ABCF->ab-angle b

Average Error: 52.2 → 35.3

Time: 36.2s

Precision: binary64

Cost: 20432

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-236}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]

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
         (/
          (- (sqrt (* -8.0 (* A (* F (* C (+ A A)))))))
          (fma B B (* A (* C -4.0))))))
   (if (<= B -1e-61)
     (/ (sqrt (* (- A (hypot A B)) (* F 2.0))) B)
     (if (<= B 5.4e-236)
       t_0
       (if (<= B 7.8e-174)
         (sqrt (/ (- F) C))
         (if (<= B 1.35e-50)
           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 = -sqrt((-8.0 * (A * (F * (C * (A + A)))))) / fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= -1e-61) {
		tmp = sqrt(((A - hypot(A, B)) * (F * 2.0))) / B;
	} else if (B <= 5.4e-236) {
		tmp = t_0;
	} else if (B <= 7.8e-174) {
		tmp = sqrt((-F / C));
	} else if (B <= 1.35e-50) {
		tmp = 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(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(F * Float64(C * Float64(A + A))))))) / fma(B, B, Float64(A * Float64(C * -4.0))))
	tmp = 0.0
	if (B <= -1e-61)
		tmp = Float64(sqrt(Float64(Float64(A - hypot(A, B)) * Float64(F * 2.0))) / B);
	elseif (B <= 5.4e-236)
		tmp = t_0;
	elseif (B <= 7.8e-174)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (B <= 1.35e-50)
		tmp = t_0;
	else
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-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[((-N[Sqrt[N[(-8.0 * N[(A * N[(F * N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1e-61], N[(N[Sqrt[N[(N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 5.4e-236], t$95$0, If[LessEqual[B, 7.8e-174], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[B, 1.35e-50], t$95$0, 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 4 regimes

2. if B < -1e-61

   1. Initial program 52.5

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

   2. Simplified 50.2

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      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)))): 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 (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)))): 31 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)))): 15 points increase in error, 3 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)))): 3 points increase in error, 3 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. Applied egg-rr 62.5

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

   4. Simplified 62.1

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(B, B, \left(A \cdot -4\right) \cdot C\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot -4\right) \cdot C\right)\right)}^{2}}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right) \cdot \left(2 \cdot F\right)}}}} \]
      Proof
      (sqrt.f64 (/.f64 (fma.f64 B B (*.f64 (*.f64 A -4) C)) (/.f64 (pow.f64 (fma.f64 B B (*.f64 (*.f64 A -4) C)) 2) (*.f64 (+.f64 (-.f64 C (hypot.f64 B (-.f64 A C))) A) (*.f64 2 F))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 -4 C)))) (/.f64 (pow.f64 (fma.f64 B B (*.f64 (*.f64 A -4) C)) 2) (*.f64 (+.f64 (-.f64 C (hypot.f64 B (-.f64 A C))) A) (*.f64 2 F))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) (/.f64 (pow.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 -4 C)))) 2) (*.f64 (+.f64 (-.f64 C (hypot.f64 B (-.f64 A C))) A) (*.f64 2 F))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) (/.f64 (pow.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) 2) (*.f64 (Rewrite<= associate--r-_binary64 (-.f64 C (-.f64 (hypot.f64 B (-.f64 A C)) A))) (*.f64 2 F))))): 6 points increase in error, 7 points decrease in error
      (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) (*.f64 (-.f64 C (-.f64 (hypot.f64 B (-.f64 A C)) A)) (*.f64 2 F))) (pow.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) 2)))): 20 points increase in error, 7 points decrease in error

   5. Taylor expanded in C around 0 50.3

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

   6. Simplified 34.1

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 F (-.f64 A (hypot.f64 A B))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 F (-.f64 A (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 A A) (*.f64 B B)))))))): 67 points increase in error, 3 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 F (-.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 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 F (-.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 (/.f64 (sqrt.f64 2) B) (sqrt.f64 (*.f64 F (-.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
      (*.f64 (/.f64 (sqrt.f64 2) B) (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<= *-commutative_binary64 (*.f64 (sqrt.f64 (*.f64 (-.f64 A (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 A 2)))) F)) (/.f64 (sqrt.f64 2) B))): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 34.1

      \[\leadsto \color{blue}{0 + \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)}}{B}} \]

   8. Simplified 34.1

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

   if -1e-61 < B < 5.4e-236 or 7.7999999999999997e-174 < B < 1.35e-50

   1. Initial program 51.0

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

   2. Simplified 44.2

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 29 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)))): 16 points increase in error, 4 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 A around inf 50.3

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

   4. Taylor expanded in C around inf 41.5

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot \left(C \cdot F\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   5. Simplified 36.4

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(\left(\left(A - \left(-A\right)\right) \cdot C\right) \cdot F\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (*.f64 -8 (*.f64 A (*.f64 (*.f64 (-.f64 A (neg.f64 A)) C) F))): 0 points increase in error, 0 points decrease in error
      (*.f64 -8 (*.f64 A (*.f64 (*.f64 (-.f64 A (Rewrite<= mul-1-neg_binary64 (*.f64 -1 A))) C) F))): 0 points increase in error, 0 points decrease in error
      (*.f64 -8 (*.f64 A (Rewrite<= associate-*r*_binary64 (*.f64 (-.f64 A (*.f64 -1 A)) (*.f64 C F))))): 30 points increase in error, 22 points decrease in error

   if 5.4e-236 < B < 7.7999999999999997e-174

   1. Initial program 54.7

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

   2. Simplified 47.1

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      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)))): 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 (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)))): 31 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)))): 15 points increase in error, 3 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)))): 3 points increase in error, 3 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. Applied egg-rr 58.6

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

   4. Simplified 57.9

      \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{fma}\left(B, B, \left(A \cdot -4\right) \cdot C\right)}{\frac{{\left(\mathsf{fma}\left(B, B, \left(A \cdot -4\right) \cdot C\right)\right)}^{2}}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right) \cdot \left(2 \cdot F\right)}}}} \]
      Proof
      (sqrt.f64 (/.f64 (fma.f64 B B (*.f64 (*.f64 A -4) C)) (/.f64 (pow.f64 (fma.f64 B B (*.f64 (*.f64 A -4) C)) 2) (*.f64 (+.f64 (-.f64 C (hypot.f64 B (-.f64 A C))) A) (*.f64 2 F))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 -4 C)))) (/.f64 (pow.f64 (fma.f64 B B (*.f64 (*.f64 A -4) C)) 2) (*.f64 (+.f64 (-.f64 C (hypot.f64 B (-.f64 A C))) A) (*.f64 2 F))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) (/.f64 (pow.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 -4 C)))) 2) (*.f64 (+.f64 (-.f64 C (hypot.f64 B (-.f64 A C))) A) (*.f64 2 F))))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) (/.f64 (pow.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) 2) (*.f64 (Rewrite<= associate--r-_binary64 (-.f64 C (-.f64 (hypot.f64 B (-.f64 A C)) A))) (*.f64 2 F))))): 6 points increase in error, 7 points decrease in error
      (sqrt.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) (*.f64 (-.f64 C (-.f64 (hypot.f64 B (-.f64 A C)) A)) (*.f64 2 F))) (pow.f64 (fma.f64 B B (*.f64 A (*.f64 -4 C))) 2)))): 20 points increase in error, 7 points decrease in error

   5. Taylor expanded in A around -inf 39.9

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

   6. Simplified 39.9

      \[\leadsto \sqrt{\color{blue}{\frac{-F}{C}}} \]
      Proof
      (/.f64 (neg.f64 F) C): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 F)) C): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 F C))): 0 points increase in error, 0 points decrease in error

   if 1.35e-50 < B

   1. Initial program 52.9

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

   2. Simplified 50.0

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(-8, A \cdot C, 2 \cdot \left(B \cdot B\right)\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      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)))): 29 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)))): 16 points increase in error, 4 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 50.1

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

   4. Simplified 34.4

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)} \]
      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))))))))): 67 points increase in error, 3 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 4 regimes into one program.

4. Final simplification 35.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-174}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	36.9
Cost	14608

\[\begin{array}{l} t_0 := \frac{-\sqrt{-8 \cdot \left(A \cdot \left(F \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -8.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-233}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 8.4 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \left(-0.5 \cdot \left(C \cdot \frac{C}{B}\right) - B\right)\right)}\right)\\ \end{array} \]

Alternative 2
Error	39.0
Cost	14348

\[\begin{array}{l} \mathbf{if}\;B \leq -2.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 7.8 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.04 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \left(-0.5 \cdot \left(C \cdot \frac{C}{B}\right) - B\right)\right)}\right)\\ \end{array} \]

Alternative 3
Error	39.0
Cost	14216

\[\begin{array}{l} \mathbf{if}\;B \leq -6.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \left(-0.5 \cdot \left(C \cdot \frac{C}{B}\right) - B\right)\right)}\right)\\ \end{array} \]

Alternative 4
Error	40.7
Cost	13704

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -7.4 \cdot 10^{-96}:\\ \;\;\;\;t_0 \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - B\right)} \cdot \left(-t_0\right)\\ \end{array} \]

Alternative 5
Error	39.1
Cost	13704

\[\begin{array}{l} \mathbf{if}\;B \leq -1.3 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(F \cdot 2\right)}}{B}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 6
Error	45.8
Cost	13380

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;B \leq -6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{elif}\;B \leq 1.62 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 51000000:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right) \cdot \left(B - \left(A + C\right)\right)}}{B \cdot B - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	49.5
Cost	10128

\[\begin{array}{l} t_0 := \frac{B \cdot B}{A}\\ t_1 := \sqrt{\frac{-F}{C}}\\ t_2 := 4 \cdot \left(A \cdot C\right)\\ t_3 := B \cdot B - t_2\\ t_4 := 2 \cdot \left(F \cdot t_3\right)\\ \mathbf{if}\;C \leq -4.8 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.65 \cdot 10^{-221}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(B + \left(A + C\right)\right)}}{t_3}\\ \mathbf{elif}\;C \leq 2.1 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(t_2 - B \cdot B\right)\right)\right) \cdot \left(B - \left(A + C\right)\right)}}{t_3}\\ \mathbf{elif}\;C \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{t_4 \cdot \left(\left(A + C\right) - \left(C + \left(-0.5 \cdot \left(t_0 + t_0 \cdot \frac{C}{A}\right) - A\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	49.5
Cost	8976

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := 4 \cdot \left(A \cdot C\right)\\ t_2 := B \cdot B - t_1\\ t_3 := 2 \cdot \left(F \cdot t_2\right)\\ \mathbf{if}\;C \leq -1.76 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 6.2 \cdot 10^{-221}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(B + \left(A + C\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 3 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right) \cdot \left(B - \left(A + C\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(A + C\right) + \left(A - C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	47.1
Cost	8716

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := 4 \cdot \left(A \cdot C\right)\\ t_2 := B \cdot B - t_1\\ \mathbf{if}\;A \leq -2.4 \cdot 10^{+160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -3.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq -2.3 \cdot 10^{-289}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right) \cdot \left(B - \left(A + C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	49.5
Cost	8716

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := 4 \cdot \left(A \cdot C\right)\\ t_2 := B \cdot B - t_1\\ \mathbf{if}\;C \leq -3.35 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 3.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_2\right)\right) \cdot \left(B + \left(A + C\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 1.38 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right) \cdot \left(B - \left(A + C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	49.7
Cost	8584

\[\begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ t_1 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;C \leq -5.9 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 1.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right) \cdot \left(B - \left(A + C\right)\right)}}{B \cdot B - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	62.8
Cost	6656

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

Alternative 13
Error	51.2
Cost	6656

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

Alternative 14
Error	63.1
Cost	6592

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

Reproduce

herbie shell --seed 2022329 
(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))))