ABCF->ab-angle a

Average Error: 81.62% → 68.68%

Time: 44.9s

Precision: binary64

Cost: 46988

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{C + \left(A + t_1\right)}\\ t_3 := -\sqrt{2}\\ t_4 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_5 := -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;F \leq 3.15 \cdot 10^{-229}:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, t_5\right)} \cdot t_2\right) \cdot t_3}{t_4}\\ \mathbf{elif}\;F \leq 80000000:\\ \;\;\;\;\frac{\sqrt{t_0} \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(A + \left(C + t_1\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{+275}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(t_2 \cdot \left(\mathsf{hypot}\left(B, \sqrt{t_5}\right) \cdot \left(-\sqrt{F}\right)\right)\right)}{t_4}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+304}:\\ \;\;\;\;t_2 \cdot \left(\frac{B}{t_4} \cdot \left(\sqrt{F} \cdot t_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(-\sqrt{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)}\right)}{t_0}\\ \end{array} \]

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 (fma -4.0 (* A C) (* B B)))
        (t_1 (hypot B (- A C)))
        (t_2 (sqrt (+ C (+ A t_1))))
        (t_3 (- (sqrt 2.0)))
        (t_4 (fma B B (* C (* -4.0 A))))
        (t_5 (* -4.0 (* A C))))
   (if (<= F 3.15e-229)
     (/ (* (* (sqrt (* F (fma B B t_5))) t_2) t_3) t_4)
     (if (<= F 80000000.0)
       (/ (* (sqrt t_0) (- (sqrt (* (* F 2.0) (+ A (+ C t_1)))))) t_0)
       (if (<= F 1.95e+275)
         (/ (* (sqrt 2.0) (* t_2 (* (hypot B (sqrt t_5)) (- (sqrt F))))) t_4)
         (if (<= F 4.1e+304)
           (* t_2 (* (/ B t_4) (* (sqrt F) t_3)))
           (/ (* (sqrt F) (- (sqrt (* -16.0 (* A (* C 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 = fma(-4.0, (A * C), (B * B));
	double t_1 = hypot(B, (A - C));
	double t_2 = sqrt((C + (A + t_1)));
	double t_3 = -sqrt(2.0);
	double t_4 = fma(B, B, (C * (-4.0 * A)));
	double t_5 = -4.0 * (A * C);
	double tmp;
	if (F <= 3.15e-229) {
		tmp = ((sqrt((F * fma(B, B, t_5))) * t_2) * t_3) / t_4;
	} else if (F <= 80000000.0) {
		tmp = (sqrt(t_0) * -sqrt(((F * 2.0) * (A + (C + t_1))))) / t_0;
	} else if (F <= 1.95e+275) {
		tmp = (sqrt(2.0) * (t_2 * (hypot(B, sqrt(t_5)) * -sqrt(F)))) / t_4;
	} else if (F <= 4.1e+304) {
		tmp = t_2 * ((B / t_4) * (sqrt(F) * t_3));
	} else {
		tmp = (sqrt(F) * -sqrt((-16.0 * (A * (C * 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 = fma(-4.0, Float64(A * C), Float64(B * B))
	t_1 = hypot(B, Float64(A - C))
	t_2 = sqrt(Float64(C + Float64(A + t_1)))
	t_3 = Float64(-sqrt(2.0))
	t_4 = fma(B, B, Float64(C * Float64(-4.0 * A)))
	t_5 = Float64(-4.0 * Float64(A * C))
	tmp = 0.0
	if (F <= 3.15e-229)
		tmp = Float64(Float64(Float64(sqrt(Float64(F * fma(B, B, t_5))) * t_2) * t_3) / t_4);
	elseif (F <= 80000000.0)
		tmp = Float64(Float64(sqrt(t_0) * Float64(-sqrt(Float64(Float64(F * 2.0) * Float64(A + Float64(C + t_1)))))) / t_0);
	elseif (F <= 1.95e+275)
		tmp = Float64(Float64(sqrt(2.0) * Float64(t_2 * Float64(hypot(B, sqrt(t_5)) * Float64(-sqrt(F))))) / t_4);
	elseif (F <= 4.1e+304)
		tmp = Float64(t_2 * Float64(Float64(B / t_4) * Float64(sqrt(F) * t_3)));
	else
		tmp = Float64(Float64(sqrt(F) * Float64(-sqrt(Float64(-16.0 * Float64(A * Float64(C * 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[(-4.0 * N[(A * C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(C + N[(A + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$4 = N[(B * B + N[(C * N[(-4.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 3.15e-229], N[(N[(N[(N[Sqrt[N[(F * N[(B * B + t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[F, 80000000.0], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * (-N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[F, 1.95e+275], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$2 * N[(N[Sqrt[B ^ 2 + N[Sqrt[t$95$5], $MachinePrecision] ^ 2], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[F, 4.1e+304], N[(t$95$2 * N[(N[(B / t$95$4), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(-16.0 * N[(A * N[(C * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < 3.14999999999999993e-229

   1. Initial program 76.91

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

   2. Simplified 64.94

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

      [Start] 76.91	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 64.89

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

   4. Applied egg-rr 56.61

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

   5. Simplified 56.43

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

      [Start] 56.61	\[ \frac{-\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 56.61	\[ \frac{-\left(\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r* [=>] 56.62	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right) \cdot -4}\right)} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [<=] 56.62	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right)} \cdot -4\right)} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [<=] 56.62	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 56.62	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\color{blue}{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r+ [<=] 56.43	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 3.14999999999999993e-229 < F < 8e7

   1. Initial program 78.88

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

   2. Simplified 72.61

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

      [Start] 78.88	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 66.78

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

   4. Simplified 66.14

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

      [Start] 66.78	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 66.78	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, \color{blue}{C \cdot A}, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      *-commutative [=>] 66.78	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\color{blue}{\left(2 \cdot F\right)} \cdot \left(\mathsf{hypot}\left(B, A - C\right) + \left(A + C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      +-commutative [=>] 66.78	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-+r+ [<=] 66.14	\[ \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \color{blue}{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   if 8e7 < F < 1.95e275

   1. Initial program 85.9

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

   2. Simplified 85.31

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

      [Start] 85.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 83.98

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

   4. Applied egg-rr 78.71

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

   5. Simplified 78.01

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

      [Start] 78.71	\[ \frac{-\left(\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [=>] 78.71	\[ \frac{-\left(\sqrt{\color{blue}{F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-*r* [=>] 78.71	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(C \cdot A\right) \cdot -4}\right)} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [<=] 78.71	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, \color{blue}{\left(A \cdot C\right)} \cdot -4\right)} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      *-commutative [<=] 78.71	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)} \cdot \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      +-commutative [=>] 78.71	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\color{blue}{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)}}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]
      associate-+r+ [<=] 78.01	\[ \frac{-\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}\right) \cdot \sqrt{2}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   6. Applied egg-rr 76.94

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

   if 1.95e275 < F < 4.09999999999999985e304

   1. Initial program 91.63

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

   2. Simplified 91.63

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

      [Start] 91.63	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 89.48

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

   4. Simplified 89.48

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

      [Start] 89.48	\[ \frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 89.48	\[ \frac{-\color{blue}{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot F\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]
      *-commutative [=>] 89.48	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{2 \cdot \color{blue}{\left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)}}}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \]

   5. Taylor expanded in B around -inf 93.22

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

   6. Applied egg-rr 85.79

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

   7. Simplified 85

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

      [Start] 85.79	\[ \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{1} \cdot \frac{B \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      /-rgt-identity [=>] 85.79	\[ \color{blue}{\left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)} \cdot \frac{B \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      associate-+r+ [=>] 85.63	\[ \left(-\sqrt{\color{blue}{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}}\right) \cdot \frac{B \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      +-commutative [=>] 85.63	\[ \left(-\sqrt{\color{blue}{\left(C + A\right)} + \mathsf{hypot}\left(B, A - C\right)}\right) \cdot \frac{B \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      associate-+l+ [=>] 85.79	\[ \left(-\sqrt{\color{blue}{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}}\right) \cdot \frac{B \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \]
      associate-/l* [=>] 86.98	\[ \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \color{blue}{\frac{B}{\frac{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)}{\sqrt{2} \cdot \sqrt{F}}}} \]
      associate-/r/ [=>] 85	\[ \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \color{blue}{\left(\frac{B}{\mathsf{fma}\left(B, B, \left(-4 \cdot A\right) \cdot C\right)} \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)\right)} \]
      *-commutative [=>] 85	\[ \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \left(\frac{B}{\mathsf{fma}\left(B, B, \color{blue}{C \cdot \left(-4 \cdot A\right)}\right)} \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)\right) \]
      *-commutative [=>] 85	\[ \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \left(\frac{B}{\mathsf{fma}\left(B, B, C \cdot \color{blue}{\left(A \cdot -4\right)}\right)} \cdot \left(\sqrt{2} \cdot \sqrt{F}\right)\right) \]

   if 4.09999999999999985e304 < F

   1. Initial program 93.68

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

   2. Simplified 93.68

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

      [Start] 93.68	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around inf 96.04

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

   4. Taylor expanded in A around -inf 99.32

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

   5. Simplified 99.23

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

      [Start] 99.32	\[ \frac{-\sqrt{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      associate-*r* [=>] 99.23	\[ \frac{-\sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot {C}^{2}\right) \cdot F\right)}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
      unpow2 [=>] 99.23	\[ \frac{-\sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]

   6. Applied egg-rr 93.03

      \[\leadsto \frac{-\color{blue}{\sqrt{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)} \cdot \sqrt{F}}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.68

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.15 \cdot 10^{-229}:\\ \;\;\;\;\frac{\left(\sqrt{F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot \sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right) \cdot \left(-\sqrt{2}\right)}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;F \leq 80000000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)} \cdot \left(-\sqrt{\left(F \cdot 2\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{+275}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\mathsf{hypot}\left(B, \sqrt{-4 \cdot \left(A \cdot C\right)}\right) \cdot \left(-\sqrt{F}\right)\right)\right)}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\frac{B}{\mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(-\sqrt{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	64.4%
Cost	27652

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ \mathbf{if}\;A \leq -5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(F \cdot \mathsf{fma}\left(2, C, \frac{\left(B \cdot B\right) \cdot -0.5}{A}\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ \end{array} \]

Alternative 2
Error	71.75%
Cost	21520

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{if}\;A \leq -1.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(-\sqrt{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq 1.2 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 1.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{A + \left(A + C\right)} \cdot \sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(-4 \cdot C\right)\right)\right)}}{t_1}\\ \end{array} \]

Alternative 3
Error	66.97%
Cost	21444

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;A \leq -1.15 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(-\sqrt{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_0\right)}\right)}{t_0}\\ \end{array} \]

Alternative 4
Error	73.83%
Cost	21133

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(-4 \cdot A\right)\right)\\ t_1 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;A \leq -1.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(-\sqrt{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq 1.2 \cdot 10^{-140} \lor \neg \left(A \leq 5 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_1\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(A \cdot 2\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 5
Error	73.15%
Cost	20484

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;A \leq -1.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{F} \cdot \left(-\sqrt{-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)}\right)}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 6
Error	78.65%
Cost	15056

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := \frac{-\sqrt{\left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(2 \cdot \left(B \cdot \left(F \cdot B\right)\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -1 \cdot 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq 7.8 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;A \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 7
Error	73.25%
Cost	15044

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;A \leq -8.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 8
Error	83.28%
Cost	14084

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -7.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq -3.4 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq -9 \cdot 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 5.5 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 1.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 9
Error	82.22%
Cost	14084

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot C\right)\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq -1.05 \cdot 10^{-163}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq -1.02 \cdot 10^{-227}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 4.2 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 10
Error	81.41%
Cost	14084

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ t_2 := \frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{if}\;A \leq -1.02 \cdot 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot C\right)\right)}}{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}\\ \mathbf{elif}\;A \leq -2.5 \cdot 10^{-134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq -9.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;A \leq 1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 11
Error	86.51%
Cost	8716

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -6 \cdot 10^{-169}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-277}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 12
Error	86.63%
Cost	8716

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;B \leq -3.9 \cdot 10^{-146}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) - B\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-277}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(\left(A + C\right) + \left(C - A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 13
Error	87.33%
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;A \leq 2.9 \cdot 10^{-141}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;A \leq 1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 14
Error	83.4%
Cost	8584

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ t_1 := F \cdot t_0\\ \mathbf{if}\;C \leq -5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;C \leq 3.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(B + \left(A + C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_1 \cdot \left(C + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 15
Error	86.86%
Cost	8452

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;C \leq -5.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 16
Error	89.18%
Cost	8196

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;C \leq -8.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(A + \left(A + C\right)\right) \cdot \left(-4 \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 17
Error	88.27%
Cost	8196

\[\begin{array}{l} t_0 := -4 \cdot \left(A \cdot C\right) + B \cdot B\\ \mathbf{if}\;C \leq -8.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(-4 \cdot \left(F \cdot \left(A \cdot C\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)\right)}}{t_0}\\ \end{array} \]

Alternative 18
Error	89.02%
Cost	7940

\[\begin{array}{l} \mathbf{if}\;C \leq 7.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(-8 \cdot \left(\left(A \cdot A\right) \cdot \left(F \cdot C\right)\right)\right)}}{-4 \cdot \left(A \cdot C\right) + B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{F}{\frac{B \cdot B}{C}}}\\ \end{array} \]

Alternative 19
Error	95.53%
Cost	6976

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

Alternative 20
Error	97.02%
Cost	6848

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

Reproduce

herbie shell --seed 2023089 
(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))))