ABCF->ab-angle a

Average Accuracy: 17.7% → 52.1%

Time: 52.8s

Precision: binary64

Cost: 27088

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ \mathbf{if}\;B \leq -4.4 \cdot 10^{-21}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;\left(C \cdot \sqrt{A \cdot -16}\right) \cdot \frac{\sqrt{F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot 4\right)\right)}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

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 2.0) B)) (t_1 (fma C (* A -4.0) (* B B))))
   (if (<= B -4.4e-21)
     (* t_0 (* (sqrt (+ C (hypot C B))) (sqrt F)))
     (if (<= B -7e-236)
       (/
        (- (sqrt (* 2.0 (* (* F t_1) (+ C (+ C (* -0.5 (/ (* B B) A))))))))
        t_1)
       (if (<= B 1.52e-156)
         (* (* C (sqrt (* A -16.0))) (/ (sqrt F) (fma B B (* A (* C 4.0)))))
         (if (<= B 3.2e-93)
           (/
            (*
             (sqrt (* 2.0 (* F (fma B B (* (* C A) -4.0)))))
             (- (sqrt (* 2.0 C))))
            (- (* B B) (* (* C A) 4.0)))
           (* (sqrt (+ (hypot B (- A C)) (+ C A))) (* t_0 (- (sqrt F))))))))))

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(2.0) / B;
	double t_1 = fma(C, (A * -4.0), (B * B));
	double tmp;
	if (B <= -4.4e-21) {
		tmp = t_0 * (sqrt((C + hypot(C, B))) * sqrt(F));
	} else if (B <= -7e-236) {
		tmp = -sqrt((2.0 * ((F * t_1) * (C + (C + (-0.5 * ((B * B) / A))))))) / t_1;
	} else if (B <= 1.52e-156) {
		tmp = (C * sqrt((A * -16.0))) * (sqrt(F) / fma(B, B, (A * (C * 4.0))));
	} else if (B <= 3.2e-93) {
		tmp = (sqrt((2.0 * (F * fma(B, B, ((C * A) * -4.0))))) * -sqrt((2.0 * C))) / ((B * B) - ((C * A) * 4.0));
	} else {
		tmp = sqrt((hypot(B, (A - C)) + (C + A))) * (t_0 * -sqrt(F));
	}
	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(sqrt(2.0) / B)
	t_1 = fma(C, Float64(A * -4.0), Float64(B * B))
	tmp = 0.0
	if (B <= -4.4e-21)
		tmp = Float64(t_0 * Float64(sqrt(Float64(C + hypot(C, B))) * sqrt(F)));
	elseif (B <= -7e-236)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(F * t_1) * Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A)))))))) / t_1);
	elseif (B <= 1.52e-156)
		tmp = Float64(Float64(C * sqrt(Float64(A * -16.0))) * Float64(sqrt(F) / fma(B, B, Float64(A * Float64(C * 4.0)))));
	elseif (B <= 3.2e-93)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B, B, Float64(Float64(C * A) * -4.0))))) * Float64(-sqrt(Float64(2.0 * C)))) / Float64(Float64(B * B) - Float64(Float64(C * A) * 4.0)));
	else
		tmp = Float64(sqrt(Float64(hypot(B, Float64(A - C)) + Float64(C + A))) * Float64(t_0 * Float64(-sqrt(F))));
	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[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -4.4e-21], N[(t$95$0 * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -7e-236], N[((-N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1.52e-156], N[(N[(C * N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] / N[(B * B + N[(A * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.2e-93], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B * B + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + N[(C + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < -4.4000000000000001e-21

   1. Initial program 15.2%

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

   2. Simplified 15.2%

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

      [Start] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 24.0%

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

      [Start] 15.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 15.2	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 18.7	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 18.7	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 18.7	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 24.0	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Taylor expanded in B around -inf 26.7%

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

   5. Simplified 26.7%

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

      [Start] 26.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-1 \cdot \left(\left(\sqrt{2} \cdot B\right) \cdot \sqrt{F}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      mul-1-neg [=>] 26.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(-\left(\sqrt{2} \cdot B\right) \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 26.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\color{blue}{\sqrt{2} \cdot \left(B \cdot \sqrt{F}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-rgt-neg-in [=>] 26.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(-B \cdot \sqrt{F}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Taylor expanded in A around 0 20.0%

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

   7. Simplified 47.4%

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

      [Start] 20.0	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F} \]
      unpow2 [=>] 20.0	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{\color{blue}{B \cdot B} + {C}^{2}}\right) \cdot F} \]
      unpow2 [=>] 20.0	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{B \cdot B + \color{blue}{C \cdot C}}\right) \cdot F} \]
      hypot-def [=>] 47.4	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \color{blue}{\mathsf{hypot}\left(B, C\right)}\right) \cdot F} \]

   8. Applied egg-rr 68.4%

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

      [Start] 47.4	\[ \frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot F} \]
      sqrt-prod [=>] 68.4	\[ \frac{\sqrt{2}}{B} \cdot \color{blue}{\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right)} \]
      hypot-udef [=>] 23.2	\[ \frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}} \cdot \sqrt{F}\right) \]
      +-commutative [=>] 23.2	\[ \frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \sqrt{\color{blue}{C \cdot C + B \cdot B}}} \cdot \sqrt{F}\right) \]
      hypot-def [=>] 68.4	\[ \frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \color{blue}{\mathsf{hypot}\left(C, B\right)}} \cdot \sqrt{F}\right) \]

   if -4.4000000000000001e-21 < B < -6.99999999999999988e-236

   1. Initial program 22.6%

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

   2. Simplified 33.5%

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

      [Start] 22.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 44.7%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \color{blue}{\left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]

   4. Simplified 44.7%

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)}\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]
      Proof

      [Start] 44.7	\[ \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]
      unpow2 [=>] 44.7	\[ \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right) \cdot F\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \]

   if -6.99999999999999988e-236 < B < 1.52000000000000008e-156

   1. Initial program 15.7%

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

   2. Simplified 15.6%

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

      [Start] 15.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 15.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 15.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 15.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 15.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 15.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 15.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Taylor expanded in A around -inf 27.8%

      \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Simplified 27.8%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 27.8	\[ \frac{-\sqrt{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*r* [=>] 27.8	\[ \frac{-\sqrt{\color{blue}{\left(-16 \cdot A\right) \cdot \left({C}^{2} \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 27.8	\[ \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Applied egg-rr 19.4%

      \[\leadsto \frac{-\color{blue}{\left(0 - \sqrt{-16 \cdot A} \cdot \left(C \cdot \sqrt{F}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 27.8	\[ \frac{-\sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      add-sqr-sqrt [=>] 27.8	\[ \frac{-\sqrt{\color{blue}{\sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqr-neg [<=] 27.8	\[ \frac{-\sqrt{\color{blue}{\left(-\sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}\right) \cdot \left(-\sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-unprod [<=] 1.0	\[ \frac{-\color{blue}{\sqrt{-\sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}} \cdot \sqrt{-\sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      add-sqr-sqrt [<=] 1.6	\[ \frac{-\color{blue}{\left(-\sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      neg-sub0 [=>] 1.6	\[ \frac{-\color{blue}{\left(0 - \sqrt{\left(-16 \cdot A\right) \cdot \left(\left(C \cdot C\right) \cdot F\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 1.5	\[ \frac{-\left(0 - \color{blue}{\sqrt{-16 \cdot A} \cdot \sqrt{\left(C \cdot C\right) \cdot F}}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 1.5	\[ \frac{-\left(0 - \sqrt{-16 \cdot A} \cdot \color{blue}{\left(\sqrt{C \cdot C} \cdot \sqrt{F}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 0.9	\[ \frac{-\left(0 - \sqrt{-16 \cdot A} \cdot \left(\color{blue}{\left(\sqrt{C} \cdot \sqrt{C}\right)} \cdot \sqrt{F}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      add-sqr-sqrt [<=] 19.4	\[ \frac{-\left(0 - \sqrt{-16 \cdot A} \cdot \left(\color{blue}{C} \cdot \sqrt{F}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Simplified 19.4%

      \[\leadsto \frac{-\color{blue}{\sqrt{A \cdot -16} \cdot \left(\left(-C\right) \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      Proof

      [Start] 19.4	\[ \frac{-\left(0 - \sqrt{-16 \cdot A} \cdot \left(C \cdot \sqrt{F}\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      neg-sub0 [<=] 19.4	\[ \frac{-\color{blue}{\left(-\sqrt{-16 \cdot A} \cdot \left(C \cdot \sqrt{F}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-rgt-neg-in [=>] 19.4	\[ \frac{-\color{blue}{\sqrt{-16 \cdot A} \cdot \left(-C \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-lft-neg-in [=>] 19.4	\[ \frac{-\sqrt{-16 \cdot A} \cdot \color{blue}{\left(\left(-C\right) \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 19.4	\[ \frac{-\sqrt{\color{blue}{A \cdot -16}} \cdot \left(\left(-C\right) \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   7. Applied egg-rr 33.7%

      \[\leadsto \color{blue}{\frac{\sqrt{A \cdot -16} \cdot C}{1} \cdot \frac{\sqrt{F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot 4\right)\right)}} \]
      Proof

      [Start] 19.4	\[ \frac{-\sqrt{A \cdot -16} \cdot \left(\left(-C\right) \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      distribute-rgt-neg-in [=>] 19.4	\[ \frac{\color{blue}{\sqrt{A \cdot -16} \cdot \left(-\left(-C\right) \cdot \sqrt{F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-/l* [=>] 18.0	\[ \color{blue}{\frac{\sqrt{A \cdot -16}}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{-\left(-C\right) \cdot \sqrt{F}}}} \]
      distribute-lft-neg-in [=>] 18.0	\[ \frac{\sqrt{A \cdot -16}}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\color{blue}{\left(-\left(-C\right)\right) \cdot \sqrt{F}}}} \]
      add-sqr-sqrt [=>] 17.0	\[ \frac{\sqrt{A \cdot -16}}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\left(-\color{blue}{\sqrt{-C} \cdot \sqrt{-C}}\right) \cdot \sqrt{F}}} \]
      sqrt-unprod [=>] 30.6	\[ \frac{\sqrt{A \cdot -16}}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\left(-\color{blue}{\sqrt{\left(-C\right) \cdot \left(-C\right)}}\right) \cdot \sqrt{F}}} \]
      sqr-neg [=>] 30.6	\[ \frac{\sqrt{A \cdot -16}}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\left(-\sqrt{\color{blue}{C \cdot C}}\right) \cdot \sqrt{F}}} \]
      sqrt-unprod [<=] 31.5	\[ \frac{\sqrt{A \cdot -16}}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\left(-\color{blue}{\sqrt{C} \cdot \sqrt{C}}\right) \cdot \sqrt{F}}} \]
      add-sqr-sqrt [<=] 32.3	\[ \frac{\sqrt{A \cdot -16}}{\frac{B \cdot B - 4 \cdot \left(A \cdot C\right)}{\left(-\color{blue}{C}\right) \cdot \sqrt{F}}} \]
      associate-/l* [<=] 33.7	\[ \color{blue}{\frac{\sqrt{A \cdot -16} \cdot \left(\left(-C\right) \cdot \sqrt{F}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      associate-*r* [=>] 35.1	\[ \frac{\color{blue}{\left(\sqrt{A \cdot -16} \cdot \left(-C\right)\right) \cdot \sqrt{F}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-un-lft-identity [=>] 35.1	\[ \frac{\left(\sqrt{A \cdot -16} \cdot \left(-C\right)\right) \cdot \sqrt{F}}{\color{blue}{1 \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)}} \]

   if 1.52000000000000008e-156 < B < 3.1999999999999999e-93

   1. Initial program 21.0%

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

   2. Simplified 21.0%

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

      [Start] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 35.0%

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

      [Start] 21.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 21.0	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 22.7	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 22.7	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 22.7	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 35.0	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Taylor expanded in A around -inf 42.5%

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

   5. Simplified 42.5%

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

      [Start] 42.5	\[ \frac{-\sqrt{2 \cdot C} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 42.5	\[ \frac{-\sqrt{\color{blue}{C \cdot 2}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 3.1999999999999999e-93 < B

   1. Initial program 17.7%

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

   2. Simplified 17.7%

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

      [Start] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      +-commutative [=>] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      unpow2 [=>] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      associate-*l* [=>] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \color{blue}{4 \cdot \left(A \cdot C\right)}} \]
      unpow2 [=>] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{\color{blue}{B \cdot B} - 4 \cdot \left(A \cdot C\right)} \]

   3. Applied egg-rr 27.7%

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

      [Start] 17.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 17.7	\[ \frac{-\sqrt{\color{blue}{\left(\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right) \cdot \left(2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      sqrt-prod [=>] 20.7	\[ \frac{-\color{blue}{\sqrt{\left(A + C\right) + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 20.7	\[ \frac{-\sqrt{\color{blue}{A + \left(C + \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 20.7	\[ \frac{-\sqrt{A + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      hypot-def [=>] 27.7	\[ \frac{-\sqrt{A + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)} \cdot \sqrt{2 \cdot \left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   4. Applied egg-rr 27.7%

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

      [Start] 27.7	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      div-inv [=>] 27.7	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}} \]
      distribute-rgt-neg-in [=>] 27.7	\[ \color{blue}{\left(\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right)\right)} \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 27.7	\[ \color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right)} \]
      *-commutative [=>] 27.7	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      associate-*l* [=>] 27.7	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right)}\right) \cdot \frac{1}{B \cdot B - 4 \cdot \left(A \cdot C\right)}\right) \]
      fma-neg [=>] 27.7	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\right) \]
      distribute-lft-neg-in [=>] 27.7	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{\left(-4\right) \cdot \left(A \cdot C\right)}\right)}\right) \]
      metadata-eval [=>] 27.7	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \color{blue}{-4} \cdot \left(A \cdot C\right)\right)}\right) \]

   5. Simplified 27.7%

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

      [Start] 27.7	\[ \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-+r+ [=>] 27.7	\[ \sqrt{\color{blue}{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      +-commutative [=>] 27.7	\[ \sqrt{\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)} \cdot \left(\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right) \]
      associate-*r/ [=>] 27.7	\[ \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \color{blue}{\frac{\left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right) \cdot 1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]
      *-rgt-identity [=>] 27.7	\[ \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \frac{\color{blue}{-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
      associate-*r* [=>] 27.7	\[ \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Taylor expanded in B around inf 54.7%

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

   7. Simplified 54.7%

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

      [Start] 54.7	\[ \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \left(-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\right) \]
      mul-1-neg [=>] 54.7	\[ \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \color{blue}{\left(-\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)} \]
      distribute-rgt-neg-in [=>] 54.7	\[ \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -4.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -7 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.52 \cdot 10^{-156}:\\ \;\;\;\;\left(C \cdot \sqrt{A \cdot -16}\right) \cdot \frac{\sqrt{F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot 4\right)\right)}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	53.6%
Cost	27912

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-18}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	49.1%
Cost	26372

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot 4\right)\right)\\ t_2 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_2}\\ t_4 := \sqrt{A \cdot -16}\\ t_5 := C + \mathsf{hypot}\left(C, B\right)\\ t_6 := \sqrt{t_5 \cdot F}\\ \mathbf{if}\;B \leq -1.22 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{t_5} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-235}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.1 \cdot 10^{-287}:\\ \;\;\;\;C \cdot \left(\sqrt{F} \cdot \frac{t_4}{t_1}\right)\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-265}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.38 \cdot 10^{-157}:\\ \;\;\;\;\left(C \cdot t_4\right) \cdot \frac{\sqrt{F}}{t_1}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{-86}:\\ \;\;\;\;t_4 \cdot \left(\sqrt{F} \cdot \frac{-C}{t_1}\right)\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_6 \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq 10^{+31}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3
Accuracy	43.5%
Cost	22436

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_0}\\ t_2 := \sqrt{A \cdot -16}\\ t_3 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_4 := C + \mathsf{hypot}\left(C, B\right)\\ t_5 := \sqrt{t_4 \cdot F}\\ t_6 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot 4\right)\right)\\ \mathbf{if}\;B \leq -5.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_4\right)}}{B}\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-287}:\\ \;\;\;\;C \cdot \left(\sqrt{F} \cdot \frac{t_2}{t_6}\right)\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-266}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-158}:\\ \;\;\;\;\left(C \cdot t_2\right) \cdot \frac{\sqrt{F}}{t_6}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_3}\\ \mathbf{elif}\;B \leq 8.6 \cdot 10^{-88}:\\ \;\;\;\;t_2 \cdot \left(\sqrt{F} \cdot \frac{-C}{t_6}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_5 \cdot \left(\sqrt{2} \cdot \left(-B\right)\right)}{t_3}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_5 \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4
Accuracy	44.8%
Cost	21128

\[\begin{array}{l} t_0 := C + \mathsf{hypot}\left(C, B\right)\\ \mathbf{if}\;B \leq -3.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_0\right)}}{B}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(C \cdot A\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0 \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5
Accuracy	45.3%
Cost	21000

\[\begin{array}{l} t_0 := \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)\\ t_1 := C + \mathsf{hypot}\left(C, B\right)\\ \mathbf{if}\;B \leq -2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_1\right)}}{B}\\ \mathbf{elif}\;B \leq -6.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -6.4 \cdot 10^{-288}:\\ \;\;\;\;C \cdot \left(\sqrt{F} \cdot \frac{\sqrt{A \cdot -16}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot 4\right)\right)}\right)\\ \mathbf{elif}\;B \leq 2.45 \cdot 10^{-88}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6
Accuracy	44.5%
Cost	20556

\[\begin{array}{l} t_0 := \frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ t_1 := C + \mathsf{hypot}\left(C, B\right)\\ \mathbf{if}\;B \leq -2.15 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_1\right)}}{B}\\ \mathbf{elif}\;B \leq -2.2 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -4.6 \cdot 10^{-288}:\\ \;\;\;\;C \cdot \left(\sqrt{F} \cdot \frac{\sqrt{A \cdot -16}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot 4\right)\right)}\right)\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_1 \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Accuracy	43.6%
Cost	20432

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := C + \mathsf{hypot}\left(C, B\right)\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_1\right)}}{B}\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{+296}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 8
Accuracy	45.0%
Cost	20168

\[\begin{array}{l} t_0 := C + \mathsf{hypot}\left(C, B\right)\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_0\right)}}{B}\\ \mathbf{elif}\;B \leq 9 \cdot 10^{-91}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t_0 \cdot F} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9
Accuracy	41.1%
Cost	13712

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B} \cdot \sqrt{B \cdot \left(-F\right)}\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq -4.9 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -10000000000:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(\left(C + A\right) + \left(C - \left(A + 0.5 \cdot \frac{0 \cdot \left(A + A\right) - B \cdot B}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 10
Accuracy	43.4%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;B \leq -8.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{B}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-63}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 11
Accuracy	32.8%
Cost	13316

\[\begin{array}{l} \mathbf{if}\;B \leq 3.9 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{\frac{F}{B}}\\ \end{array} \]

Alternative 12
Accuracy	26.3%
Cost	9476

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;A \leq -1.69 \cdot 10^{-80}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\left(C + A\right) + \left(C - \left(A + 0.5 \cdot \frac{0 \cdot \left(A + A\right) - B \cdot B}{C}\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 13
Accuracy	24.8%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq 7.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(B + \left(C + A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Accuracy	26.2%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;A \leq -1.69 \cdot 10^{-80}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C + \left(C + A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 15
Accuracy	23.6%
Cost	7680

\[-\frac{\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4} \]

Alternative 16
Accuracy	5.0%
Cost	6848

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

Reproduce

herbie shell --seed 2023151 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))