ABCF->ab-angle a

Average Error: 52.4 → 39.0

Time: 1.2min

Precision: binary64

Cost: 40520

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(B, A - C\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -1.08 \cdot 10^{+63}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq 7.5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + t_0\right)\right)}}{t_1} \cdot \left(\sqrt{t_1} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_1}}{\frac{-t_1}{\sqrt{2 \cdot \left(A + \left(C + t_0\right)\right)}}}\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (hypot B (- A C)))
        (t_1 (fma -4.0 (* A C) (* B B)))
        (t_2 (fma B B (* C (* A -4.0)))))
   (if (<= A -1.08e+63)
     (/
      (- (sqrt (* 2.0 (* t_2 (* F (fma 2.0 C (/ (* -0.5 (* B B)) A)))))))
      t_2)
     (if (<= A 7.5e-268)
       (* (/ (sqrt (* 2.0 (+ C (+ A t_0)))) t_1) (* (sqrt t_1) (- (sqrt F))))
       (/ (sqrt (* F t_1)) (/ (- t_1) (sqrt (* 2.0 (+ A (+ C t_0))))))))))

C

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

↓

double code(double A, double B, double C, double F) {
	double t_0 = hypot(B, (A - C));
	double t_1 = fma(-4.0, (A * C), (B * B));
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (A <= -1.08e+63) {
		tmp = -sqrt((2.0 * (t_2 * (F * fma(2.0, C, ((-0.5 * (B * B)) / A)))))) / t_2;
	} else if (A <= 7.5e-268) {
		tmp = (sqrt((2.0 * (C + (A + t_0)))) / t_1) * (sqrt(t_1) * -sqrt(F));
	} else {
		tmp = sqrt((F * t_1)) / (-t_1 / sqrt((2.0 * (A + (C + t_0)))));
	}
	return tmp;
}

Julia

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

↓

function code(A, B, C, F)
	t_0 = hypot(B, Float64(A - C))
	t_1 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (A <= -1.08e+63)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_2 * Float64(F * fma(2.0, C, Float64(Float64(-0.5 * Float64(B * B)) / A))))))) / t_2);
	elseif (A <= 7.5e-268)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(C + Float64(A + t_0)))) / t_1) * Float64(sqrt(t_1) * Float64(-sqrt(F))));
	else
		tmp = Float64(sqrt(Float64(F * t_1)) / Float64(Float64(-t_1) / sqrt(Float64(2.0 * Float64(A + Float64(C + t_0))))));
	end
	return tmp
end

Wolfram

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

↓

code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -1.08e+63], N[((-N[Sqrt[N[(2.0 * N[(t$95$2 * N[(F * N[(2.0 * C + N[(N[(-0.5 * N[(B * B), $MachinePrecision]), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[A, 7.5e-268], N[(N[(N[Sqrt[N[(2.0 * N[(C + N[(A + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] / N[((-t$95$1) / N[Sqrt[N[(2.0 * N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if A < -1.08e63

   1. Initial program 63.0

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

   2. Simplified 62.0

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

      [Start] 63.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around -inf 45.0

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

   4. Simplified 45.0

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

      [Start] 45.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot C + -0.5 \cdot \frac{{B}^{2}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      fma-def [=>] 45.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \color{blue}{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r/ [=>] 45.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \color{blue}{\frac{-0.5 \cdot {B}^{2}}{A}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      unpow2 [=>] 45.0	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \color{blue}{\left(B \cdot B\right)}}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if -1.08e63 < A < 7.4999999999999999e-268

   1. Initial program 49.8

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

   2. Simplified 46.5

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

      [Start] 49.8	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 43.3

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

   4. Applied egg-rr 43.1

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

   5. Simplified 43.1

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

      [Start] 43.1	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 43.1	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 43.1	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 43.1	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 43.1	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \]

   6. Applied egg-rr 37.7

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

   if 7.4999999999999999e-268 < A

   1. Initial program 49.5

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

   2. Simplified 44.0

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

      [Start] 49.5	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 38.1

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

   4. Applied egg-rr 38.1

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

   5. Simplified 38.1

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

      [Start] 38.1	\[ \frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}}{1} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      /-rgt-identity [=>] 38.1	\[ \color{blue}{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right) \cdot F}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 38.1	\[ \sqrt{\color{blue}{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 38.1	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 38.1	\[ \sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{-\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \]

   6. Applied egg-rr 37.3

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

   7. Simplified 37.3

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

      [Start] 37.3	\[ \frac{\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
      associate-/l* [=>] 37.3	\[ \color{blue}{\frac{\sqrt{F \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\frac{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}} \]
      *-commutative [=>] 37.3	\[ \frac{\sqrt{F \cdot \mathsf{fma}\left(-4, \color{blue}{A \cdot C}, B \cdot B\right)}}{\frac{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}} \]
      *-commutative [=>] 37.3	\[ \frac{\sqrt{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\frac{-\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, B \cdot B\right)}{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 39.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.08 \cdot 10^{+63}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;A \leq 7.5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}}{\frac{-\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	40.8
Cost	33988

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ \mathbf{if}\;A \leq -1.26 \cdot 10^{+54}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot t_1}}{\frac{-t_1}{\sqrt{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}\\ \end{array} \]

Alternative 2
Error	43.6
Cost	28048

\[\begin{array}{l} t_0 := C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\\ t_1 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_2 := F \cdot t_1\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := \sqrt{2 \cdot \left(t_0 \cdot t_2\right)} \cdot \frac{-1}{t_1}\\ \mathbf{if}\;C \leq -1.9 \cdot 10^{-21}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;C \leq 8.5 \cdot 10^{-107}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;C \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0}}{t_1} \cdot \left(-\sqrt{F \cdot \left(-4 \cdot \left(A \cdot C\right)\right)}\right)\\ \mathbf{elif}\;C \leq 1.35 \cdot 10^{+99}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \left(-\sqrt{t_2}\right)}{t_1}\\ \end{array} \]

Alternative 3
Error	44.1
Cost	27724

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := -4 \cdot \left(A \cdot C\right)\\ t_2 := B \cdot B + t_1\\ t_3 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_2}\\ t_4 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ \mathbf{if}\;C \leq -2.7 \cdot 10^{-27}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 2.45 \cdot 10^{-118}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 2.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{t_4} \cdot \left(-\sqrt{F \cdot t_1}\right)\\ \mathbf{elif}\;C \leq 3.8 \cdot 10^{+92}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \left(-\sqrt{F \cdot t_4}\right)}{t_4}\\ \end{array} \]

Alternative 4
Error	44.0
Cost	27724

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := -4 \cdot \left(A \cdot C\right)\\ t_4 := B \cdot B + t_3\\ \mathbf{if}\;C \leq -5.6 \cdot 10^{-29}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 6 \cdot 10^{-110}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(A + \left(C + t_2\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 1.95 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + \left(A + t_2\right)\right)}}{t_0} \cdot \left(-\sqrt{F \cdot t_3}\right)\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{+95}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_4\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \left(-\sqrt{F \cdot t_0}\right)}{t_0}\\ \end{array} \]

Alternative 5
Error	40.8
Cost	27716

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -6 \cdot 10^{+53}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{-0.5 \cdot \left(B \cdot B\right)}{A}\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B + t_0}\\ \end{array} \]

Alternative 6
Error	44.2
Cost	27665

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;C \leq -2.2 \cdot 10^{-24}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 3 \cdot 10^{-118} \lor \neg \left(C \leq 3.2 \cdot 10^{-65}\right) \land C \leq 1.8 \cdot 10^{+94}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot C\right)} \cdot \left(-\sqrt{F \cdot t_0}\right)}{t_0}\\ \end{array} \]

Alternative 7
Error	45.6
Cost	21520

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := F \cdot t_1\\ \mathbf{if}\;C \leq -4.5 \cdot 10^{-22}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot -16\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;C \leq 1.38 \cdot 10^{-154}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 1.42 \cdot 10^{+45}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(C + \sqrt{B \cdot B + C \cdot C}\right)\right)}}{t_1}\\ \mathbf{elif}\;C \leq 1.9 \cdot 10^{+70}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \left(F \cdot t_0\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{A + 2 \cdot C}\right)}{t_1}\\ \end{array} \]

Alternative 8
Error	44.2
Cost	21520

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{if}\;C \leq -8 \cdot 10^{-29}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(A + \left(A + \frac{-0.5 \cdot \left(B \cdot B\right)}{C}\right)\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;C \leq 2.3 \cdot 10^{-108}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;C \leq 8.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;C \leq 1.05 \cdot 10^{+67}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{A + 2 \cdot C}\right)}{t_1}\\ \end{array} \]

Alternative 9
Error	45.3
Cost	21452

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -1.18 \cdot 10^{+55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;A \leq 1.08 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 5.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;A \leq 1.96 \cdot 10^{+127}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{A + A}\right)}{t_1}\\ \end{array} \]

Alternative 10
Error	44.9
Cost	21128

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right)\\ t_1 := B \cdot B + t_0\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -8.8 \cdot 10^{+53}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_2 \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;A \leq 1.5 \cdot 10^{+127}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, t_0\right)\right)} \cdot \left(-\sqrt{A + A}\right)}{t_1}\\ \end{array} \]

Alternative 11
Error	45.5
Cost	20868

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;A \leq -3.4 \cdot 10^{+60}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{+120}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot C\right) \cdot \left(F \cdot -16\right)} \cdot \sqrt{A}}{t_0}\\ \end{array} \]

Alternative 12
Error	45.5
Cost	15176

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -2.45 \cdot 10^{+55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.45 \cdot 10^{+121}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot C\right) \cdot \left(F \cdot -16\right)} \cdot \sqrt{A}}{t_0}\\ \end{array} \]

Alternative 13
Error	49.9
Cost	14856

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -7.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 5.6 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(A + \sqrt{B \cdot B + A \cdot A}\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot C\right) \cdot \left(F \cdot -16\right)} \cdot \sqrt{A}}{t_0}\\ \end{array} \]

Alternative 14
Error	50.0
Cost	14856

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;A \leq -5.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot \left(F \cdot t_0\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.5 \cdot 10^{-278}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(C + \sqrt{B \cdot B + C \cdot C}\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot C\right) \cdot \left(F \cdot -16\right)} \cdot \sqrt{A}}{t_0}\\ \end{array} \]

Alternative 15
Error	51.2
Cost	14740

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -6.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.58 \cdot 10^{-40}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.05 \cdot 10^{-221}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(C + 0.3333333333333333 \cdot \left(C \cdot 3\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.12 \cdot 10^{+21}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot -16\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{elif}\;A \leq 1.32 \cdot 10^{+72}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(A \cdot C\right) \cdot \left(F \cdot -16\right)} \cdot \sqrt{A}}{t_0}\\ \end{array} \]

Alternative 16
Error	51.4
Cost	8844

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq -2.65 \cdot 10^{+54}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot t_1\right)}}{t_0}\\ \mathbf{elif}\;A \leq -8.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.8 \cdot 10^{-103}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_1 \cdot \left(C + 0.3333333333333333 \cdot \left(C \cdot 3\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A \cdot 2\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	53.5
Cost	8720

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -1.16 \cdot 10^{+55}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -5.2 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -1.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(C \cdot \left(A \cdot \left(F \cdot -16\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;A \leq 6.5 \cdot 10^{-233}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A \cdot 2\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	51.5
Cost	8588

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(2 \cdot C\right) \cdot t_1\right)}}{t_0}\\ \mathbf{if}\;A \leq -4.2 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq -1.75 \cdot 10^{-43}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.12 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A \cdot 2\right)\right)}}{t_0}\\ \end{array} \]

Alternative 19
Error	54.2
Cost	8464

\[\begin{array}{l} t_0 := B \cdot B + -4 \cdot \left(A \cdot C\right)\\ t_1 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -6 \cdot 10^{+53}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq -1.15 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq -2.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(C \cdot \left(A \cdot \left(F \cdot -16\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;A \leq 7.8 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-{\left(-16 \cdot \left(A \cdot \left(F \cdot \left(A \cdot C\right)\right)\right)\right)}^{0.5}}{t_0}\\ \end{array} \]

Alternative 20
Error	54.9
Cost	8072

\[\begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(C \cdot \left(A \cdot \left(F \cdot -16\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \left(A + B\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot -16\right)}}{B \cdot B}\\ \end{array} \]

Alternative 21
Error	54.9
Cost	8072

\[\begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(C \cdot \left(A \cdot \left(F \cdot -16\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(\left(B \cdot B\right) \cdot \left(B + C\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot -16\right)}}{B \cdot B}\\ \end{array} \]

Alternative 22
Error	53.9
Cost	7940

\[\begin{array}{l} \mathbf{if}\;C \leq 2.9 \cdot 10^{-154}:\\ \;\;\;\;\frac{\sqrt{A \cdot \left(C \cdot \left(A \cdot \left(F \cdot -16\right)\right)\right)}}{A \cdot \left(C \cdot 4\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(A \cdot -8\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)\right)}}{B \cdot B + -4 \cdot \left(A \cdot C\right)}\\ \end{array} \]

Alternative 23
Error	56.0
Cost	7616

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

Alternative 24
Error	56.8
Cost	7424

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

Alternative 25
Error	63.0
Cost	7104

\[-0.25 \cdot \left(B \cdot \left(2 \cdot \frac{\sqrt{\frac{F}{A}}}{C}\right)\right) \]

Reproduce

herbie shell --seed 2023038 
(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))))