ABCF->ab-angle a

Average Error: 53.0 → 42.0

Time: 59.8s

Precision: binary64

Cost: 41444

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_3 := A + \left(C + t_0\right)\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_5 := -\sqrt{t_4}\\ t_6 := \frac{\sqrt{\left(2 \cdot F\right) \cdot t_3} \cdot t_5}{t_4}\\ t_7 := \frac{\left(\sqrt{C + \left(A + t_0\right)} \cdot \sqrt{2 \cdot F}\right) \cdot t_5}{t_4}\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-178}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \left(B \cdot \frac{B}{A}\right)\right) \cdot \left(2 \cdot \left(F \cdot t_2\right)\right)} \cdot \frac{1}{-t_2}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-254}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot t_3\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-95}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{t_3} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_1\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

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 (hypot B (- A C)))
        (t_1 (- (* B B) (* (* C A) 4.0)))
        (t_2 (fma B B (* C (* -4.0 A))))
        (t_3 (+ A (+ C t_0)))
        (t_4 (fma -4.0 (* C A) (* B B)))
        (t_5 (- (sqrt t_4)))
        (t_6 (/ (* (sqrt (* (* 2.0 F) t_3)) t_5) t_4))
        (t_7 (/ (* (* (sqrt (+ C (+ A t_0))) (sqrt (* 2.0 F))) t_5) t_4)))
   (if (<= B -1.65e-178)
     t_7
     (if (<= B 1.25e-278)
       (*
        (sqrt (* (fma 2.0 C (* -0.5 (* B (/ B A)))) (* 2.0 (* F t_2))))
        (/ 1.0 (- t_2)))
       (if (<= B 2.2e-254)
         t_6
         (if (<= B 4.8e-220)
           (sqrt (/ (- F) A))
           (if (<= B 5.2e-127)
             (/ (- (sqrt (* 2.0 (* t_2 (* F t_3))))) t_2)
             (if (<= B 8.8e-95)
               t_6
               (if (<= B 8.5e-20)
                 (/ (* (sqrt t_3) (- (sqrt (* 2.0 (* F t_1))))) t_1)
                 (if (<= B 1.15e+16)
                   (/
                    (-
                     (sqrt
                      (*
                       2.0
                       (* t_2 (* F (fma 2.0 A (* -0.5 (/ (* B B) C))))))))
                    t_2)
                   (if (<= B 9.5e+153)
                     t_7
                     (* (sqrt (/ F B)) (- (sqrt 2.0))))))))))))))

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 = hypot(B, (A - C));
	double t_1 = (B * B) - ((C * A) * 4.0);
	double t_2 = fma(B, B, (C * (-4.0 * A)));
	double t_3 = A + (C + t_0);
	double t_4 = fma(-4.0, (C * A), (B * B));
	double t_5 = -sqrt(t_4);
	double t_6 = (sqrt(((2.0 * F) * t_3)) * t_5) / t_4;
	double t_7 = ((sqrt((C + (A + t_0))) * sqrt((2.0 * F))) * t_5) / t_4;
	double tmp;
	if (B <= -1.65e-178) {
		tmp = t_7;
	} else if (B <= 1.25e-278) {
		tmp = sqrt((fma(2.0, C, (-0.5 * (B * (B / A)))) * (2.0 * (F * t_2)))) * (1.0 / -t_2);
	} else if (B <= 2.2e-254) {
		tmp = t_6;
	} else if (B <= 4.8e-220) {
		tmp = sqrt((-F / A));
	} else if (B <= 5.2e-127) {
		tmp = -sqrt((2.0 * (t_2 * (F * t_3)))) / t_2;
	} else if (B <= 8.8e-95) {
		tmp = t_6;
	} else if (B <= 8.5e-20) {
		tmp = (sqrt(t_3) * -sqrt((2.0 * (F * t_1)))) / t_1;
	} else if (B <= 1.15e+16) {
		tmp = -sqrt((2.0 * (t_2 * (F * fma(2.0, A, (-0.5 * ((B * B) / C))))))) / t_2;
	} else if (B <= 9.5e+153) {
		tmp = t_7;
	} else {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	}
	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 = hypot(B, Float64(A - C))
	t_1 = Float64(Float64(B * B) - Float64(Float64(C * A) * 4.0))
	t_2 = fma(B, B, Float64(C * Float64(-4.0 * A)))
	t_3 = Float64(A + Float64(C + t_0))
	t_4 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_5 = Float64(-sqrt(t_4))
	t_6 = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_3)) * t_5) / t_4)
	t_7 = Float64(Float64(Float64(sqrt(Float64(C + Float64(A + t_0))) * sqrt(Float64(2.0 * F))) * t_5) / t_4)
	tmp = 0.0
	if (B <= -1.65e-178)
		tmp = t_7;
	elseif (B <= 1.25e-278)
		tmp = Float64(sqrt(Float64(fma(2.0, C, Float64(-0.5 * Float64(B * Float64(B / A)))) * Float64(2.0 * Float64(F * t_2)))) * Float64(1.0 / Float64(-t_2)));
	elseif (B <= 2.2e-254)
		tmp = t_6;
	elseif (B <= 4.8e-220)
		tmp = sqrt(Float64(Float64(-F) / A));
	elseif (B <= 5.2e-127)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * t_3))))) / t_2);
	elseif (B <= 8.8e-95)
		tmp = t_6;
	elseif (B <= 8.5e-20)
		tmp = Float64(Float64(sqrt(t_3) * Float64(-sqrt(Float64(2.0 * Float64(F * t_1))))) / t_1);
	elseif (B <= 1.15e+16)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * fma(2.0, A, Float64(-0.5 * Float64(Float64(B * B) / C)))))))) / t_2);
	elseif (B <= 9.5e+153)
		tmp = t_7;
	else
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(B * B), $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-N[Sqrt[t$95$4], $MachinePrecision])}, Block[{t$95$6 = N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[Sqrt[N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[B, -1.65e-178], t$95$7, If[LessEqual[B, 1.25e-278], N[(N[Sqrt[N[(N[(2.0 * C + N[(-0.5 * N[(B * N[(B / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.2e-254], t$95$6, If[LessEqual[B, 4.8e-220], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision], If[LessEqual[B, 5.2e-127], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 8.8e-95], t$95$6, If[LessEqual[B, 8.5e-20], N[(N[(N[Sqrt[t$95$3], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B, 1.15e+16], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(2.0 * A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[B, 9.5e+153], t$95$7, N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 8 regimes

2. if B < -1.6500000000000001e-178 or 1.15e16 < B < 9.4999999999999995e153

   1. Initial program 52.0

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

   2. Simplified 48.4

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

      [Start] 52.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 44.7

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

   4. Simplified 44.4

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

      [Start] 44.7	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 44.7	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 44.7	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 44.7	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 44.4	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Applied egg-rr 41.5

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

   if -1.6500000000000001e-178 < B < 1.24999999999999996e-278

   1. Initial program 53.6

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

   2. Simplified 53.6

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

      [Start] 53.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 44.7

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

   4. Simplified 44.7

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

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

   5. Taylor expanded in F around 0 44.7

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

   6. Simplified 44.7

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

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

   7. Taylor expanded in A around -inf 49.5

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

   8. Simplified 49.5

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

      [Start] 49.5	\[ \frac{-\sqrt{2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-def [=>] 49.5	\[ \frac{-\sqrt{\color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 49.5	\[ \frac{-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-/l* [=>] 49.5	\[ \frac{-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \color{blue}{\frac{B}{\frac{A}{B}}}\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   9. Applied egg-rr 49.0

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \left(B \cdot \frac{B}{A}\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   if 1.24999999999999996e-278 < B < 2.2000000000000001e-254 or 5.19999999999999982e-127 < B < 8.7999999999999995e-95

   1. Initial program 51.1

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

   2. Simplified 43.7

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

      [Start] 51.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 49.1

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

   4. Simplified 48.3

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

      [Start] 49.1	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 49.1	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 49.1	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 49.1	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 48.3	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if 2.2000000000000001e-254 < B < 4.8000000000000003e-220

   1. Initial program 52.6

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

   2. Simplified 46.7

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

      [Start] 52.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 58.0

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

   4. Simplified 57.1

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

      [Start] 58.0	\[ \sqrt{\frac{2 \cdot \left(\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)\right)}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-*r* [=>] 58.0	\[ \sqrt{\frac{\color{blue}{\left(2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)}}{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}} \]
      associate-/l* [=>] 57.1	\[ \sqrt{\color{blue}{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F}}}} \]
      *-commutative [=>] 57.1	\[ \sqrt{\frac{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\frac{{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right)}^{2}}{\color{blue}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}}}} \]

   5. Taylor expanded in C around inf 50.7

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

   6. Simplified 50.7

      \[\leadsto \sqrt{\color{blue}{\frac{-F}{A}}} \]
      Proof

      [Start] 50.7	\[ \sqrt{-1 \cdot \frac{F}{A}} \]
      associate-*r/ [=>] 50.7	\[ \sqrt{\color{blue}{\frac{-1 \cdot F}{A}}} \]
      neg-mul-1 [<=] 50.7	\[ \sqrt{\frac{\color{blue}{-F}}{A}} \]

   if 4.8000000000000003e-220 < B < 5.19999999999999982e-127

   1. Initial program 51.5

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

   2. Simplified 45.5

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

      [Start] 51.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 8.7999999999999995e-95 < B < 8.5000000000000005e-20

   1. Initial program 47.2

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

   2. Simplified 47.2

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

      [Start] 47.2	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 36.6

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

   4. Simplified 36.6

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

      [Start] 36.6	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 36.6	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 36.6	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in F around 0 36.6

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

   6. Simplified 36.6

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

      [Start] 36.6	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 36.6	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 36.6	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 36.6	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(B \cdot B + -4 \cdot \color{blue}{\left(C \cdot A\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if 8.5000000000000005e-20 < B < 1.15e16

   1. Initial program 37.8

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

   2. Simplified 32.8

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

      [Start] 37.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around -inf 52.9

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

   4. Simplified 52.9

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

      [Start] 52.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 52.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 52.9	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 9.4999999999999995e153 < B

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around inf 64.0

      \[\leadsto \frac{-\sqrt{\color{blue}{2 \cdot \left(F \cdot {B}^{3}\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   4. Simplified 64.0

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot {B}^{3}}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      Proof

      [Start] 64.0	\[ \frac{-\sqrt{2 \cdot \left(F \cdot {B}^{3}\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-*r* [=>] 64.0	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot F\right) \cdot {B}^{3}}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   5. Taylor expanded in A around 0 30.6

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

   6. Simplified 30.6

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

      [Start] 30.6	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      mul-1-neg [=>] 30.6	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      *-commutative [=>] 30.6	\[ -\color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
      distribute-rgt-neg-in [=>] 30.6	\[ \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)} \]

3. Recombined 8 regimes into one program.

4. Final simplification 42.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.65 \cdot 10^{-178}:\\ \;\;\;\;\frac{\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot F}\right) \cdot \left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \left(B \cdot \frac{B}{A}\right)\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-254}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \left(B \cdot B - \left(C \cdot A\right) \cdot 4\right)\right)}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot F}\right) \cdot \left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	43.0
Cost	34648

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_4 := \sqrt{t_2}\\ t_5 := A + \left(C + t_0\right)\\ t_6 := \frac{\sqrt{\left(2 \cdot F\right) \cdot t_5} \cdot \left(-t_4\right)}{t_2}\\ t_7 := \sqrt{t_5}\\ \mathbf{if}\;B \leq -6.6 \cdot 10^{-176}:\\ \;\;\;\;-\frac{\sqrt{F \cdot \left(\left(C + \left(A + t_0\right)\right) \cdot 2\right)}}{t_4}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \left(B \cdot \frac{B}{A}\right)\right) \cdot \left(2 \cdot \left(F \cdot t_3\right)\right)} \cdot \frac{1}{-t_3}\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-254}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-220}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-130}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot t_5\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 2.65 \cdot 10^{-95}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{t_7 \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_1\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{t_7 \cdot \left(-\left(B \cdot \sqrt{2}\right) \cdot \sqrt{F}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 2
Error	42.0
Cost	28180

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)}}{t_1}\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_4 := \sqrt{t_3}\\ t_5 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;C \leq -1.5 \cdot 10^{+239}:\\ \;\;\;\;\frac{t_4 \cdot \left(-\sqrt{\left(2 \cdot F\right) \cdot \left(A \cdot 2\right)}\right)}{t_3}\\ \mathbf{elif}\;C \leq -5.7 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;C \leq -6.5 \cdot 10^{+159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -3.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;C \leq -4.25 \cdot 10^{+15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 1.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(t_0 + \left(C + A\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_5\right)}\right)}{t_5}\\ \end{array} \]

Alternative 3
Error	44.1
Cost	27736

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := \sqrt{2 \cdot \left(F \cdot t_1\right)}\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := -\frac{\sqrt{F \cdot \left(\left(C + \left(A + t_3\right)\right) \cdot 2\right)}}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_3\right)} \cdot \left(-t_2\right)}{t_1}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+24}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;F \leq 1.66 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+121}:\\ \;\;\;\;\frac{t_2 \cdot \left(-\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B}{\frac{A}{B}}\right)}\right)}{t_1}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+260}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	42.0
Cost	27472

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \frac{-\sqrt{\left(2 \cdot F\right) \cdot \left(t_1 + \left(C + A\right)\right)}}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}\\ \mathbf{if}\;A \leq -6.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq -2.35 \cdot 10^{-89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq -1.2 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq 4 \cdot 10^{-224}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_1\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ \end{array} \]

Alternative 5
Error	43.5
Cost	21841

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ \mathbf{if}\;C \leq -2.3 \cdot 10^{+243}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)}\right)}{t_0}\\ \mathbf{elif}\;C \leq -1.8 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;C \leq -5.2 \cdot 10^{-199} \lor \neg \left(C \leq -2.1 \cdot 10^{-266}\right):\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-t_1\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 6
Error	43.4
Cost	21709

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;C \leq -9 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;C \leq -3 \cdot 10^{-199} \lor \neg \left(C \leq -2.5 \cdot 10^{-266}\right):\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7
Error	45.1
Cost	21516

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_2 := \sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\\ \mathbf{if}\;B \leq -1.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{B \cdot t_2}{t_1}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{t_2 \cdot \left(2 \cdot \frac{C \cdot A}{B} - B\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 8
Error	46.1
Cost	21332

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq -1 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq -3.3 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_1\right)} \cdot \left(-\sqrt{A \cdot 2}\right)}{t_1}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;B \leq 1.95 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_0 \cdot \left(F \cdot \left(C \cdot -2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+151}:\\ \;\;\;\;\frac{B \cdot \left(-\sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9
Error	45.8
Cost	21200

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot 4\\ t_1 := B \cdot B - t_0\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)}\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_1\right)} \cdot \left(-\sqrt{A \cdot 2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{B \cdot \left(-\sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 10
Error	47.3
Cost	21004

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := \left(C \cdot A\right) \cdot 4\\ t_2 := B \cdot B - t_1\\ t_3 := \frac{-\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{if}\;A \leq -1.6 \cdot 10^{+85}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq -3.6 \cdot 10^{-39}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot t_2\right)} \cdot \sqrt{C \cdot 2}}{t_2}\\ \mathbf{elif}\;A \leq -8.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{B \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;A \leq -8 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -2.7 \cdot 10^{-208}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq 2.2 \cdot 10^{-244}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{+188}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	48.9
Cost	15384

\[\begin{array}{l} t_0 := \sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := F \cdot t_1\\ t_3 := -\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(C + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{if}\;A \leq -7.6 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq -1.55 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;A \leq -4.2 \cdot 10^{-208}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq 1.52 \cdot 10^{-204}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 1.32 \cdot 10^{-73}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_2} \cdot \left(-\sqrt{A \cdot 2}\right)}{t_1}\\ \end{array} \]

Alternative 12
Error	47.6
Cost	15308

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot 4\\ t_1 := B \cdot B - t_0\\ \mathbf{if}\;C \leq -7.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;C \leq -9.5 \cdot 10^{-267}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;C \leq 5.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot t_1\right)} \cdot \sqrt{C \cdot 2}}{t_1}\\ \end{array} \]

Alternative 13
Error	48.4
Cost	15120

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt{2 \cdot \left(F \cdot t_0\right)}\\ t_2 := \sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{if}\;C \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;C \leq 2.7 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq 1.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{A \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;C \leq 1.95 \cdot 10^{-87}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_1 \cdot \sqrt{C \cdot 2}}{t_0}\\ \end{array} \]

Alternative 14
Error	50.9
Cost	13976

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := \left(C \cdot A\right) \cdot 4\\ t_2 := \sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ t_3 := B \cdot B - t_1\\ t_4 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(C + \left(C + A\right)\right)\right)}}{t_3}\\ \mathbf{if}\;A \leq -4.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq -7.6 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -4.4 \cdot 10^{-208}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;A \leq 1.52 \cdot 10^{-204}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 1.02 \cdot 10^{-73}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;A \leq 4.1 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 5.7 \cdot 10^{+186}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	51.5
Cost	13188

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := F \cdot t_1\\ t_3 := -\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(C + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{if}\;A \leq -1.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-A}}\\ \mathbf{elif}\;A \leq -7.6 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -5.5 \cdot 10^{-203}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq 9.5 \cdot 10^{-206}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(B + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq 3.6 \cdot 10^{+185}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A - C\right) + \left(C + A\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	52.6
Cost	9240

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := F \cdot t_0\\ t_2 := -\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(C + A\right)\right)\right)}}{t_0}\\ t_3 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -8.5 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;A \leq -8 \cdot 10^{-134}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;A \leq -9.6 \cdot 10^{-203}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 4.5 \cdot 10^{-209}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 8 \cdot 10^{-44}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 1.05 \cdot 10^{+188}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A - C\right) + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 17
Error	52.5
Cost	9112

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ t_1 := \sqrt{\frac{-F}{C}}\\ t_2 := \left(C \cdot A\right) \cdot 4\\ t_3 := B \cdot B - t_2\\ \mathbf{if}\;A \leq -2.55 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -1.05 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.3 \cdot 10^{-200}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_3\right) \cdot \left(B + \left(C + A\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 10^{+52}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{t_3}\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{+188}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot \left(t_2 - B \cdot B\right)\right)\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	52.6
Cost	9112

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{C}}\\ t_1 := \left(C \cdot A\right) \cdot 4\\ t_2 := B \cdot B - t_1\\ t_3 := F \cdot t_2\\ t_4 := -\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(C + \left(C + A\right)\right)\right)}}{t_2}\\ \mathbf{if}\;A \leq -2.5 \cdot 10^{+60}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \mathbf{elif}\;A \leq -7.6 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-203}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(B + \left(C + A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{-44}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;A \leq 8.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot \left(t_1 - B \cdot B\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	52.7
Cost	8848

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt{\frac{-F}{A}}\\ t_2 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -1.1 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.65 \cdot 10^{-298}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{+188}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(F \cdot \left(t_0 - B \cdot B\right)\right)\right)}}{B \cdot B - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 20
Error	52.7
Cost	8336

\[\begin{array}{l} t_0 := \sqrt{\frac{-F}{A}}\\ t_1 := \sqrt{\frac{-F}{C}}\\ \mathbf{if}\;A \leq -1.05 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 1.95 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq 2.6 \cdot 10^{+146}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(A \cdot A\right) \cdot \left(C \cdot F\right)\right) \cdot 8\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	54.1
Cost	7053

\[\begin{array}{l} \mathbf{if}\;C \leq -8.4 \cdot 10^{-10} \lor \neg \left(C \leq 6.5 \cdot 10^{-246}\right) \land C \leq 3 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]

Alternative 22
Error	56.7
Cost	6656

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

Alternative 23
Error	63.2
Cost	6592

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

Reproduce

herbie shell --seed 2023066 
(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))))