ABCF->ab-angle b

Average Error: 52.3 → 34.1

Time: 46.2s

Precision: binary64

Cost: 20300

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \mathbf{if}\;B \leq -6.2 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{B}{t_0}}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(A \cdot -4\right) + B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t_0}{-B}\\ \end{array} \]

FPCore

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

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* F (- A (hypot A B))))))
   (if (<= B -6.2e+239)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B -7e-14)
       (/ (sqrt 2.0) (/ B t_0))
       (if (<= B 5.1e-41)
         (/
          (- (sqrt (* 2.0 (* 2.0 (* (* F A) (+ (* C (* A -4.0)) (* B B)))))))
          (fma B B (* A (* C -4.0))))
         (/ (* (sqrt 2.0) t_0) (- B)))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = sqrt((F * (A - hypot(A, B))));
	double tmp;
	if (B <= -6.2e+239) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= -7e-14) {
		tmp = sqrt(2.0) / (B / t_0);
	} else if (B <= 5.1e-41) {
		tmp = -sqrt((2.0 * (2.0 * ((F * A) * ((C * (A * -4.0)) + (B * B)))))) / fma(B, B, (A * (C * -4.0)));
	} else {
		tmp = (sqrt(2.0) * t_0) / -B;
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = sqrt(Float64(F * Float64(A - hypot(A, B))))
	tmp = 0.0
	if (B <= -6.2e+239)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= -7e-14)
		tmp = Float64(sqrt(2.0) / Float64(B / t_0));
	elseif (B <= 5.1e-41)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(2.0 * Float64(Float64(F * A) * Float64(Float64(C * Float64(A * -4.0)) + Float64(B * B))))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(sqrt(2.0) * t_0) / Float64(-B));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, -6.2e+239], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -7e-14], N[(N[Sqrt[2.0], $MachinePrecision] / N[(B / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 5.1e-41], N[((-N[Sqrt[N[(2.0 * N[(2.0 * N[(N[(F * A), $MachinePrecision] * N[(N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision] / (-B)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -6.20000000000000001e239

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{-\sqrt{F \cdot \left(2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 F (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 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 F (*.f64 2 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.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 F (*.f64 2 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.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 F (*.f64 2 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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)))): 22 points increase in error, 1 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (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)))): 12 points increase in error, 4 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (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 (sqrt.f64 (*.f64 F (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 F (*.f64 2 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))) (-.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)))): 6 points increase in error, 10 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 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.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 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 B around -inf 64.0

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

   4. Taylor expanded in A around 0 26.0

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

   5. Simplified 26.0

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)} \]
      Proof
      (*.f64 (sqrt.f64 2) (neg.f64 (sqrt.f64 (/.f64 F B)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.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 -6.20000000000000001e239 < B < -7.0000000000000005e-14

   1. Initial program 51.8

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

   2. Taylor expanded in C around 0 63.3

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

   3. Simplified 62.7

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)} \]
      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))))))))): 64 points increase in error, 5 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

   4. Applied egg-rr 34.0

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

   if -7.0000000000000005e-14 < B < 5.09999999999999976e-41

   1. Initial program 50.0

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

   2. Simplified 45.5

      \[\leadsto \color{blue}{\frac{-\sqrt{F \cdot \left(2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 F (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (+.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 F (*.f64 2 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 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 F (*.f64 2 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.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 F (*.f64 2 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.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 F (*.f64 2 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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)))): 22 points increase in error, 1 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (+.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 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (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)))): 12 points increase in error, 4 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 F (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (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 (sqrt.f64 (*.f64 F (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 1 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 F (*.f64 2 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))) (-.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)))): 6 points increase in error, 10 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 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.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 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 F around 0 50.0

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

   4. Simplified 42.7

      \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(\left(\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (*.f64 (*.f64 -4 A) C)) F) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 -4 (*.f64 A C)))) F) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 B B) (*.f64 -4 (*.f64 A C)))) F) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 -4 (*.f64 A C))) F) (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C))) F) (+.f64 A (-.f64 C (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))))): 29 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C))) F) (+.f64 A (-.f64 C (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C))) F) (+.f64 A (-.f64 C (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C))) F) (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 (-.f64 A C) 2))))))): 4 points increase in error, 11 points decrease in error
      (*.f64 2 (Rewrite<= associate-*r*_binary64 (*.f64 (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C))) (*.f64 F (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 B 2) (pow.f64 (-.f64 A C) 2)))))))): 7 points increase in error, 13 points decrease in error

   5. Taylor expanded in C around inf 34.9

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

   6. Taylor expanded in F around 0 34.9

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

   7. Simplified 34.8

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(2 \cdot \left(\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      Proof
      (*.f64 2 (*.f64 (*.f64 A F) (fma.f64 -4 (*.f64 A C) (*.f64 B B)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 A F) (fma.f64 -4 (*.f64 A C) (Rewrite<= unpow2_binary64 (pow.f64 B 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 A F) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -4 (*.f64 A C)) (pow.f64 B 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (*.f64 A F) (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 F (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C))))))): 31 points increase in error, 14 points decrease in error
      (*.f64 2 (*.f64 A (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 (pow.f64 B 2) (*.f64 -4 (*.f64 A C))) F)))): 0 points increase in error, 0 points decrease in error

   8. Applied egg-rr 34.8

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

   if 5.09999999999999976e-41 < B

   1. Initial program 54.0

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

   2. Taylor expanded in C around 0 50.8

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

   3. Simplified 34.8

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)} \]
      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))))))))): 64 points increase in error, 5 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

   4. Applied egg-rr 34.8

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

4. Final simplification 34.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -6.2 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{B}{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(A \cdot -4\right) + B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}{-B}\\ \end{array} \]

Alternatives

Alternative 1
Error	35.6
Cost	20104

\[\begin{array}{l} \mathbf{if}\;B \leq -6.2 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}{B}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(A \cdot -4\right) + B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 2
Error	35.6
Cost	20104

\[\begin{array}{l} \mathbf{if}\;B \leq -6.2 \cdot 10^{+239}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{B}{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}}}\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(A \cdot -4\right) + B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 3
Error	37.6
Cost	14728

\[\begin{array}{l} \mathbf{if}\;B \leq -7.2 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 5.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot A\right) \cdot \left(C \cdot \left(A \cdot -4\right) + B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 4
Error	37.9
Cost	14472

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ \mathbf{if}\;B \leq -7.2 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(F \cdot A\right) \cdot t_0\right)\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 5
Error	39.3
Cost	14216

\[\begin{array}{l} \mathbf{if}\;B \leq -7.2 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 6
Error	37.9
Cost	14216

\[\begin{array}{l} \mathbf{if}\;B \leq -7.2 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 7
Error	41.9
Cost	13704

\[\begin{array}{l} \mathbf{if}\;B \leq -3.6 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - B\right)}\right)\\ \end{array} \]

Alternative 8
Error	42.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;B \leq -3.6 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]

Alternative 9
Error	46.3
Cost	13316

\[\begin{array}{l} \mathbf{if}\;B \leq -3.6 \cdot 10^{-107}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot A} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 10
Error	51.7
Cost	7688

\[\begin{array}{l} t_0 := \sqrt{F \cdot A}\\ \mathbf{if}\;B \leq -1.62 \cdot 10^{+66}:\\ \;\;\;\;t_0 \cdot \frac{2}{B}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 11
Error	58.4
Cost	6980

\[\begin{array}{l} t_0 := \sqrt{F \cdot A}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{-243}:\\ \;\;\;\;\frac{2}{\frac{B}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{B}{-2}}\\ \end{array} \]

Alternative 12
Error	60.9
Cost	6848

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

Reproduce

herbie shell --seed 2022291 
(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))))