ABCF->ab-angle b

Average Error: 52.5 → 38.5

Time: 37.1s

Precision: binary64

Cost: 27720

Math

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

↓

\[\begin{array}{l} t_0 := C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\\ t_1 := \sqrt{t_0 \cdot \left(2 \cdot F\right)}\\ t_2 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.35 \cdot 10^{+14}:\\ \;\;\;\;\frac{t_1}{B}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot t_2\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{-B}\\ \end{array} \]

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 (+ C (- A (hypot B (- A C)))))
        (t_1 (sqrt (* t_0 (* 2.0 F))))
        (t_2 (fma B B (* A (* C -4.0)))))
   (if (<= B -2.35e+14)
     (/ t_1 B)
     (if (<= B 3.1e-47)
       (/ (- (sqrt (* 2.0 (* (* t_0 F) t_2)))) t_2)
       (/ t_1 (- 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 = C + (A - hypot(B, (A - C)));
	double t_1 = sqrt((t_0 * (2.0 * F)));
	double t_2 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= -2.35e+14) {
		tmp = t_1 / B;
	} else if (B <= 3.1e-47) {
		tmp = -sqrt((2.0 * ((t_0 * F) * t_2))) / t_2;
	} else {
		tmp = t_1 / -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 = Float64(C + Float64(A - hypot(B, Float64(A - C))))
	t_1 = sqrt(Float64(t_0 * Float64(2.0 * F)))
	t_2 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= -2.35e+14)
		tmp = Float64(t_1 / B);
	elseif (B <= 3.1e-47)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * t_2)))) / t_2);
	else
		tmp = Float64(t_1 / 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[(C + N[(A - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.35e+14], N[(t$95$1 / B), $MachinePrecision], If[LessEqual[B, 3.1e-47], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(t$95$1 / (-B)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -2.35e14

   1. Initial program 55.4

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

   2. Simplified 53.3

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F))) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 9 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 14 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 27 points decrease in error

   3. Applied egg-rr 63.1

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

   4. Simplified 63.1

      \[\leadsto \color{blue}{\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot C\right) \cdot -4}\right)}}} \]
      Proof
      (/.f64 (sqrt.f64 (*.f64 (+.f64 C (-.f64 A (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 C A) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4)))) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 8 points increase in error, 1 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4))))))): 9 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))) (fma.f64 B B (*.f64 A (*.f64 C -4))))): 1 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1)) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F)))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1) (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (fma.f64 B B (*.f64 A (*.f64 C -4)))))): 0 points increase in error, 9 points decrease in error

   5. Taylor expanded in B around inf 31.9

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

   if -2.35e14 < B < 3.0999999999999998e-47

   1. Initial program 50.9

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

   2. Simplified 44.8

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F))) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 9 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 14 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 27 points decrease in error

   if 3.0999999999999998e-47 < B

   1. Initial program 52.7

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

   2. Simplified 50.2

      \[\leadsto \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      Proof
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C))))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C)))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C))) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 C (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (Rewrite<= +-rgt-identity_binary64 (+.f64 C 0)) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 18 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 B B) (*.f64 (-.f64 A C) (-.f64 A C))))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (-.f64 A C) (-.f64 A C)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (+.f64 (pow.f64 B 2) (Rewrite<= unpow2_binary64 (pow.f64 (-.f64 A C) 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (+.f64 A (-.f64 (+.f64 C 0) (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 A (+.f64 C 0)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A (Rewrite=> +-rgt-identity_binary64 C)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))) (Rewrite<= *-commutative_binary64 (*.f64 F 2)))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 F 2) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) (*.f64 F 2)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F) 2)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F))) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 9 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (*.f64 A (*.f64 C (Rewrite<= metadata-eval (neg.f64 4)))))): 9 points increase in error, 14 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 A C) (neg.f64 4))))): 14 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 A C) 4))))): 0 points increase in error, 27 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 4 (*.f64 A C)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (fma.f64 B B (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 4 A) C))))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 B B) (*.f64 (*.f64 4 A) C)))): 27 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 B 2)) (*.f64 (*.f64 4 A) C))): 0 points increase in error, 27 points decrease in error

   3. Applied egg-rr 63.1

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

   4. Simplified 63.1

      \[\leadsto \color{blue}{\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{\frac{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}{\mathsf{hypot}\left(B, \sqrt{\left(A \cdot C\right) \cdot -4}\right)}}} \]
      Proof
      (/.f64 (sqrt.f64 (*.f64 (+.f64 C (-.f64 A (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 C A) (hypot.f64 B (-.f64 A C)))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 A C)) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 (*.f64 A C) -4)) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 0 points increase in error, 9 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4)))) (hypot.f64 B (sqrt.f64 (*.f64 (*.f64 A C) -4))))): 8 points increase in error, 1 points decrease in error
      (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (/.f64 (fma.f64 B B (*.f64 A (*.f64 C -4))) (hypot.f64 B (sqrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 A (*.f64 C -4))))))): 9 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4))))) (fma.f64 B B (*.f64 A (*.f64 C -4))))): 1 points increase in error, 8 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 9 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1)) (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F)))) (fma.f64 B B (*.f64 A (*.f64 C -4)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 (hypot.f64 B (sqrt.f64 (*.f64 A (*.f64 C -4)))) 1) (/.f64 (sqrt.f64 (*.f64 (-.f64 (+.f64 A C) (hypot.f64 B (-.f64 A C))) (*.f64 2 F))) (fma.f64 B B (*.f64 A (*.f64 C -4)))))): 0 points increase in error, 9 points decrease in error

   5. Taylor expanded in B around -inf 33.5

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

   6. Simplified 33.5

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

4. Final simplification 38.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.35 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternatives

Alternative 1
Error	40.6
Cost	21000

\[\begin{array}{l} t_0 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -0.42:\\ \;\;\;\;\frac{t_0}{B}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(C \cdot 2\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \]

Alternative 2
Error	40.6
Cost	20936

\[\begin{array}{l} t_0 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.8:\\ \;\;\;\;\frac{t_0}{B}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{t_1 \cdot \left(F \cdot \left(C \cdot 4\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \]

Alternative 3
Error	40.9
Cost	14408

\[\begin{array}{l} t_0 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -0.55:\\ \;\;\;\;\frac{t_0}{B}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-91}:\\ \;\;\;\;\frac{-{\left(\left(C \cdot F\right) \cdot \left(\left(A \cdot -8\right) \cdot \left(C + C\right)\right)\right)}^{0.5}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \]

Alternative 4
Error	41.9
Cost	14288

\[\begin{array}{l} t_0 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -38:\\ \;\;\;\;\frac{t_0}{B}\\ \mathbf{elif}\;B \leq -5.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(A + \left(C + C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{A \cdot \left(C \cdot -4\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \]

Alternative 5
Error	41.8
Cost	14288

\[\begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}\\ \mathbf{if}\;B \leq -0.42:\\ \;\;\;\;\frac{t_1}{B}\\ \mathbf{elif}\;B \leq -8.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{-\sqrt{\left(C \cdot \left(C \cdot A\right)\right) \cdot \left(F \cdot -16\right)}}{\mathsf{fma}\left(B, B, t_0\right)}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{t_0 + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{-B}\\ \end{array} \]

Alternative 6
Error	48.1
Cost	13828

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -0.42:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)}}{B}\\ \mathbf{elif}\;B \leq -4.3 \cdot 10^{-100}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{A \cdot \left(C \cdot -4\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 7
Error	54.9
Cost	8848

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{A \cdot \left(C \cdot -4\right) + B \cdot B}\\ \mathbf{if}\;B \leq -26000000:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C}{\frac{B}{\frac{F}{B}}}}\\ \mathbf{elif}\;B \leq -1.75 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.46 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 8
Error	54.9
Cost	8848

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{+16}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C}{\frac{B}{\frac{F}{B}}}}\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{A \cdot \left(C \cdot -4\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C - B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 9
Error	52.6
Cost	8848

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ t_2 := F \cdot t_1\\ \mathbf{if}\;A \leq -3 \cdot 10^{-136}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -8 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;A \leq -8.8 \cdot 10^{-256}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C \cdot F}{B \cdot B}}\\ \mathbf{elif}\;A \leq 6.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \end{array} \]

Alternative 10
Error	52.2
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ t_2 := F \cdot t_1\\ \mathbf{if}\;A \leq -3.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 8.2 \cdot 10^{-89}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \end{array} \]

Alternative 11
Error	51.7
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(C \cdot A\right)\\ t_1 := B \cdot B + t_0\\ t_2 := F \cdot t_1\\ \mathbf{if}\;A \leq -155000:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + \left(A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \end{array} \]

Alternative 12
Error	56.4
Cost	8072

\[\begin{array}{l} t_0 := -\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}\\ \mathbf{if}\;B \leq -17000:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C}{\frac{B}{\frac{F}{B}}}}\\ \mathbf{elif}\;B \leq -5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{t_0}{A \cdot \left(C \cdot -4\right) + B \cdot B}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-4 \cdot \left(C \cdot A\right)}\\ \end{array} \]

Alternative 13
Error	56.5
Cost	7949

\[\begin{array}{l} \mathbf{if}\;B \leq -26000000:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C}{\frac{B}{\frac{F}{B}}}}\\ \mathbf{elif}\;B \leq -4.3 \cdot 10^{-50} \lor \neg \left(B \leq 6.2 \cdot 10^{-244}\right):\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(C \cdot F\right) \cdot \left(C + C\right)\right)}}{-4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 14
Error	56.3
Cost	7108

\[\begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{B} \cdot \frac{C}{B}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 15
Error	56.5
Cost	7108

\[\begin{array}{l} \mathbf{if}\;B \leq -2 \cdot 10^{-38}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{C}{\frac{B}{\frac{F}{B}}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 16
Error	56.8
Cost	6656

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

Alternative 17
Error	61.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022340 
(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))))