ABCF->ab-angle a

Average Error: 52.6 → 41.2

Time: 50.6s

Precision: binary64

Cost: 34120

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_2 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_3 := t_2 \cdot F\\ \mathbf{if}\;C \leq -5.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq -5.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + t_0\right)\right)} \cdot \left(-\sqrt{t_1}\right)}{t_1}\\ \mathbf{elif}\;C \leq -3.2 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + t_0\right)} \cdot \left(-\sqrt{2 \cdot t_3}\right)}{t_2}\\ \end{array} \]

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 -4.0 (* C A) (* B B)))
        (t_2 (+ (* B B) (* (* C A) -4.0)))
        (t_3 (* t_2 F)))
   (if (<= C -5.8e+26)
     (/ (- (sqrt (* 2.0 (* t_3 (fma 2.0 A (* -0.5 (/ (* B B) C))))))) t_2)
     (if (<= C -5.8e-148)
       (/ (* (sqrt (* (* 2.0 F) (+ A (+ C t_0)))) (- (sqrt t_1))) t_1)
       (if (<= C -3.2e-179)
         (* (sqrt (/ F B)) (- (sqrt 2.0)))
         (/ (* (sqrt (+ C (+ A t_0))) (- (sqrt (* 2.0 t_3)))) t_2))))))

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(-4.0, (C * A), (B * B));
	double t_2 = (B * B) + ((C * A) * -4.0);
	double t_3 = t_2 * F;
	double tmp;
	if (C <= -5.8e+26) {
		tmp = -sqrt((2.0 * (t_3 * fma(2.0, A, (-0.5 * ((B * B) / C)))))) / t_2;
	} else if (C <= -5.8e-148) {
		tmp = (sqrt(((2.0 * F) * (A + (C + t_0)))) * -sqrt(t_1)) / t_1;
	} else if (C <= -3.2e-179) {
		tmp = sqrt((F / B)) * -sqrt(2.0);
	} else {
		tmp = (sqrt((C + (A + t_0))) * -sqrt((2.0 * t_3))) / t_2;
	}
	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(-4.0, Float64(C * A), Float64(B * B))
	t_2 = Float64(Float64(B * B) + Float64(Float64(C * A) * -4.0))
	t_3 = Float64(t_2 * F)
	tmp = 0.0
	if (C <= -5.8e+26)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * fma(2.0, A, Float64(-0.5 * Float64(Float64(B * B) / C))))))) / t_2);
	elseif (C <= -5.8e-148)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A + Float64(C + t_0)))) * Float64(-sqrt(t_1))) / t_1);
	elseif (C <= -3.2e-179)
		tmp = Float64(sqrt(Float64(F / B)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(C + Float64(A + t_0))) * Float64(-sqrt(Float64(2.0 * t_3)))) / t_2);
	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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(B * B), $MachinePrecision] + N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * F), $MachinePrecision]}, If[LessEqual[C, -5.8e+26], N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(2.0 * A + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[C, -5.8e-148], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$1], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[C, -3.2e-179], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if C < -5.8e26

   1. Initial program 61.6

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

   2. Simplified 61.6

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

      [Start] 61.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 61.4

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

   4. Taylor expanded in C around -inf 45.1

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

   5. Simplified 45.1

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

      [Start] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \left(2 \cdot A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      fma-def [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \color{blue}{\mathsf{fma}\left(2, A, -0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 45.1	\[ \frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right) \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{\color{blue}{B \cdot B}}{C}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   if -5.8e26 < C < -5.7999999999999997e-148

   1. Initial program 49.7

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

   2. Simplified 46.8

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

      [Start] 49.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 42.0

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

   4. Simplified 41.5

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

      [Start] 42.0	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 42.0	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 42.0	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 42.0	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 41.5	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if -5.7999999999999997e-148 < C < -3.2000000000000001e-179

   1. Initial program 48.7

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

   2. Simplified 48.7

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

      [Start] 48.7	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around 0 54.6

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

   4. Simplified 54.6

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

      [Start] 54.6	\[ \frac{-\left(\sqrt{2} \cdot B\right) \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 54.6	\[ \frac{-\color{blue}{\sqrt{2} \cdot \left(B \cdot \sqrt{\left(A + \sqrt{{B}^{2} + {A}^{2}}\right) \cdot F}\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 54.6	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{\color{blue}{F \cdot \left(A + \sqrt{{B}^{2} + {A}^{2}}\right)}}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 54.6	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(A + \sqrt{\color{blue}{B \cdot B} + {A}^{2}}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 54.6	\[ \frac{-\sqrt{2} \cdot \left(B \cdot \sqrt{F \cdot \left(A + \sqrt{B \cdot B + \color{blue}{A \cdot A}}\right)}\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in A around 0 51.5

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

   6. Simplified 51.5

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

      [Start] 51.5	\[ -1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{F}{B}}\right) \]
      associate-*r* [=>] 51.5	\[ \color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
      neg-mul-1 [<=] 51.5	\[ \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]
      *-commutative [=>] 51.5	\[ \color{blue}{\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)} \]

   if -3.2000000000000001e-179 < C

   1. Initial program 49.9

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

   2. Simplified 49.9

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

      [Start] 49.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 46.2

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

   4. Applied egg-rr 39.8

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

   5. Simplified 39.1

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

      [Start] 39.8	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 39.8	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+r+ [=>] 39.8	\[ \frac{-\sqrt{\color{blue}{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      +-commutative [=>] 39.8	\[ \frac{-\sqrt{\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-+l+ [=>] 39.1	\[ \frac{-\sqrt{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 39.1	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      associate-*l* [=>] 39.1	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \color{blue}{A \cdot \left(C \cdot -4\right)}\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   6. Taylor expanded in F around 0 39.1

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

   7. Simplified 39.1

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

      [Start] 39.1	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot F\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 39.1	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      unpow2 [=>] 39.1	\[ \frac{-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left(\color{blue}{B \cdot B} + -4 \cdot \left(A \cdot C\right)\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 41.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -5.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot F\right) \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{B \cdot B + \left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;C \leq -5.8 \cdot 10^{-148}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\right)}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;C \leq -3.2 \cdot 10^{-179}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(\left(B \cdot B + \left(C \cdot A\right) \cdot -4\right) \cdot F\right)}\right)}{B \cdot B + \left(C \cdot A\right) \cdot -4}\\ \end{array} \]

Alternatives

Alternative 1
Error	41.1
Cost	34120

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_2 := t_1 \cdot F\\ \mathbf{if}\;C \leq -3 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq -4.4 \cdot 10^{-152}:\\ \;\;\;\;-\frac{\sqrt{F \cdot \left(A + \left(C + t_0\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;C \leq -4.9 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + t_0\right)} \cdot \left(-\sqrt{2 \cdot t_2}\right)}{t_1}\\ \end{array} \]

Alternative 2
Error	41.2
Cost	27976

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_2 := t_1 \cdot F\\ \mathbf{if}\;C \leq -1.65 \cdot 10^{+27}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq -8.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(-4, \left(C \cdot A\right) \cdot F, B \cdot \left(B \cdot F\right)\right)} \cdot \left(-\sqrt{A + \left(C + t_0\right)}\right)}{t_1}\\ \mathbf{elif}\;C \leq -2.9 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + t_0\right)} \cdot \left(-\sqrt{2 \cdot t_2}\right)}{t_1}\\ \end{array} \]

Alternative 3
Error	41.3
Cost	27848

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot -4\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := B \cdot B + t_0\\ t_3 := t_2 \cdot F\\ \mathbf{if}\;C \leq -1.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq -1 \cdot 10^{-142}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{A + \left(C + t_1\right)}\right)}{t_2}\\ \mathbf{elif}\;C \leq -3.2 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C + \left(A + t_1\right)} \cdot \left(-\sqrt{2 \cdot t_3}\right)}{t_2}\\ \end{array} \]

Alternative 4
Error	41.4
Cost	21709

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ \mathbf{if}\;C \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -1.3 \cdot 10^{-152} \lor \neg \left(C \leq -3.1 \cdot 10^{-176}\right):\\ \;\;\;\;\frac{\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot t_1}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 5
Error	45.0
Cost	21588

\[\begin{array}{l} t_0 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_2 := t_1 \cdot F\\ t_3 := \frac{t_0 \cdot \left(-\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}\right)}{t_1}\\ \mathbf{if}\;B \leq -1.18 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-294}:\\ \;\;\;\;\frac{t_0 \cdot \left(-\sqrt{-2 \cdot \left(4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(C \cdot 2\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.55 \cdot 10^{-19}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{+81}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 6
Error	44.0
Cost	21588

\[\begin{array}{l} t_0 := \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\\ t_1 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_2 := t_1 \cdot F\\ t_3 := \frac{t_0 \cdot \left(-\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}\right)}{t_1}\\ \mathbf{if}\;B \leq -9.2 \cdot 10^{-66}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-129}:\\ \;\;\;\;-\frac{t_0 \cdot \sqrt{2 \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot F\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(C \cdot 2\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{-19}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(2 \cdot A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{+80}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7
Error	45.0
Cost	21388

\[\begin{array}{l} t_0 := \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot B + t_0\\ t_2 := t_1 \cdot F\\ \mathbf{if}\;A \leq -1.3 \cdot 10^{+49}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\mathsf{fma}\left(2, C, -0.5 \cdot \frac{B \cdot B}{A}\right) \cdot \left(F \cdot \left(4 \cdot \left(C \cdot A\right) - B \cdot B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq -4.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 6.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{A + \left(C + A\right)}\right)}{t_1}\\ \end{array} \]

Alternative 8
Error	45.8
Cost	21324

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := \frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}\right)}{t_0}\\ \mathbf{if}\;B \leq -5.2 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-20}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9
Error	47.0
Cost	15637

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;C \leq -4.2 \cdot 10^{+26}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -1.4 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -1.08 \cdot 10^{-178}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;C \leq 1.4 \cdot 10^{-131} \lor \neg \left(C \leq 1.85 \cdot 10^{+51}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)\right)}}{t_0}\\ \end{array} \]

Alternative 10
Error	46.0
Cost	15637

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;C \leq -1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \mathsf{fma}\left(2, A, -0.5 \cdot \frac{B \cdot B}{C}\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq -9 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;C \leq -1 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;C \leq 1.55 \cdot 10^{-131} \lor \neg \left(C \leq 1.85 \cdot 10^{+51}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)\right)}}{t_0}\\ \end{array} \]

Alternative 11
Error	46.2
Cost	15044

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ t_2 := -\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -6.5 \cdot 10^{-257}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-20}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 12
Error	48.0
Cost	14240

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ t_2 := -\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;-\sqrt{4 \cdot \left(\frac{F}{B} \cdot \frac{A}{B}\right)}\\ \mathbf{elif}\;B \leq -7.2 \cdot 10^{-113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-260}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.38 \cdot 10^{-196}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.55 \cdot 10^{-55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 13
Error	53.1
Cost	9376

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ t_2 := -\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;-\sqrt{4 \cdot \left(\frac{F}{B} \cdot \frac{A}{B}\right)}\\ \mathbf{elif}\;B \leq -1.12 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -8.5 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 4.3 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-196}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 8.5 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(C + A\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Error	53.2
Cost	9248

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ t_2 := -\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ t_3 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C \cdot 2\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -9.4 \cdot 10^{+23}:\\ \;\;\;\;-\sqrt{4 \cdot \left(\frac{F}{B} \cdot \frac{A}{B}\right)}\\ \mathbf{elif}\;B \leq -2.8 \cdot 10^{-115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -1.4 \cdot 10^{-255}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-265}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.45 \cdot 10^{-197}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-108}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 10^{-54}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot t_1\right)}}{t_0}\\ \end{array} \]

Alternative 15
Error	53.4
Cost	8456

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := t_0 \cdot F\\ \mathbf{if}\;B \leq -3.1 \cdot 10^{+24}:\\ \;\;\;\;-\sqrt{4 \cdot \left(\frac{F}{B} \cdot \frac{A}{B}\right)}\\ \mathbf{elif}\;B \leq 5.4 \cdot 10^{-20}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(2 \cdot A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot t_1\right)}}{t_0}\\ \end{array} \]

Alternative 16
Error	56.9
Cost	8328

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := \frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{if}\;A \leq -4.4 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 4.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(B \cdot \left(t_0 \cdot F\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 7.5 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{4 \cdot \left(\frac{F}{B} \cdot \frac{A}{B}\right)}\\ \end{array} \]

Alternative 17
Error	55.3
Cost	8328

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ t_1 := B \cdot \left(t_0 \cdot F\right)\\ \mathbf{if}\;B \leq -5.4 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\sqrt{t_1 \cdot -2}}{t_0}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{-53}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot F\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot t_1}}{t_0}\\ \end{array} \]

Alternative 18
Error	56.7
Cost	8196

\[\begin{array}{l} t_0 := B \cdot B + \left(C \cdot A\right) \cdot -4\\ \mathbf{if}\;C \leq -7.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right) \cdot 8\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A + \left(C + A\right)\right) \cdot \left(-4 \cdot \left(\left(C \cdot A\right) \cdot F\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	57.6
Cost	7940

\[\begin{array}{l} \mathbf{if}\;A \leq 3.8 \cdot 10^{+100}:\\ \;\;\;\;\frac{-\sqrt{-2 \cdot \left(\left(\left(C \cdot F\right) \cdot \left(A \cdot A\right)\right) \cdot 8\right)}}{B \cdot B + \left(C \cdot A\right) \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{4 \cdot \left(\frac{F}{B} \cdot \frac{A}{B}\right)}\\ \end{array} \]

Alternative 20
Error	60.6
Cost	7040

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

Alternative 21
Error	62.0
Cost	6848

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

Reproduce

herbie shell --seed 2023187 
(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))))