ABCF->ab-angle b

Average Error: 18.7% → 39.0%

Time: 1.5min

Precision: binary64

Cost: 34516.00

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_2 := -\sqrt{2 \cdot t_1}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - t_0\right)\right)} \cdot \frac{t_2}{t_1}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A \cdot -8\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\frac{t_2 \cdot \sqrt{F \cdot \left(A - \left(t_0 - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

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 (fma A (* C -4.0) (* B B)))
        (t_2 (- (sqrt (* 2.0 t_1)))))
   (if (<= B -9.2e+153)
     (* (sqrt 2.0) (- (sqrt (/ F B))))
     (if (<= B -4.2e-133)
       (* (sqrt (* F (+ C (- A t_0)))) (/ t_2 t_1))
       (if (<= B -2e-186)
         (/ (- (sqrt (* F (* (* A -8.0) (* C (+ A A)))))) t_1)
         (if (<= B 1e-121)
           (/ (- (sqrt (* t_1 (* F (* C 4.0))))) t_1)
           (if (<= B 2e+100)
             (/
              (* t_2 (sqrt (* F (- A (- t_0 C)))))
              (fma B B (* A (* C -4.0))))
             (* (sqrt (* F (- A (hypot B A)))) (- (/ (sqrt 2.0) B))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = fma(A, (C * -4.0), (B * B));
	double t_2 = -sqrt((2.0 * t_1));
	double tmp;
	if (B <= -9.2e+153) {
		tmp = sqrt(2.0) * -sqrt((F / B));
	} else if (B <= -4.2e-133) {
		tmp = sqrt((F * (C + (A - t_0)))) * (t_2 / t_1);
	} else if (B <= -2e-186) {
		tmp = -sqrt((F * ((A * -8.0) * (C * (A + A))))) / t_1;
	} else if (B <= 1e-121) {
		tmp = -sqrt((t_1 * (F * (C * 4.0)))) / t_1;
	} else if (B <= 2e+100) {
		tmp = (t_2 * sqrt((F * (A - (t_0 - C))))) / fma(B, B, (A * (C * -4.0)));
	} else {
		tmp = sqrt((F * (A - hypot(B, A)))) * -(sqrt(2.0) / B);
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = fma(A, Float64(C * -4.0), Float64(B * B))
	t_2 = Float64(-sqrt(Float64(2.0 * t_1)))
	tmp = 0.0
	if (B <= -9.2e+153)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	elseif (B <= -4.2e-133)
		tmp = Float64(sqrt(Float64(F * Float64(C + Float64(A - t_0)))) * Float64(t_2 / t_1));
	elseif (B <= -2e-186)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(Float64(A * -8.0) * Float64(C * Float64(A + A)))))) / t_1);
	elseif (B <= 1e-121)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(F * Float64(C * 4.0))))) / t_1);
	elseif (B <= 2e+100)
		tmp = Float64(Float64(t_2 * sqrt(Float64(F * Float64(A - Float64(t_0 - C))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B, A)))) * Float64(-Float64(sqrt(2.0) / B)));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[B, -9.2e+153], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B, -4.2e-133], N[(N[Sqrt[N[(F * N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -2e-186], N[((-N[Sqrt[N[(F * N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 1e-121], N[((-N[Sqrt[N[(t$95$1 * N[(F * N[(C * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 2e+100], N[(N[(t$95$2 * N[Sqrt[N[(F * N[(A - N[(t$95$0 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -9.2000000000000005e153

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 0.1

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

      [Start] 0.1	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      frac-2neg [=>] 0.1	\[ \color{blue}{\frac{-\left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}\right)}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      remove-double-neg [=>] 0.1	\[ \frac{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      div-inv [=>] 0.1	\[ \color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate-+r- [=>] 0.1	\[ \sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   4. Taylor expanded in B around -inf 0.1

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

   5. Taylor expanded in A around 0 51.0

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

   6. Simplified 51.0

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

      [Start] 51.0	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      mul-1-neg [=>] 51.0	\[ \color{blue}{-\sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]
      distribute-rgt-neg-in [=>] 51.0	\[ \color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)} \]

   if -9.2000000000000005e153 < B < -4.2000000000000002e-133

   1. Initial program 29.3

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

   2. Simplified 35.3

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

      [Start] 29.3	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 43.8

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

      [Start] 35.3	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 35.3	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      sqrt-prod [=>] 43.7	\[ \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 43.8	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 43.8	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Simplified 43.8

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

      [Start] 43.8	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 43.8	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 43.8	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 43.8	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 43.8	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 43.8	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 43.8	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 43.8	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 43.6

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

      [Start] 43.8	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      div-inv [=>] 43.8	\[ \color{blue}{\left(-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      distribute-rgt-neg-in [=>] 43.8	\[ \color{blue}{\left(\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)\right)} \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*l* [=>] 43.8	\[ \color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right)} \]
      +-commutative [=>] 43.8	\[ \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      associate-+l- [=>] 43.6	\[ \sqrt{F \cdot \color{blue}{\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}\right) \]
      fma-udef [=>] 43.6	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\color{blue}{B \cdot B + A \cdot \left(-4 \cdot C\right)}}\right) \]
      +-commutative [<=] 43.6	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\color{blue}{A \cdot \left(-4 \cdot C\right) + B \cdot B}}\right) \]
      *-commutative [<=] 43.6	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{A \cdot \color{blue}{\left(C \cdot -4\right)} + B \cdot B}\right) \]
      fma-udef [<=] 43.6	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\right) \]

   6. Simplified 43.7

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

      [Start] 43.6	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-lft-neg-out [=>] 43.6	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \frac{1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right)} \]
      associate-*r/ [=>] 43.7	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\color{blue}{\frac{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot 1}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}\right) \]
      *-rgt-identity [=>] 43.7	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \left(-\frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \]
      distribute-frac-neg [<=] 43.7	\[ \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)} \cdot \color{blue}{\frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   if -4.2000000000000002e-133 < B < -1.9999999999999998e-186

   1. Initial program 17.4

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

   2. Simplified 26.4

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

      [Start] 17.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 19.6

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

      [Start] 26.4	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 26.4	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      sqrt-prod [=>] 19.5	\[ \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 19.6	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 19.6	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Simplified 19.6

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

      [Start] 19.6	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.6	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 19.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 19.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 19.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 19.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 19.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 19.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   5. Applied egg-rr 21.7

      \[\leadsto \color{blue}{0 - \frac{\sqrt{F \cdot \left(\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      Proof

      [Start] 19.6	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      distribute-frac-neg [=>] 19.6	\[ \color{blue}{-\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      neg-sub0 [=>] 19.6	\[ \color{blue}{0 - \frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}} \]
      *-commutative [=>] 19.6	\[ 0 - \frac{\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   6. Simplified 21.7

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

      [Start] 21.7	\[ 0 - \frac{\sqrt{F \cdot \left(\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      sub0-neg [=>] 21.7	\[ \color{blue}{-\frac{\sqrt{F \cdot \left(\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      distribute-neg-frac [=>] 21.7	\[ \color{blue}{\frac{-\sqrt{F \cdot \left(\left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right) \cdot \left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      *-commutative [=>] 21.7	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right) \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   7. Taylor expanded in C around inf 24.6

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

   8. Simplified 24.6

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

      [Start] 24.6	\[ \frac{-\sqrt{F \cdot \left(-8 \cdot \left(A \cdot \left(\left(A - -1 \cdot A\right) \cdot C\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      associate-*r* [=>] 24.6	\[ \frac{-\sqrt{F \cdot \color{blue}{\left(\left(-8 \cdot A\right) \cdot \left(\left(A - -1 \cdot A\right) \cdot C\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      *-commutative [=>] 24.6	\[ \frac{-\sqrt{F \cdot \left(\left(-8 \cdot A\right) \cdot \color{blue}{\left(C \cdot \left(A - -1 \cdot A\right)\right)}\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      mul-1-neg [=>] 24.6	\[ \frac{-\sqrt{F \cdot \left(\left(-8 \cdot A\right) \cdot \left(C \cdot \left(A - \color{blue}{\left(-A\right)}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if -1.9999999999999998e-186 < B < 9.9999999999999998e-122

   1. Initial program 17.8

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

   2. Simplified 26.6

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

      [Start] 17.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 24.2

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

      [Start] 26.6	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      frac-2neg [=>] 26.6	\[ \color{blue}{\frac{-\left(-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}\right)}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      remove-double-neg [=>] 26.6	\[ \frac{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      div-inv [=>] 26.3	\[ \color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      associate-+r- [=>] 24.2	\[ \sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\color{blue}{\left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   4. Taylor expanded in C around -inf 23.2

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

   5. Applied egg-rr 23.4

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

      [Start] 23.2	\[ \sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{1}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      un-div-inv [=>] 23.4	\[ \color{blue}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]
      frac-2neg [=>] 23.4	\[ \color{blue}{\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot C\right) \cdot \left(2 \cdot F\right)\right)}}{-\left(-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}} \]
      associate-*r* [=>] 23.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(\left(\left(2 \cdot C\right) \cdot 2\right) \cdot F\right)}}}{-\left(-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
      *-commutative [=>] 23.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(F \cdot \left(\left(2 \cdot C\right) \cdot 2\right)\right)}}}{-\left(-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
      *-commutative [=>] 23.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(F \cdot \left(\color{blue}{\left(C \cdot 2\right)} \cdot 2\right)\right)}}{-\left(-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
      associate-*l* [=>] 23.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(F \cdot \color{blue}{\left(C \cdot \left(2 \cdot 2\right)\right)}\right)}}{-\left(-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
      metadata-eval [=>] 23.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(F \cdot \left(C \cdot \color{blue}{4}\right)\right)}}{-\left(-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)} \]
      remove-double-neg [=>] 23.4	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}{\color{blue}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   if 9.9999999999999998e-122 < B < 2.00000000000000003e100

   1. Initial program 31.1

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

   2. Simplified 40.0

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

      [Start] 31.1	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 40.1

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

      [Start] 40.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate-*r* [=>] 40.0	\[ \frac{-\sqrt{\color{blue}{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right) \cdot \left(F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      sqrt-prod [=>] 39.8	\[ \frac{-\color{blue}{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(C - \left(\mathsf{hypot}\left(B, A - C\right) - A\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      associate--r- [=>] 40.1	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(\left(C - \mathsf{hypot}\left(B, A - C\right)\right) + A\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 40.1	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   4. Simplified 40.1

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

      [Start] 40.1	\[ \frac{-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 40.1	\[ \frac{-\color{blue}{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [<=] 40.1	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(B \cdot B + A \cdot \left(-4 \cdot C\right)\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [<=] 40.1	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \left(\color{blue}{{B}^{2}} + A \cdot \left(-4 \cdot C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      +-commutative [=>] 40.1	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(A \cdot \left(-4 \cdot C\right) + {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      fma-def [=>] 40.1	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(A, -4 \cdot C, {B}^{2}\right)}}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      *-commutative [=>] 40.1	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, \color{blue}{C \cdot -4}, {B}^{2}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]
      unpow2 [=>] 40.1	\[ \frac{-\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, \color{blue}{B \cdot B}\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)} \]

   if 2.00000000000000003e100 < B

   1. Initial program 5.7

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

   2. Simplified 6.7

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

      [Start] 5.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 5.9

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}} \]
      Proof

      [Start] 6.7	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]
      expm1-log1p-u [=>] 5.9	\[ \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)\right)}} \]

   4. Taylor expanded in C around 0 10.7

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

   5. Simplified 49.4

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

      [Start] 10.7	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right) \]
      mul-1-neg [=>] 10.7	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}} \]
      distribute-rgt-neg-in [=>] 10.7	\[ \color{blue}{\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\left(A - \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)} \]
      *-commutative [=>] 10.7	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}\right) \]
      unpow2 [=>] 10.7	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)}\right) \]
      unpow2 [=>] 10.7	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)}\right) \]
      hypot-def [=>] 49.4	\[ \frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \color{blue}{\mathsf{hypot}\left(B, A\right)}\right)}\right) \]

3. Recombined 6 regimes into one program.

4. Final simplification 39.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A \cdot -8\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 10^{-121}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\right) \cdot \sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	39.0%
Cost	34516.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \frac{-\sqrt{2 \cdot t_0}}{t_0}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -7.5 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -1.35 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A \cdot -8\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-122}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.9 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 2
Error	26.2%
Cost	34128.00

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := \frac{\sqrt{2 \cdot t_2}}{t_2}\\ \mathbf{if}\;C \leq -4.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(C \cdot A\right) - B \cdot B\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(F \cdot -2\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq -2.1 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;C \leq -7.8 \cdot 10^{-202}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{elif}\;C \leq 8.5 \cdot 10^{-144}:\\ \;\;\;\;\left(-t_1\right) \cdot t_3\\ \mathbf{elif}\;C \leq 6 \cdot 10^{-30}:\\ \;\;\;\;t_1 \cdot \left(-t_0\right)\\ \mathbf{elif}\;C \leq 3 \cdot 10^{+92}:\\ \;\;\;\;\frac{-1}{\frac{t_2}{\sqrt{t_2 \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}}\\ \mathbf{elif}\;C \leq 6 \cdot 10^{+207}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_2}\\ \mathbf{elif}\;C \leq 6.5 \cdot 10^{+285}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 3
Error	26.0%
Cost	33996.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \sqrt{2 \cdot t_0}\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;C \leq -4.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(C \cdot A\right) - B \cdot B\right) \cdot \left(\left(A - \left(t_2 - C\right)\right) \cdot \left(F \cdot -2\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -2.85 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;C \leq -3.15 \cdot 10^{-195}:\\ \;\;\;\;\frac{t_1}{t_0} \cdot \left(-\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - t_2\right)\right)} \cdot \frac{-t_1}{B \cdot B}\\ \mathbf{elif}\;C \leq 4.4 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_3\right)\\ \mathbf{elif}\;C \leq 6 \cdot 10^{+207}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_0}\\ \mathbf{elif}\;C \leq 6 \cdot 10^{+285}:\\ \;\;\;\;t_3 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 4
Error	25.4%
Cost	28576.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := C + \left(A - t_3\right)\\ t_5 := F \cdot t_4\\ t_6 := 2 \cdot t_2\\ \mathbf{if}\;C \leq -1.18 \cdot 10^{+19}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(C \cdot A\right) - B \cdot B\right) \cdot \left(\left(A - \left(t_3 - C\right)\right) \cdot \left(F \cdot -2\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq -1.28 \cdot 10^{-48}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;C \leq -1.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot t_0\right)}}{t_0}\\ \mathbf{elif}\;C \leq -1.5 \cdot 10^{-88}:\\ \;\;\;\;\left(-\frac{\sqrt{t_2}}{t_2}\right) \cdot \sqrt{F \cdot \left(C \cdot 4\right)}\\ \mathbf{elif}\;C \leq -4 \cdot 10^{-144}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(t_4 \cdot t_6\right)}}{t_2}\\ \mathbf{elif}\;C \leq 2.3 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{t_5} \cdot \frac{-\sqrt{t_6}}{B \cdot B}\\ \mathbf{elif}\;C \leq 1.15 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_1\right)\\ \mathbf{elif}\;C \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right)\right)\right)}}{-t_2}\\ \mathbf{elif}\;C \leq 4.5 \cdot 10^{+286}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 5
Error	25.7%
Cost	28116.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_4 := F \cdot \left(C + \left(A - t_2\right)\right)\\ t_5 := \frac{-\sqrt{2 \cdot \left(t_4 \cdot t_0\right)}}{t_0}\\ \mathbf{if}\;C \leq -7.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(C \cdot A\right) - B \cdot B\right) \cdot \left(\left(A - \left(t_2 - C\right)\right) \cdot \left(F \cdot -2\right)\right)}}{t_3}\\ \mathbf{elif}\;C \leq -2.55 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;C \leq -7.5 \cdot 10^{-78}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;C \leq -2.1 \cdot 10^{-88}:\\ \;\;\;\;\left(-\frac{\sqrt{t_3}}{t_3}\right) \cdot \sqrt{F \cdot \left(C \cdot 4\right)}\\ \mathbf{elif}\;C \leq -2.9 \cdot 10^{-144}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;C \leq 3.25 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{t_4} \cdot \frac{-\sqrt{2 \cdot t_3}}{B \cdot B}\\ \mathbf{elif}\;C \leq 1.3 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_1\right)\\ \mathbf{elif}\;C \leq 1.8 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)} \cdot \frac{1}{-t_3}\\ \mathbf{elif}\;C \leq 4.4 \cdot 10^{+286}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_3}\\ \end{array} \]

Alternative 6
Error	25.5%
Cost	28116.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := \mathsf{hypot}\left(B, A - C\right)\\ t_4 := C + \left(A - t_3\right)\\ t_5 := F \cdot t_4\\ t_6 := 2 \cdot t_2\\ \mathbf{if}\;C \leq -8.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(C \cdot A\right) - B \cdot B\right) \cdot \left(\left(A - \left(t_3 - C\right)\right) \cdot \left(F \cdot -2\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq -4.1 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;C \leq -1.35 \cdot 10^{-78}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_5 \cdot t_0\right)}}{t_0}\\ \mathbf{elif}\;C \leq -1.7 \cdot 10^{-88}:\\ \;\;\;\;\left(-\frac{\sqrt{t_2}}{t_2}\right) \cdot \sqrt{F \cdot \left(C \cdot 4\right)}\\ \mathbf{elif}\;C \leq -4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(t_4 \cdot t_6\right)}}{t_2}\\ \mathbf{elif}\;C \leq 6.6 \cdot 10^{-143}:\\ \;\;\;\;\sqrt{t_5} \cdot \frac{-\sqrt{t_6}}{B \cdot B}\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_1\right)\\ \mathbf{elif}\;C \leq 1.8 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)} \cdot \frac{1}{-t_2}\\ \mathbf{elif}\;C \leq 1.55 \cdot 10^{+286}:\\ \;\;\;\;t_1 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_2}\\ \end{array} \]

Alternative 7
Error	25.0%
Cost	27992.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ t_3 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_4 := \frac{-\sqrt{\left(4 \cdot \left(C \cdot A\right) - B \cdot B\right) \cdot \left(\left(A - \left(t_1 - C\right)\right) \cdot \left(F \cdot -2\right)\right)}}{t_3}\\ \mathbf{if}\;C \leq -1.18 \cdot 10^{+19}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;C \leq -3 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;C \leq -2.3 \cdot 10^{-77}:\\ \;\;\;\;-\frac{\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(\sqrt{B \cdot B + {\left(A - C\right)}^{2}} - C\right) - A\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -2.75 \cdot 10^{-124}:\\ \;\;\;\;\left(-\frac{\sqrt{t_3}}{t_3}\right) \cdot \sqrt{F \cdot \left(C \cdot 4\right)}\\ \mathbf{elif}\;C \leq -3.3 \cdot 10^{-144}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;C \leq 7.5 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \left(A - t_1\right)\right)} \cdot \frac{-\sqrt{2 \cdot t_3}}{B \cdot B}\\ \mathbf{elif}\;C \leq 2.7 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_2\right)\\ \mathbf{elif}\;C \leq 2.7 \cdot 10^{+207}:\\ \;\;\;\;\sqrt{t_3 \cdot \left(\left(2 \cdot F\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{B \cdot B}{C}\right)\right)\right)} \cdot \frac{1}{-t_3}\\ \mathbf{elif}\;C \leq 4.6 \cdot 10^{+286}:\\ \;\;\;\;t_2 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{B}{\frac{C}{B}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(A + A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_3}\\ \end{array} \]

Alternative 8
Error	39.2%
Cost	21448.00

\[\begin{array}{l} \mathbf{if}\;B \leq -2.08 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 3 \cdot 10^{+100}:\\ \;\;\;\;\frac{-\sqrt{\left(4 \cdot \left(C \cdot A\right) - B \cdot B\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(F \cdot -2\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 9
Error	35.5%
Cost	21400.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \frac{\sqrt{2}}{B}\\ t_2 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot t_1\\ t_3 := \frac{-\sqrt{t_0 \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -2.35 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -8.8 \cdot 10^{-53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.3 \cdot 10^{-113}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -1.7 \cdot 10^{-127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A \cdot -8\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-124}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_1\right)\\ \end{array} \]

Alternative 10
Error	36.3%
Cost	21328.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\\ t_2 := \frac{\sqrt{2}}{B}\\ \mathbf{if}\;B \leq -1.45 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -4 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{F \cdot t_1} \cdot t_2\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(F \cdot \left(C \cdot 4\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{-t_0} \cdot \left(B \cdot \sqrt{t_1 \cdot \left(2 \cdot F\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_2\right)\\ \end{array} \]

Alternative 11
Error	35.3%
Cost	20696.00

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot t_0\\ t_2 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_3 := \sqrt{t_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)} \cdot \frac{1}{A \cdot \left(C \cdot 4\right)}\\ \mathbf{if}\;B \leq -2.3 \cdot 10^{+166}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.02 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.4 \cdot 10^{-114}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq -2.1 \cdot 10^{-127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.45 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A \cdot -8\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-123}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-t_0\right)\\ \end{array} \]

Alternative 12
Error	33.6%
Cost	20168.00

\[\begin{array}{l} \mathbf{if}\;B \leq -3.95 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)} \cdot \frac{1}{A \cdot \left(C \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 13
Error	30.5%
Cost	14612.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ t_1 := \sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ t_2 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq -2.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq -1.2 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.2 \cdot 10^{-288}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\left(A \cdot -8\right) \cdot \left(C \cdot \left(A + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.4 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{F \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot C\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \end{array} \]

Alternative 14
Error	32.1%
Cost	14536.00

\[\begin{array}{l} \mathbf{if}\;B \leq -1.9 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(2 \cdot C\right)\right)} \cdot \frac{1}{A \cdot \left(C \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \end{array} \]

Alternative 15
Error	29.7%
Cost	14348.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -1.65 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.3 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{F \cdot \left(\left(A \cdot -16\right) \cdot \left(C \cdot C\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \end{array} \]

Alternative 16
Error	30.8%
Cost	14344.00

\[\begin{array}{l} \mathbf{if}\;B \leq -6.5 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{A \cdot \left(-16 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \end{array} \]

Alternative 17
Error	30.9%
Cost	14344.00

\[\begin{array}{l} \mathbf{if}\;B \leq -3.95 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(\left(F \cdot C\right) \cdot \left(C + C\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \end{array} \]

Alternative 18
Error	29.2%
Cost	13836.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -8.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -2.15 \cdot 10^{-182}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4 \cdot 10^{-124}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C - B\right)}\right)\\ \end{array} \]

Alternative 19
Error	22.3%
Cost	13316.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -4.8 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{elif}\;B \leq -1.15 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(B - C\right) - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 20
Error	14.0%
Cost	8716.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq -9.5 \cdot 10^{-178}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(B - C\right) - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 21
Error	11.7%
Cost	8584.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -3650000000:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{C}{B \cdot B}}\\ \mathbf{elif}\;B \leq 1.66 \cdot 10^{-261}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(B - C\right) - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Error	13.4%
Cost	8584.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(B + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 10^{-249}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(C + A\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(t_1 \cdot \left(\left(B - C\right) - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 23
Error	9.1%
Cost	8452.00

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B \leq -6.8 \cdot 10^{-13}:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{C}{B \cdot B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(B - C\right) - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 24
Error	5.7%
Cost	8072.00

\[\begin{array}{l} \mathbf{if}\;B \leq -4800000:\\ \;\;\;\;-2 \cdot \sqrt{F \cdot \frac{C}{B \cdot B}}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{-124}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(A \cdot \left(-4 \cdot \left(C \cdot \left(F \cdot C\right)\right)\right)\right)}}{B \cdot B + -4 \cdot \left(C \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot C} \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 25
Error	4.6%
Cost	6980.00

\[\begin{array}{l} t_0 := \frac{\sqrt{F \cdot C}}{B}\\ \mathbf{if}\;B \leq -4 \cdot 10^{-308}:\\ \;\;\;\;2 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot t_0\\ \end{array} \]

Alternative 26
Error	4.6%
Cost	6980.00

\[\begin{array}{l} t_0 := \sqrt{F \cdot C}\\ \mathbf{if}\;B \leq 1.2 \cdot 10^{-308}:\\ \;\;\;\;2 \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-2}{B}\\ \end{array} \]

Alternative 27
Error	4.5%
Cost	6976.00

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

Alternative 28
Error	2.9%
Cost	6848.00

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

Reproduce

herbie shell --seed 2023104 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))