ABCF->ab-angle a

Average Accuracy: 17.5% → 50.9%

Time: 56.6s

Precision: binary64

Cost: 34644

\[ \begin{array}{c}[A, C] = \mathsf{sort}([A, C])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ t_4 := \sqrt{t_1 + \left(C + A\right)}\\ t_5 := \frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_2}\\ t_6 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;t_4 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-176}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, \frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(-8 \cdot \left(C \cdot A\right)\right)}}{t_6}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-153}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 5500:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_1\right)} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2 \cdot t_6}\right)\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{+39}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

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 (/ (sqrt 2.0) B))
        (t_1 (hypot B (- A C)))
        (t_2 (fma B B (* C (* A -4.0))))
        (t_3 (* F t_2))
        (t_4 (sqrt (+ t_1 (+ C A))))
        (t_5
         (/ (- (sqrt (* 2.0 (* t_3 (+ C (+ C (* -0.5 (/ (* B B) A)))))))) t_2))
        (t_6 (fma B B (* A (* C -4.0)))))
   (if (<= B -2.4e+18)
     (* t_4 (* t_0 (sqrt F)))
     (if (<= B -4.5e-176)
       t_5
       (if (<= B 4.1e-248)
         (*
          (sqrt (fma 2.0 C (* (/ B A) (* B -0.5))))
          (/ (- (sqrt (* F (* -8.0 (* C A))))) t_6))
         (if (<= B 1.02e-153)
           (- (/ (sqrt (* 2.0 (* t_3 (+ C C)))) t_2))
           (if (<= B 5500.0)
             (/
              (* (sqrt (+ A (+ C t_1))) (* (sqrt F) (- (sqrt (* 2.0 t_6)))))
              (- (* B B) (* (* C A) 4.0)))
             (if (<= B 3.3e+39) t_5 (* t_4 (* t_0 (- (sqrt F))))))))))))

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 = sqrt(2.0) / B;
	double t_1 = hypot(B, (A - C));
	double t_2 = fma(B, B, (C * (A * -4.0)));
	double t_3 = F * t_2;
	double t_4 = sqrt((t_1 + (C + A)));
	double t_5 = -sqrt((2.0 * (t_3 * (C + (C + (-0.5 * ((B * B) / A))))))) / t_2;
	double t_6 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= -2.4e+18) {
		tmp = t_4 * (t_0 * sqrt(F));
	} else if (B <= -4.5e-176) {
		tmp = t_5;
	} else if (B <= 4.1e-248) {
		tmp = sqrt(fma(2.0, C, ((B / A) * (B * -0.5)))) * (-sqrt((F * (-8.0 * (C * A)))) / t_6);
	} else if (B <= 1.02e-153) {
		tmp = -(sqrt((2.0 * (t_3 * (C + C)))) / t_2);
	} else if (B <= 5500.0) {
		tmp = (sqrt((A + (C + t_1))) * (sqrt(F) * -sqrt((2.0 * t_6)))) / ((B * B) - ((C * A) * 4.0));
	} else if (B <= 3.3e+39) {
		tmp = t_5;
	} else {
		tmp = t_4 * (t_0 * -sqrt(F));
	}
	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 = Float64(sqrt(2.0) / B)
	t_1 = hypot(B, Float64(A - C))
	t_2 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_3 = Float64(F * t_2)
	t_4 = sqrt(Float64(t_1 + Float64(C + A)))
	t_5 = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_3 * Float64(C + Float64(C + Float64(-0.5 * Float64(Float64(B * B) / A)))))))) / t_2)
	t_6 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= -2.4e+18)
		tmp = Float64(t_4 * Float64(t_0 * sqrt(F)));
	elseif (B <= -4.5e-176)
		tmp = t_5;
	elseif (B <= 4.1e-248)
		tmp = Float64(sqrt(fma(2.0, C, Float64(Float64(B / A) * Float64(B * -0.5)))) * Float64(Float64(-sqrt(Float64(F * Float64(-8.0 * Float64(C * A))))) / t_6));
	elseif (B <= 1.02e-153)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_3 * Float64(C + C)))) / t_2));
	elseif (B <= 5500.0)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + t_1))) * Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 * t_6))))) / Float64(Float64(B * B) - Float64(Float64(C * A) * 4.0)));
	elseif (B <= 3.3e+39)
		tmp = t_5;
	else
		tmp = Float64(t_4 * Float64(t_0 * Float64(-sqrt(F))));
	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[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(F * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$1 + N[(C + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sqrt[N[(2.0 * N[(t$95$3 * N[(C + N[(C + N[(-0.5 * N[(N[(B * B), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -2.4e+18], N[(t$95$4 * N[(t$95$0 * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -4.5e-176], t$95$5, If[LessEqual[B, 4.1e-248], N[(N[Sqrt[N[(2.0 * C + N[(N[(B / A), $MachinePrecision] * N[(B * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[N[(F * N[(-8.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.02e-153], (-N[(N[Sqrt[N[(2.0 * N[(t$95$3 * N[(C + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), If[LessEqual[B, 5500.0], N[(N[(N[Sqrt[N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$6), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / N[(N[(B * B), $MachinePrecision] - N[(N[(C * A), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 3.3e+39], t$95$5, N[(t$95$4 * N[(t$95$0 * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if B < -2.4e18

   1. Initial program 14.9%

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

   2. Simplified 14.9%

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

   3. Applied egg-rr 28.5%

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

   4. Applied egg-rr 28.4%

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

   5. Simplified 28.3%

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

   6. Taylor expanded in B around -inf 66.8%

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

   if -2.4e18 < B < -4.5e-176 or 5500 < B < 3.30000000000000021e39

   1. Initial program 17.8%

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

   2. Simplified 29.3%

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

   3. Taylor expanded in A around -inf 32.5%

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

   4. Simplified 32.5%

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

   if -4.5e-176 < B < 4.10000000000000033e-248

   1. Initial program 19.5%

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

   2. Simplified 19.5%

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

   3. Applied egg-rr 36.2%

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

   4. Taylor expanded in B around 0 32.3%

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

   5. Simplified 36.1%

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

   6. Taylor expanded in A around -inf 37.8%

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

   7. Simplified 37.8%

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

   8. Applied egg-rr 31.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \left(\frac{B}{A} \cdot B\right)\right)} \cdot \left(\left(-\sqrt{\left(A \cdot \left(C \cdot F\right)\right) \cdot -8}\right) \cdot \frac{1}{\mathsf{fma}\left(B, B, \left(-4 \cdot C\right) \cdot A\right)}\right)} \]

   9. Simplified 37.8%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, C, \frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)} \cdot \left(-\frac{\sqrt{\left(-8 \cdot \left(A \cdot C\right)\right) \cdot F}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)} \]

   if 4.10000000000000033e-248 < B < 1.02e-153

   1. Initial program 24.3%

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

   2. Simplified 36.0%

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

   3. Taylor expanded in A around -inf 22.2%

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

   if 1.02e-153 < B < 5500

   1. Initial program 27.6%

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

   2. Simplified 27.6%

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

   3. Applied egg-rr 37.5%

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

   4. Applied egg-rr 48.7%

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

   if 3.30000000000000021e39 < B

   1. Initial program 9.8%

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

   2. Simplified 9.8%

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

   3. Applied egg-rr 23.6%

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

   4. Applied egg-rr 23.6%

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

   5. Simplified 23.7%

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

   6. Taylor expanded in B around inf 63.8%

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

   7. Simplified 63.8%

      \[\leadsto \sqrt{\left(C + A\right) + \mathsf{hypot}\left(B, A - C\right)} \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, \frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(-8 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-153}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + C\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 5500:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	49.7%
Cost	34132

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ t_4 := \sqrt{t_1 + \left(C + A\right)}\\ t_5 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;t_4 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -5.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{-248}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, \frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(-8 \cdot \left(C \cdot A\right)\right)}}{t_5}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-153}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-68}:\\ \;\;\;\;t_4 \cdot \frac{\left(B \cdot \sqrt{F}\right) \cdot \left(-\sqrt{2}\right)}{t_5}\\ \mathbf{elif}\;B \leq 9.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \left(A + t_1\right)\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	49.8%
Cost	28249

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \sqrt{t_1 + \left(C + A\right)}\\ t_3 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_4 := F \cdot t_3\\ \mathbf{if}\;B \leq -1.1 \cdot 10^{+20}:\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 2.75 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, \frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(-8 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-153}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_4 \cdot \left(C + C\right)\right)}}{t_3}\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-68} \lor \neg \left(B \leq 8.5 \cdot 10^{+81}\right):\\ \;\;\;\;t_2 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_4 \cdot \left(C + \left(A + t_1\right)\right)\right)}}{t_3}\\ \end{array} \]

Alternative 3
Accuracy	50.0%
Cost	27660

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ \mathbf{if}\;B \leq -6 \cdot 10^{+19}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -5.3 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 6.2 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2, C, \frac{B}{A} \cdot \left(B \cdot -0.5\right)\right)} \cdot \frac{-\sqrt{F \cdot \left(-8 \cdot \left(C \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-153}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	49.9%
Cost	27088

\[\begin{array}{l} t_0 := \frac{\sqrt{2}}{B}\\ t_1 := \sqrt{\mathsf{hypot}\left(B, A - C\right) + \left(C + A\right)}\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := F \cdot t_2\\ \mathbf{if}\;B \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_3 \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_2}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{B \cdot B - \left(C \cdot A\right) \cdot 4}\\ \mathbf{elif}\;B \leq 1.02 \cdot 10^{-153}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_3 \cdot \left(C + C\right)\right)}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(t_0 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	46.5%
Cost	26628

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_3 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_2\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_2}\\ \mathbf{if}\;B \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{t_1 + \left(C + A\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F}\right)\\ \mathbf{elif}\;B \leq -4.5 \cdot 10^{-176}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{-33}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 2600:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_1\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 6
Accuracy	36.5%
Cost	22040

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := \frac{-\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + \left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot B\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq -8 \cdot 10^{-176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-247}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 5000:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{+94}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 7
Accuracy	34.6%
Cost	21656

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := \frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot B\right)}\right)}{t_1}\\ t_3 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ t_4 := \frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_1}\\ \mathbf{if}\;B \leq -7 \cdot 10^{+97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-248}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 1.6 \cdot 10^{-55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 1000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+69}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3 \cdot 10^{+106}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{+247}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 8
Accuracy	34.5%
Cost	21656

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ t_3 := \frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_1}\\ \mathbf{if}\;B \leq -7 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot B\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 2.25 \cdot 10^{-247}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-53}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 270000:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot \left(B \cdot F\right)\right)}\right)}{t_1}\\ \mathbf{elif}\;B \leq 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+105}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+270}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 9
Accuracy	36.9%
Cost	21528

\[\begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_2 := \frac{\sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(B \cdot B\right)}\right)}{t_1}\\ t_3 := -\frac{\sqrt{2 \cdot \left(\left(F \cdot t_0\right) \cdot \left(C + C\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -600000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq -4.8 \cdot 10^{-176}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 7 \cdot 10^{-248}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_1}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-53}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 165000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+67}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 10
Accuracy	36.6%
Cost	20804

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}\\ \mathbf{if}\;B \leq -3.55 \cdot 10^{+56}:\\ \;\;\;\;\frac{t_1 \cdot \left(-\sqrt{2 \cdot \left(B \cdot B\right)}\right)}{t_0}\\ \mathbf{elif}\;B \leq 3.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{+258}:\\ \;\;\;\;\frac{\left(-\sqrt{2}\right) \cdot t_1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11
Accuracy	36.4%
Cost	20740

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt[3]{F \cdot -2}\\ \mathbf{if}\;B \leq -3.55 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sqrt{B \cdot t_1} \cdot \left(B \cdot \left(-t_1\right)\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.8 \cdot 10^{+276}:\\ \;\;\;\;\frac{\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12
Accuracy	34.8%
Cost	20300

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq -2.4 \cdot 10^{+57}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\mathsf{hypot}\left(C, B\right) + \left(C + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 6.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+263}:\\ \;\;\;\;\frac{\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 13
Accuracy	34.5%
Cost	15180

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := -\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\mathsf{hypot}\left(C, B\right) + \left(C + A\right)\right)}}{t_0}\\ \mathbf{if}\;B \leq -5.5 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{elif}\;B \leq 2.95 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 14
Accuracy	33.4%
Cost	14916

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq -3.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\mathsf{hypot}\left(A, B\right) + \left(C + A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 9.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 15
Accuracy	33.2%
Cost	14600

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq -2.7 \cdot 10^{+62}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\left(C + A\right) - B\right)}}{t_0}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(F \cdot \left(C \cdot A\right)\right)\right)} \cdot \left(-\sqrt{C \cdot 2}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 16
Accuracy	24.7%
Cost	14480

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ t_2 := 2 \cdot \left(F \cdot t_0\right)\\ t_3 := -\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{if}\;B \leq -2.32 \cdot 10^{+21}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(C + A\right) - B\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 7600:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+38}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Accuracy	24.4%
Cost	14480

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ t_3 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -1.42 \cdot 10^{+18}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(C + A\right) - B\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 72000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{+38}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Accuracy	24.4%
Cost	14480

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := 2 \cdot \left(F \cdot t_0\right)\\ t_3 := \sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{if}\;B \leq -1.42 \cdot 10^{+18}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(C + A\right) - B\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot \left(C \cdot C\right)\right)} \cdot \frac{1}{-t_1}\\ \mathbf{elif}\;B \leq 210000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{+38}:\\ \;\;\;\;-\frac{\sqrt{-16 \cdot \left(F \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 19
Accuracy	24.2%
Cost	13448

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ t_1 := 2 \cdot \left(F \cdot t_0\right)\\ \mathbf{if}\;B \leq -1.75 \cdot 10^{+98}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(C + A\right) - B\right)}}{t_0}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 20
Accuracy	16.3%
Cost	8580

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;C \leq 1.04 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(B + C\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(\left(C + A\right) + \left(C - A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 21
Accuracy	15.9%
Cost	8452

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;C \leq 2.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(B + C\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(C + \left(C + A\right)\right)}}{t_0}\\ \end{array} \]

Alternative 22
Accuracy	7.0%
Cost	8320

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ -\frac{\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(B + \left(C + A\right)\right)}}{t_0} \end{array} \]

Alternative 23
Accuracy	7.9%
Cost	7940

\[\begin{array}{l} t_0 := B \cdot B - \left(C \cdot A\right) \cdot 4\\ \mathbf{if}\;B \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;-\frac{\sqrt{\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot -8\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(B + C\right)\right) \cdot \left(F \cdot \left(B \cdot B\right)\right)}}{t_0}\\ \end{array} \]

Alternative 24
Accuracy	3.0%
Cost	7680

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

Alternative 25
Accuracy	4.7%
Cost	7680

\[-\frac{\sqrt{\left(F \cdot \left(C \cdot C\right)\right) \cdot \left(A \cdot -8\right)}}{B \cdot B - \left(C \cdot A\right) \cdot 4} \]

Alternative 26
Accuracy	2.0%
Cost	6976

\[B \cdot \left(\sqrt{\frac{F}{C}} \cdot \frac{0.5}{A}\right) \]

Alternative 27
Accuracy	2.0%
Cost	6976

\[\frac{B \cdot \left(0.5 \cdot \sqrt{\frac{F}{C}}\right)}{A} \]

Reproduce

herbie shell --seed 2023157 -o generate:proofs
(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))))