (B <= -1.8e-41) {
		tmp = t_2;
	} else if (B <= 3.5e-14) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (A + A))) / t_1;
	} else if (B <= 4.4e+47) {
		tmp = t_2;
	} else if (B <= 4e+87) {
		tmp = -sqrt((((B * B) * t_3) + (t_3 * t_0))) / fma(B, B, t_0);
	} else {
		tmp = -(sqrt(2.0) / B) * sqrt((F * (A - hypot(B, A))));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = Float64(A * Float64(-4.0 * C))
	t_1 = Float64(Float64(B * B) + Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(Float64(-(Float64((fma(B, B, Float64(C * Float64(-4.0 * A))) ^ 0.25) * (Float64(Float64(F * 2.0) * Float64(A - hypot(A, B))) ^ 0.25)) ^ 2.0)) / t_1)
	t_3 = Float64(2.0 * Float64(A * Float64(F + F)))
	tmp = 0.0
	if (B <= -1.8e+134)
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	elseif (B <= -1.8e-41)
		tmp = t_2;
	elseif (B <= 3.5e-14)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(A + A)))) / t_1);
	elseif (B <= 4.4e+47)
		tmp = t_2;
	elseif (B <= 4e+87)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(B * B) * t_3) + Float64(t_3 * t_0)))) / fma(B, B, t_0));
	else
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B)) * sqrt(Float64(F * Float64(A - hypot(B, A)))));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(-4.0 * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Power[N[(N[Power[N[(B * B + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[N[(N[(F * 2.0), $MachinePrecision] * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(A * N[(F + F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.8e+134], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -1.8e-41], t$95$2, If[LessEqual[B, 3.5e-14], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 4.4e+47], t$95$2, If[LessEqual[B, 4e+87], N[((-N[Sqrt[N[(N[(N[(B * B), $MachinePrecision] * t$95$3), $MachinePrecision] + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]) * N[Sqrt[N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -1.79999999999999994e134

   1. Initial program 62.5

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

   2. Simplified 62.5

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (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 62.5

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

   4. Taylor expanded in B around -inf 62.5

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

   5. Taylor expanded in A around 0 32.2

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

   6. Simplified 32.2

      \[\leadsto \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (/.f64 F B)) (neg.f64 (sqrt.f64 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (/.f64 F B)) (sqrt.f64 2)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) (sqrt.f64 (/.f64 F B))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sqrt.f64 2) (sqrt.f64 (/.f64 F B))))): 0 points increase in error, 0 points decrease in error

   if -1.79999999999999994e134 < B < -1.8e-41 or 3.5000000000000002e-14 < B < 4.3999999999999999e47

   1. Initial program 43.0

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

   2. Simplified 43.0

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (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 40.2

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

   4. Taylor expanded in C around 0 42.6

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

   5. Simplified 38.7

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A - \mathsf{hypot}\left(A, B\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (-.f64 A (hypot.f64 A B)): 0 points increase in error, 0 points decrease in error
      (-.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 A A) (*.f64 B B))))): 124 points increase in error, 7 points decrease in error
      (-.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 A 2)) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (-.f64 A (sqrt.f64 (+.f64 (pow.f64 A 2) (Rewrite<= unpow2_binary64 (pow.f64 B 2))))): 0 points increase in error, 0 points decrease in error
      (-.f64 A (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (pow.f64 A 2))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 38.8

      \[\leadsto \frac{-\color{blue}{{\left({\left(\left(\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)\right)}^{0.25}\right)}^{2}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Applied egg-rr 35.0

      \[\leadsto \frac{-{\color{blue}{\left({\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)}^{0.25} \cdot {\left(F \cdot \left(2 \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)\right)}^{0.25}\right)}}^{2}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   8. Simplified 35.0

      \[\leadsto \frac{-{\color{blue}{\left({\left(\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)\right)}^{0.25} \cdot {\left(\left(F \cdot 2\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}^{0.25}\right)}}^{2}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (*.f64 (pow.f64 (fma.f64 B B (*.f64 (*.f64 -4 A) C)) 1/4) (pow.f64 (*.f64 (*.f64 F 2) (-.f64 A (hypot.f64 A B))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 A C)))) 1/4) (pow.f64 (*.f64 (*.f64 F 2) (-.f64 A (hypot.f64 A B))) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (fma.f64 B B (*.f64 -4 (*.f64 A C))) 1/4) (pow.f64 (Rewrite<= associate-*r*_binary64 (*.f64 F (*.f64 2 (-.f64 A (hypot.f64 A B))))) 1/4)): 0 points increase in error, 0 points decrease in error

   if -1.8e-41 < B < 3.5000000000000002e-14

   1. Initial program 51.3

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

   2. Simplified 51.4

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (*.f64 2 (*.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 (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 47.7

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

   4. Taylor expanded in A around -inf 35.6

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

   5. Simplified 35.6

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + A\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof
      (+.f64 A A): 0 points increase in error, 0 points decrease in error
      (+.f64 A (Rewrite<= *-lft-identity_binary64 (*.f64 1 A))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt1-in_binary64 (*.f64 (+.f64 1 1) A)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval 2) A): 0 points increase in error, 0 points decrease in error

   if 4.3999999999999999e47 < B < 3.9999999999999998e87

   1. Initial program 46.3

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

   2. Simplified 39.9

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) (*.f64 2 (*.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, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 (Rewrite<= metadata-eval (neg.f64 4)) C))) (*.f64 2 (*.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, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A (neg.f64 4)) C))) (*.f64 2 (*.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, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) C)) (*.f64 2 (*.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, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 A))) C)) (*.f64 2 (*.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, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (*.f64 4 A) C)))) (*.f64 2 (*.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, 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 2 (*.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, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 2 (*.f64 F (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 1 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 2 (*.f64 F (+.f64 A (-.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 41 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 2 (*.f64 F (+.f64 A (-.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 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 2 (*.f64 F (+.f64 A (-.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 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 2 (*.f64 F (+.f64 A (-.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 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 2 (*.f64 F (+.f64 A (Rewrite=> sub-neg_binary64 (+.f64 C (neg.f64 (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 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 2 (*.f64 F (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (neg.f64 (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 19 points increase in error, 7 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 2 (*.f64 F (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 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)) 2) (*.f64 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 -4 C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))) (*.f64 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 -4 C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 B 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 -4 C)))): 8 points increase in error, 6 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*r*_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 -4 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))) (fma.f64 B B (*.f64 A (*.f64 -4 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))) (fma.f64 B B (*.f64 A (*.f64 -4 C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 (Rewrite<= metadata-eval (neg.f64 4)) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A (neg.f64 4)) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 A 4))) C))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.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)) (fma.f64 B B (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in C around inf 50.1

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

   4. Applied egg-rr 50.1

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

   if 3.9999999999999998e87 < B

   1. Initial program 59.9

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

   2. Simplified 59.2

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (fma.f64 -8 (*.f64 A C) (*.f64 2 (*.f64 B B)))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (fma.f64 (Rewrite<= metadata-eval (*.f64 2 -4)) (*.f64 A C) (*.f64 2 (*.f64 B B)))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (fma.f64 (*.f64 2 (Rewrite<= metadata-eval (neg.f64 4))) (*.f64 A C) (*.f64 2 (*.f64 B B)))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (fma.f64 (*.f64 2 (neg.f64 4)) (*.f64 A C) (*.f64 2 (Rewrite<= unpow2_binary64 (pow.f64 B 2))))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 2 (neg.f64 4)) (*.f64 A C)) (*.f64 2 (pow.f64 B 2))))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (neg.f64 4) (*.f64 A C)))) (*.f64 2 (pow.f64 B 2)))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (+.f64 (*.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (neg.f64 4) A) C))) (*.f64 2 (pow.f64 B 2)))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (+.f64 (*.f64 2 (*.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 4 A))) C)) (*.f64 2 (pow.f64 B 2)))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (Rewrite<= distribute-lft-in_binary64 (*.f64 2 (+.f64 (*.f64 (neg.f64 (*.f64 4 A)) C) (pow.f64 B 2))))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 (neg.f64 (*.f64 4 A)) C))))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 F (*.f64 2 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) F)) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F))) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 A (-.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 39 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 A (-.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 A (-.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 A (-.f64 C (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 A (Rewrite=> sub-neg_binary64 (+.f64 C (neg.f64 (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 A C) (neg.f64 (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 19 points increase in error, 3 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (Rewrite<= sub-neg_binary64 (-.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, 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 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 0 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (-.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) 0) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 0 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (/.f64 (Rewrite=> --rgt-identity_binary64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 0 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in C around 0 55.7

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

   4. Simplified 29.5

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (-.f64 A (hypot.f64 B A)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (-.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 A A))))))))): 57 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (-.f64 A (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 A A)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (*.f64 F (-.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 A 2))))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (neg.f64 (sqrt.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 A 2)))) F))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 (-.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 A 2)))) F))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 (-.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 A 2)))) F))))): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 34.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.8 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{0.25} \cdot {\left(\left(F \cdot 2\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}^{0.25}\right)}^{2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(A + A\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{-{\left({\left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)}^{0.25} \cdot {\left(\left(F \cdot 2\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}^{0.25}\right)}^{2}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{+87}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot B\right) \cdot \left(2 \cdot \left(A \cdot \left(F + F\right)\right)\right) + \left(2 \cdot \left(A \cdot \left(F + F\right)\right)\right) \cdot \left(A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	35.9
Cost	34320

\[\begin{array}{l} t_0 := 2 \cdot \left(A \cdot \left(F + F\right)\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := 2 \cdot \left(F \cdot t_1\right)\\ t_3 := A \cdot \left(-4 \cdot C\right)\\ t_4 := \mathsf{fma}\left(B, B, t_3\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot t_2}}{t_1}\\ \mathbf{elif}\;B \leq 40000000000000:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(A + A\right)}}{t_1}\\ \mathbf{elif}\;B \leq 5.9 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\mathsf{hypot}\left(B, \sqrt{C \cdot \left(-4 \cdot A\right)}\right)\right)}{t_4}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+87}:\\ \;\;\;\;\frac{-\sqrt{\left(B \cdot B\right) \cdot t_0 + t_0 \cdot t_3}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	35.2
Cost	20300

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+127}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot t_1}}{t_0}\\ \mathbf{elif}\;B \leq 1250000000000:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ \end{array} \]

Alternative 3
Error	36.3
Cost	14920

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -1.05 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot t_1}}{t_0}\\ \mathbf{elif}\;B \leq 800000000000:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 4
Error	35.9
Cost	13704

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -350000:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 950000000000:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]

Alternative 5
Error	40.4
Cost	13316

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -115000:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \end{array} \]

Alternative 6
Error	45.6
Cost	8456

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B \leq -360000:\\ \;\;\;\;\frac{-\sqrt{B \cdot \left(2 \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \end{array} \]

Alternative 7
Error	50.8
Cost	8076

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -2.3 \cdot 10^{+151}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \mathbf{elif}\;A \leq -1.4 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.75 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{B \cdot \left(2 \cdot \left(B \cdot \left(B \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	51.1
Cost	7944

\[\begin{array}{l} t_0 := -2 \cdot \sqrt{\frac{F}{B} \cdot \frac{A}{B}}\\ \mathbf{if}\;B \leq -1.62 \cdot 10^{+22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 15000000000000:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	58.3
Cost	6980

\[\begin{array}{l} t_0 := \frac{\sqrt{F \cdot A}}{B}\\ \mathbf{if}\;B \leq -4 \cdot 10^{-280}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot t_0\\ \end{array} \]

Alternative 10
Error	58.1
Cost	6976

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

Alternative 11
Error	60.5
Cost	6848

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

Reproduce

herbie shell --seed 2022332 
(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))))